The local acceleration (a) can also be thought of as the "unsteady term" as this describes some change in velocity over time. The convective acceleration (b) is an acceleration caused by some change in velocity over position, for example the speeding up or slowing down of a fluid entering a constriction or an opening, respectively. Both these terms make up the inertia terms of the 1-dimensional Saint-Venant equation.


The pressure gradient term (c) describes how pressure changes with position, and since the pressure is assumed hydrostatic, this is the change in head over position. The friction term (d) accounts for losses in energy due to friction, while the gravity term (e) is the acceleration due to bed slope.


The primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:


The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined.


In general, nearly all forms of the primitive equations relate the five variables "u", "v", ω, "T", "W", and their evolution over space and time.


The equations were first written down by Vilhelm Bjerknes.


Forces that cause atmospheric motion include the pressure gradient force, gravity, and viscous friction. Together, they create the forces that accelerate our atmosphere.


The pressure gradient force causes an acceleration forcing air from regions of high pressure to regions of low pressure. Mathematically, this can be written as:


The gravitational force accelerates objects at approximately 9.8 m/s2 directly towards the center of the Earth.


The force due to viscous friction can be approximated as:


Using Newton's second law, these forces (referenced in the equations above as the accelerations due to these forces) may be summed to produce an equation of motion that describes this system. This equation can be written in the form:


Therefore, to complete the system of equations and obtain 6 equations and 6 variables:


where n is the number density in mol, and T:=RT is the temperature equivalent value in Joule/mol.


The precise form of the primitive equations depends on the vertical coordinate system chosen, such as pressure coordinates, log pressure coordinates, or sigma coordinates. Furthermore, the velocity, temperature, and geopotential variables may be decomposed into mean and perturbation components using Reynolds decomposition.


Pressure coordinate in vertical, Cartesian tangential plane.


In this form pressure is selected as the vertical coordinate and the horizontal coordinates are written for the Cartesian tangential plane (i.e. a plane tangent to some point on the surface of the Earth). This form does not take the curvature of the Earth into account, but is useful for visualizing some of the physical processes involved in formulating the equations due to its relative simplicity.


Note that the capital D time derivatives are material derivatives. Five equations in five unknowns comprise the system.


When a statement of the conservation of water vapor substance is included, these six equations form the basis for any numerical weather prediction scheme.


Primitive equations using sigma coordinate system, polar stereographic projection.


According to the "National Weather Service Handbook No. 1 – Facsimile Products", the primitive equations can be simplified into the following equations:


This equation and notation works in much the same way as the temperature equation. This equation describes the motion of water from one place to another at a point without taking into account water that changes form. Inside a given system, the total change in water with time is zero. However, concentrations are allowed to move with the wind.


These simplifications make it much easier to understand what is happening in the model. Things like the temperature (potential temperature), precipitable water, and to an extent the pressure thickness simply move from one spot on the grid to another with the wind. The wind is forecast slightly differently. It uses geopotential, specific heat, the exner function "π", and change in sigma coordinate.


The analytic solution to the linearized primitive equations involves a sinusoidal oscillation in time and longitude, modulated by coefficients related to height and latitude.


where "s" and formula_39 are the zonal wavenumber and angular frequency, respectively. The solution represents atmospheric waves and tides.


When the coefficients are separated into their height and latitude components, the height dependence takes the form of propagating or evanescent waves (depending on conditions), while the latitude dependence is given by the Hough functions.


This analytic solution is only possible when the primitive equations are linearized and simplified. Unfortunately many of these simplifications (i.e. no dissipation, isothermal atmosphere) do not correspond to conditions in the actual atmosphere. As a result, a numerical solution which takes these factors into account is often calculated using general circulation models and climate models.


Collaborative Research and Training Site, Review of the Primitive Equations.


The Kaufmann vortex, also known as the Scully model, is a mathematical model for a vortex taking account of viscosity. It uses an algebraic velocity profile.


Kaufmann and Scully's model for the velocity in the Θ direction is:


The model was suggested by Scully and Sullivan in 1972 at Massachusetts Institute of Technology, and earlier by W. Kaufmann in 1962.


The basic form of a 2-dimensional thin film equation is


and "μ" is the viscosity (or dynamic viscosity) of the liquid, "h"("x","y","t") is film thickness, "γ" is the interfacial tension between the liquid and the gas phase above it, formula_8 is the liquid density and formula_9 the surface shear. The surface shear could be caused by flow of the overlying gas or surface tension gradients. The vectors formula_10 represent the unit vector in the surface co-ordinate directions, the dot product serving to identify the gravity component in each direction. The vector formula_11 is the unit vector perpendicular to the surface.


A generalised thin film equation is discussed in


When formula_13 this may represent flow with slip at the solid surface whole formula_14 describes the thickness of a thin bridge between two masses of fluid in a Hele-Shaw cell. The value formula_15 represents surface tension driven flow.


A form frequently investigated with regard to the rupture of thin liquid films involves the addition of a disjoining pressure Π("h") in the equation, as in


where the function Π("h") is usually very small in value for moderate-large film thicknesses "h" and grows very rapidly when "h" goes very close to zero.


Physical applications, properties and solution behaviour of the thin-film equation are reviewed in. With the inclusion of phase change at the substrate a form of thin film equation for an arbitrary surface is derived in. A detailed study of the steady-flow of a thin film near a moving contact line is given in. For a yield-stress fluid flow driven by gravity and surface tension is investigated in.


For purely surface tension driven flow it is easy to see that one static (time-independent) solution is a paraboloid of revolution


and this is consistent with the experimentally observed spherical cap shape of a static sessile drop, as a "flat" spherical cap that has small height can be accurately approximated in second order with a paraboloid. This, however, does not handle correctly the circumference of the droplet where the value of the function "h"("x","y") drops to zero and below, as a real physical liquid film can't have a negative thickness. This is one reason why the disjoining pressure term Π("h") is important in the theory.


One possible realistic form of the disjoining pressure term is


where "B", "h"*, "m" and "n" are some parameters. These constants and the surface tension formula_19 can be approximately related to the equilibrium liquid-solid contact angle formula_20 through the equation


The thin film equation can be used to simulate several behaviors of liquids, such as the fingering instability in gravity driven flow.


The lack of a second-order time derivative in the thin-film equation is a result of the assumption of small Reynold's number in its derivation, which allows the ignoring of inertial terms dependent on fluid density formula_8. This is somewhat similar to the situation with Washburn's equation, which describes the capillarity-driven flow of a liquid in a thin tube.


The barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind are independent of height. In other words, there is no vertical wind shear of the geostrophic wind. It also implies that thickness contours (a proxy for temperature) are parallel to upper level height contours. In this type of atmosphere, high and low pressure areas are centers of warm and cold temperature anomalies. Warm-core highs (such as the subtropical ridge and the Bermuda-Azores high) and cold-core lows have strengthening winds with height, with the reverse true for cold-core highs (shallow Arctic highs) and warm-core lows (such as tropical cyclones).


A simplified form of the vorticity equation for an inviscid, divergence-free flow (solenoidal velocity field), the barotropic vorticity equation can simply be stated as


is "absolute vorticity", with "ζ" being "relative vorticity", defined as the vertical component of the curl of the fluid velocity and "f" is the "Coriolis parameter"


where "Ω" is the angular frequency of the planet's


rotation ("Ω" = for the earth) and "φ" is latitude.


In terms of "relative vorticity", the equation can be rewritten as


where "β" =  is the variation of the "Coriolis parameter" with distance "y" in the north–south direction and "v" is the component of velocity in this direction.


In 1950, Charney, Fjørtoft, and von Neumann integrated this equation (with an added diffusion term on the right-hand side) on a computer for the first time, using an observed field of 500 hPa geopotential height for the first timestep. This was one of the first successful instances of numerical weather prediction.


The Rankine vortex is a simple mathematical model of a vortex in a viscous fluid. It is named after its discoverer, William John Macquorn Rankine.


A swirling flow in a viscous fluid can be characterized by a central core comprising a forced vortex, surrounded by a free vortex. In an inviscid fluid, on the other hand, a swirling flow consists entirely of a free vortex with a singularity at its center point. The tangential velocity


of a Rankine vortex with circulation formula_1 and radius formula_2 is


The remainder of the velocity components are identically zero, so that the total velocity field is formula_4.


In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a fluid. The properties that are carried with the advected substance are conserved properties such as energy. An example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as water or air. In general, any substance or conserved, extensive quantity can be advected by a fluid that can hold or contain the quantity or substance.


During advection, a fluid transports some conserved quantity or material via bulk motion. The fluid's motion is described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution over space. Advection requires currents in the fluid, and so cannot happen in rigid solids. It does not include transport of substances by molecular diffusion.


Advection is sometimes confused with the more encompassing process of convection which is the combination of advective transport and diffusive transport.


In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity (see moisture) or salinity.


Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle.


The term "advection" often serves as a synonym for "convection", and this correspondence of terms is used in the literature. More technically, convection applies to the movement of a fluid (often due to density gradients created by thermal gradients), whereas advection is the movement of some material by the velocity of the fluid. Thus, somewhat confusingly, it is technically correct to think of momentum being advected by the velocity field in the Navier-Stokes equations, although the resulting motion would be considered to be convection. Because of the specific use of the term convection to indicate transport in association with thermal gradients, it is probably safer to use the term advection if one is uncertain about which terminology best describes their particular system.


In meteorology and physical oceanography, advection often refers to the horizontal transport of some property of the atmosphere or ocean, such as heat, humidity or salinity, and convection generally refers to vertical transport (vertical advection). Advection is important for the formation of orographic clouds (terrain-forced convection) and the precipitation of water from clouds, as part of the hydrological cycle.


The advection equation also applies if the quantity being advected is represented by a probability density function at each point, although accounting for diffusion is more difficult.


The advection equation is the partial differential equation that governs the motion of a conserved scalar field as it is advected by a known velocity vector field. It is derived using the scalar field's conservation law, together with Gauss's theorem, and taking the infinitesimal limit.


One easily visualized example of advection is the transport of ink dumped into a river. As the river flows, ink will move downstream in a "pulse" via advection, as the water's movement itself transports the ink. If added to a lake without significant bulk water flow, the ink would simply disperse outwards from its source in a diffusive manner, which is not advection. Note that as it moves downstream, the "pulse" of ink will also spread via diffusion. The sum of these processes is called convection.


In Cartesian coordinates the advection operator is


where formula_2 is the velocity field, and formula_3 is the del operator (note that Cartesian coordinates are used here).


The advection equation for a conserved quantity described by a scalar field formula_4 is expressed mathematically by a continuity equation:


where formula_5 is the divergence operator and again formula_6 is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies


In this case, formula_6 is said to be solenoidal. If this is so, the above equation can be rewritten as


In particular, if the flow is steady, then


which shows that formula_4 is constant along a streamline.


Hence, formula_11 so formula_4 doesn't vary in time.


If a vector quantity formula_13 (such as a magnetic field) is being advected by the solenoidal velocity field formula_6, the advection equation above becomes:


Here, formula_13 is a vector field instead of the scalar field formula_4.


The advection equation is not simple to solve numerically: the system is a hyperbolic partial differential equation, and interest typically centers on discontinuous "shock" solutions (which are notoriously difficult for numerical schemes to handle).


Even with one space dimension and a constant velocity field, the system remains difficult to simulate. The equation becomes


where formula_19 is the scalar field being advected


and formula_20 is the formula_21 component of the vector formula_22.


Treatment of the advection operator in the incompressible Navier–Stokes equations.


According to Zang, numerical simulation can be aided by considering the skew symmetric form for the advection operator.


and formula_6 is the same as above.


Since skew symmetry implies only imaginary eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities (see Boyd).


Using vector calculus identities, these operators can also be expressed in other ways, available in more software packages for more coordinate systems.


This form also makes visible that the skew symmetric operator introduces error when the velocity field diverges. Solving the advection equation by numerical methods is very challenging and there is a large scientific literature about this.


In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is:


The equation is attributed to Lord Rayleigh, who originally used "L"2 in place of "A" (with "L" being some linear dimension).


For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000. For smooth bodies, like a circular cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 107 (ten million).


The equation is easier understood for the idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area. No real object exactly corresponds to this behavior. "CD" is the ratio of drag for any real object to that of the ideal object. In practice a rough un-streamlined body (a bluff body) will have a "CD" around 1, more or less. Smoother objects can have much lower values of "CD". The equation is precise – it simply provides the definition of "CD" (drag coefficient), which varies with the Reynolds number and is found by experiment.


Of particular importance is the formula_9 dependence on flow velocity, meaning that fluid drag increases with the square of flow velocity. When flow velocity is doubled, for example, not only does the fluid strike with twice the flow velocity, but twice the mass of fluid strikes per second. Therefore, the change of momentum per second is multiplied by four. Force is equivalent to the change of momentum divided by time. This is in contrast with solid-on-solid friction, which generally has very little flow velocity dependence.


The drag force can also be specified as,


where, "Pd" is pressure exerted by fluid on area "A". Here the pressure "Pd" is referred to as dynamic pressure due to kinetic energy of fluid experiencing relative flow velocity "u". This is defined in similar form as kinetic energy equation:


The drag equation may be derived to within a multiplicative constant by the method of dimensional analysis. If a moving fluid meets an object, it exerts a force on the object. Suppose that the fluid is a liquid, and the variables involved – under some conditions – are the:


Using the algorithm of the Buckingham π theorem, these five variables can be reduced to two dimensionless groups:


Alternatively, the dimensionless groups via direct manipulation of the variables.


That this is so becomes apparent when the drag force "FD" is expressed as part of a function of the other variables in the problem:


This rather odd form of expression is used because it does not assume a one-to-one relationship. Here, "fa" is some (as-yet-unknown) function that takes five arguments. Now the right-hand side is zero in any system of units; so it should be possible to express the relationship described by "fa" in terms of only dimensionless groups.


There are many ways of combining the five arguments of "fa" to form dimensionless groups, but the Buckingham π theorem states that there will be two such groups. The most appropriate are the Reynolds number, given by


Thus the function of five variables may be replaced by another function of only two variables:


where "fb" is some function of two arguments.


The original law is then reduced to a law involving only these two numbers.


Because the only unknown in the above equation is the drag force "FD", it is possible to express it as


Thus the force is simply ½ "ρ" "A" "u2" times some (as-yet-unknown) function "fc" of the Reynolds number Re – a considerably simpler system than the original five-argument function given above.


Dimensional analysis thus makes a very complex problem (trying to determine the behavior of a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number.


If the fluid is a gas, certain properties of the gas influence the drag and those properties must also be taken into account. Those properties are conventionally considered to be the absolute temperature of the gas, and the ratio of its specific heats. These two properties determine the speed of sound in the gas at its given temperature. The Buckingham pi theorem then leads to a third dimensionless group, the ratio of the relative velocity to the speed of sound, which is known as the Mach number. Consequently when a body is moving relative to a gas, the drag coefficient varies with the Mach number and the Reynolds number.


The analysis also gives other information for free, so to speak. The analysis shows that, other things being equal, the drag force will be proportional to the density of the fluid. This kind of information often proves to be extremely valuable, especially in the early stages of a research project.


To empirically determine the Reynolds number dependence, instead of experimenting on a large body with fast-flowing fluids (such as real-size airplanes in wind tunnels), one may just as well experiment using a small model in a flow of higher velocity because these two systems deliver similitude by having the same Reynolds number. If the same Reynolds number and Mach number cannot be achieved just by using a flow of higher velocity it may be advantageous to use a fluid of greater density or lower viscosity.


In mathematical physics, the Whitham equation is a non-local model for non-linear dispersive waves.


The equation is notated as follows :


This integro-differential equation for the oscillatory variable "η"("x","t") is named after Gerald Whitham who introduced it as a model to study breaking of non-linear dispersive water waves in 1967. Wave breaking – bounded solutions with unbounded derivatives – for the Whitham equation has recently been proven.


For a certain choice of the kernel "K"("x" − "ξ") it becomes the Fornberg–Whitham equation.


Using the Fourier transform (and its inverse), with respect to the space coordinate "x" and in terms of the wavenumber "k":


From the mathematical point of view, Euler equations are notably hyperbolic conservation equations in the case without external field (i.e., in the limit of high Froude number). In fact, like any Cauchy equation, the Euler equations originally formulated in convective form (also called "Lagrangian form") can also be put in the "conservation form" (also called "Eulerian form"). The conservation form emphasizes the mathematical interpretation of the equations as conservation equations through a control volume fixed in space, and is the most important for these equations also from a numerical point of view. The convective form emphasizes changes to the state in a frame of reference moving with the fluid.


The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides", published in "Mémoires de l'Académie des Sciences de Berlin" in 1757 (in this article Euler actually published only the "general" form of the continuity equation and the momentum equation; the energy balance equation would be obtained a century later). They were among the first partial differential equations to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible fluid. An additional equation, which was later to be called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816.


During the second half of the 19th century, it was found that the equation related to the balance of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the special theory of relativity, the concepts of energy density, momentum density, and stress were unified into the concept of the stress–energy tensor, and energy and momentum were likewise unified into a single concept, the energy–momentum vector


Incompressible Euler equations with constant and uniform density.


In convective form (i.e., the form with the convective operator made explicit in the momentum equation), the incompressible Euler equations in case of density constant in time and uniform in space are:


The first equation is the Euler momentum equation with uniform density (for this equation it could also not be constant in time). By expanding the material derivative, the equations become:


In fact for a flow with uniform density formula_13 the following identity holds:


where formula_15 is the mechanic pressure. The second equation is the incompressible constraint, stating the flow velocity is a solenoidal field (the order of the equations is not causal, but underlines the fact that the incompressible constraint is not a degenerate form of the continuity equation, but rather of the energy equation, as it will become clear in the following). Notably, the continuity equation would be required also in this incompressible case as an additional third equation in case of density varying in time "or" varying in space. For example, with density uniform but varying in time, the continuity equation to be added to the above set would correspond to:


So the case of constant and uniform density is the only one not requiring the continuity equation as additional equation regardless of the presence or absence of the incompressible constraint. In fact, the case of incompressible Euler equations with constant and uniform density being analyzed is a toy model featuring only two simplified equations, so it is ideal for didactical purposes even if with limited physical relevancy.


The equations above thus represent respectively conservation of mass (1 scalar equation) and momentum (1 vector equation containing formula_17 scalar components, where formula_17 is the physical dimension of the space of interest). Flow velocity and pressure are the so-called "physical variables".


In a coordinate system given by formula_19 the velocity and external force vectors formula_1 and formula_21 have components formula_22 and formula_23, respectively. Then the equations may be expressed in subscript notation as:


where the formula_25 and formula_26 subscripts label the "N"-dimensional space components, and formula_27 is the Kroenecker delta. The use of Einstein notation (where the sum is implied by repeated indices instead of sigma notation) is also frequent.


Although Euler first presented these equations in 1755, many fundamental questions about them remain unanswered.


In three space dimensions, in certain simplified scenarios, the Euler equations produce singularities.


Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy:


In the one dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid Burgers equation:


This is a model equation giving many insights on Euler equations.


In order to make the equations dimensionless, a characteristic length formula_30, and a characteristic velocity formula_31, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:


Substitution of these inversed relations in Euler equations, defining the Froude number, yields (omitting the * at apix):


Euler equations in the Froude limit (no external field) are named free equations and are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with perturbation theory.


The conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are some advantages in using the conserved variables. This gives rise to a large class of numerical methods


The free Euler equations are conservative, in the sense they are equivalent to a conservation equation:


where the conservation quantity formula_36 in this case is a vector, and formula_37 is a flux matrix. This can be simply proved.


At last Euler equations can be recast into the particular equation:


For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by Riemann's method of characteristics. This involves finding curves in plane of independent variables (i.e., formula_38 and formula_39) along which partial differential equations (PDEs) degenerate into ordinary differential equations (ODEs). Numerical solutions of the Euler equations rely heavily on the method of characteristics.


In convective form the incompressible Euler equations in case of density variable in space are:


The first equation, which is the new one, is the incompressible continuity equation. In fact the general continuity equation would be:


but here the last term is identically zero for the incompressibility constraint.


The incompressible Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:


Here formula_36 has length formula_46 and formula_37 has size formula_48. In general (not only in the Froude limit) Euler equations are expressible as:


The variables for the equations in conservation form are not yet optimised. In fact we could define:


In differential convective form, the compressible (and most general) Euler equations can be written shortly with the material derivative notation:


The equations above thus represent conservation of mass, momentum, and energy: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form.


Mass density, flow velocity and pressure are the so-called "convective variables" (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called "conserved variables" (also called eulerian, or mathematical variables).


If one explicitates the material derivative the equations above are:


Coming back to the incompressible case, it now becomes apparent that the "incompressible constraint" typical of the former cases actually is a particular form valid for incompressible flows of the "energy equation", and not of the mass equation. In particular, the incompressible constraint corresponds to the following very simple energy equation:


Thus for an incompressible inviscid fluid the specific internal energy is constant along the flow lines, also in a time-dependent flow. The pressure in an incompressible flow acts like a Lagrange multiplier, being the multiplier of the incompressible constraint in the energy equation, and consequently in incompressible flows it has no thermodynamic meaning. In fact, thermodynamics is typical of compressible flows and degenerates in incompressible flows.


Basing on the mass conservation equation, one can put this equation in the conservation form:


meaning that for an incompressible inviscid nonconductive flow a continuity equation holds for the internal energy.


Since by definition the specific enthalpy is:


The material derivative of the specific internal energy can be expressed as:


Then by substituting the momentum equation in this expression, one obtains:


And by substituting the latter in the energy equation, one obtains that the enthalpy expression for the Euler energy equation:


In a reference frame moving with an inviscid and nonconductive flow, the variation of enthalpy directly corresponds to a variation of pressure.


In thermodynamics the independent variables are the specific volume, and the specific entropy, while the specific energy is a function of state of these two variables.


This equation is the only belonging to general continuum equations, so only this equation have the same form for example also in Navier-Stokes equations.


On the other hand, the pressure in thermodynamics is the opposite of the partial derivative of the specific internal energy with respect to the specific volume:


since the internal energy in thermodynamics is a function of the two variables aforementioned, the pressure gradient contained into the momentum equation should be explicited as:


It is convenient for brevity to switch the notation for the second order derivatives:


can be furtherly simplified in convective form by changing variable from specific energy to the specific entropy: in fact the first law of thermodynamics in local form can be written:


by substituting the material derivative of the internal energy, the energy equation becomes:


now the term between parenthesis is identically zero according to the conservation of mass, then the Euler energy equation becomes simply:


For a thermodynamic fluid, the compressible Euler equations are consequently best written as:


In the general case and not only in the incompressible case, the energy equation means that for an inviscid thermodynamic fluid the specific entropy is constant along the flow lines, also in a time-dependent flow. Basing on the mass conservation equation, one can put this equation in the conservation form:


meaning that for an inviscid nonconductive flow a continuity equation holds for the entropy.


On the other hand, the two second-order partial derivatives of the specific internal energy in the momentum equation require the specification of the fundamental equation of state of the material considered, i.e. of the specific internal energy as function of the two variables specific volume and specific entropy:


The "fundamental" equation of state contains all the thermodynamic information about the system (Callen, 1985), exactly like the couple of a "thermal" equation of state together with a "caloric" equation of state.


The Euler equations in the Froude limit are equivalent to a single conservation equation with conserved quantity and associated flux respectively:


Here formula_36 has length N + 2 and formula_37 has size N(N + 2). In general (not only in the Froude limit) Euler equations are expressible as:


We remark that also the Euler equation even when conservative (no external field, Froude limit) have no Riemann invariants in general. Some further assumptions are required


However, we already mentioned that for a thermodynamic fluid the equation for the total energy density is equivalent to the conservation equation:


Then the conservation equations in the case of a thermodynamic fluid are more simply expressed as:


Another possible form for the energy equation, being particularly useful for isobarics, is:


Expanding the fluxes can be an important part of constructing numerical solvers, for example by exploiting (approximate) solutions to the Riemann problem. In regions where the state vector y varies smoothly, the equations in conservative form can be put in quasilinear form :


where formula_85 are called the flux Jacobians defined as the matrices:


Obviously this Jacobian does not exist in discontinuity regions (e.g. contact discontinuities, shock waves in inviscid nonconductive flows). If the flux Jacobians formula_85 are not functions of the state vector formula_36, the equations reveals "linear".


The compressible Euler equations can be decoupled into a set of N+2 wave equations that describes sound in Eulerian continuum if they are expressed in characteristic variables instead of conserved variables.


In fact the tensor A is always diagonalizable. If the eigenvalues (the case of Euler equations) are all real the system is defined "hyperbolic", and physically eigenvalues represent the speeds of propagation of information. If they are all distinguished, the system is defined "strictly hyperbolic" (it will be proved to be the case of one-dimensional Euler equations). Furthermore, diagonalisation of compressible Euler equation is easier when the energy equation is expressed in the variable entropy (i.e. with equations for thermodynamic fluids) than in other energy variables. This will become clear by considering the 1D case.


If formula_89 is the right eigenvector of the matrix formula_90 corresponding to the eigenvalue formula_91, by building the projection matrix:


One can finally find the "characteristic variables" as:


Since A is constant, multiplying the original 1-D equation in flux-Jacobian form with P−1 yields the characteristic equations:


The original equations have been decoupled into N+2 characteristic equations each describing a simple wave, with the eigenvalues being the wave speeds. The variables "w"i are called the "characteristic variables" and are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In one spatial dimension it is:


Then the solution in terms of the original conservative variables is obtained by transforming back:


this computation can be explicited as the linear combination of the eigenvectors:


Now it becomes apparent that the characteristic variables act as weights in the linear combination of the jacobian eigenvectors. The solution can be seen as superposition of waves, each of which is advected independently without change in shape. Each "i"-th wave has shape "w""i""p""i" and speed of propagation "λ""i". In the following we show a very simple example of this solution procedure.


Waves in 1D inviscid, nonconductive thermodynamic fluid.


If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: "g" = 0) :


If one defines the vector of variables:


recalling that formula_69 is the specific volume, formula_101 the flow speed, formula_71 the specific entropy, the corresponding jacobian matrix is:


At first one must find the eigenvalues of this matrix by solving the characteristic equation:


This determinant is very simple: the fastest computation starts on the last row, since it has the highest number of zero elements.


This parameter is always real according to the second law of thermodynamics. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the hessian matrix of the specific energy expressed as function of specific volume and specific entropy:


is defined positive. This statement corresponds to the two conditions:


The first condition is the one ensuring the parameter "a" is defined real.


Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a strictly hyperbolic system.


At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue λ1 one obtains:


Basing on the third equation that simply has solution s1=0, the system reduces to:


The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector:


The other two eigenvectors can be found with analogous procedure as:


Then the projection matrix can be built:


Finally it becomes apparent that the real parameter "a" previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the "wave speed". It remains to be shown that the sound speed corresponds to the particular case of an isentropic transformation:


Sound speed is defined as the wavespeed of an isentropic transformation:


by the definition of the isoentropic compressibility:


the soundspeed results always the square root of ratio between the isentropic compressibility and the density:


The sound speed in an ideal gas depends only on its temperature:


Since the specific enthalpy in an ideal gas is proportional to its temperature:


the sound speed in an ideal gas can also be made dependent only on its specific enthalpy:


Bernoulli's theorem is a direct consequence of the Euler equations.


The vector calculus identity of the cross product of a curl holds:


where the Feynman subscript notation formula_127 is used, which means the subscripted gradient operates only on the factor formula_37.


Lamb in his famous classical book Hydrodynamics (1895), still in print, used this identity to change the convective term of the flow velocity in rotational form:


the Euler momentum equation in Lamb's form becomes:


the Euler momentum equation assumes a form that is optimal to demonstrate Bernoulli's theorem for steady flows:


In fact, in case of an external conservative field, by defining its potential φ:


In case of a steady flow the time derivative of the flow velocity disappears, so the momentum equation becomes:


And by projecting the momentum equation on the flow direction, i.e. along a "streamline", the cross product disappears because its result is always perpendicular to the velocity:


In the steady incompressible case the mass equation is simply:


that is the mass conservation for a steady incompressible flow states that the density along a streamline is constant. Then the Euler momentum equation in the steady incompressible case becomes:


The convenience of defining the total head for an inviscid liquid flow is now apparent:


That is, the momentum balance for a steady inviscid and incompressible flow in an external conservative field states that the total head along a streamline is constant.


In the most general steady (compressibile) case the mass equation in conservation form is:


The right-hand side appears on the energy equation in convective form, which on the steady state reads:


so that the internal specific energy now features in the head.


Since the external field potential is usually small compared to the other terms, it is convenient to group the latter ones in the total enthalpy:


and the Bernoulli invariant for an inviscid gas flow is:


That is, the energy balance for a steady inviscid flow in an external conservative field states that the sum of the total enthalpy and the external potential is constant along a streamline.


In the usual case of small potential field, simply:


By substituting the pressure gradient with the entropy and enthalpy gradient, according to the first law of thermodynamics in the enthalpy form:


in the convective form of Euler momentum equation, one arrives to:


Friedmann deduced this equation for the particular case of a perfect gas and published it in 1922. However, this equation is general for an inviscid nonconductive fluid and no equation of state is implicit in it.


On the other hand, by substituting the enthalpy form of the first law of thermodynamics in the rotational form of Euler momentum equation, one obtains:


and by defining the specific total enthalpy:


one arrives to the Crocco–Vazsonyi form (Crocco, 1937) of the Euler momentum equation:


In the steady case the two variables entropy and total enthalpy are particularly useful since Euler equations can be recast into the Crocco's form:


Finally if the flow is also isothermal:


by defining the specific total Gibbs free energy:


the Crocco's form can be reduced to:


From these relationships one deduces that the specific total free energy is uniform in a steady, irrotational, isothermal, isoentropic, inviscid flow.


Shock propagation is studied – among many other fields – in aerodynamics and rocket propulsion, where sufficiently fast flows occur.


To properly compute the continuum quantities in discontinuous zones (for example shock waves or boundary layers) from the "local" forms (all the above forms are local forms, since the variables being described are typical of one point in the space considered, i.e. they are "local variables") of Euler equations through finite difference methods generally too many space points and time steps would be necessary for the memory of computers now and in the near future. In these cases it is mandatory to avoid the local forms of the conservation equations, passing some weak forms, like the finite volume one.


Starting from the simplest case, one consider a steady free conservation equation in conservation form in the space domain:


where in general F is the flux matrix. By integrating this local equation over a fixed volume Vm, it becomes:


Then, basing on the divergence theorem, we can transform this integral in a boundary integral of the flux:


This "global form" simply states that there is no net flux of a conserved quantity passing through a region in the case steady and without source. In 1D the volume reduces to an interval, its boundary being its extrema, then the divergence theorem reduces to the fundamental theorem of calculus:


that is the simple finite difference equation, known as the "jump relation":


Or, if one performs an indefinite integral:


On the other hand, a transient conservation equation:


For one-dimensional Euler equations the conservation variables and the flux are the vectors:


In the one dimensional case the correspondent jump relations, called the Rankine–Hugoniot equations, are:<


In the steady one dimensional case the become simply:


Thanks to the mass difference equation, the energy difference equation can be simplified without any restriction:


where formula_174 is the specific total enthalpy.


These are the usually expressed in the convective variables:


The energy equation is an integral form of the Bernoulli equation in the compressible case.


The former mass and momentum equations by substitution lead to the Rayleigh equation:


Since the second term is a constant, the Rayleigh equation always describes a simple line in the pressure volume plane not depending of any equation of state, i.e. the Rayleigh line. By substitution in the Rankine–Hugoniot equations, that can be also made explicit as:


One can also obtain the kinetic equation and to the Hugoniot equation. The analytical passages are not shown here for brevity.


The Hugoniot equation, coupled with the fundamental equation of state of the material:


describes in general in the pressure volume plane a curve passing by the conditions (v0, p0), i.e. the Hugoniot curve, whose shape strongly depends on the type of material considered.


It is also customary to define a "Hugoniot function":


allowing to quantify deviations from the Hugoniot equation, similarly to the previous definition of the "hydraulic head", useful for the deviations from the Bernoulli equation.


On the other hand, by integrating a generic conservation equation:


on a fixed volume Vm, and then basing on the divergence theorem, it becomes:


By integrating this equation also over a time interval:


Now by defining the node conserved quantity:


In particular, for Euler equations, once the conserved quantities have been determined, the convective variables are deduced by back substitution:


Then the explicit finite volume expressions of the original convective variables are:<


\oint_{\partial V_m}\rho\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex]


\mathbf u_{m,n+1} &= \mathbf u_{m,n} - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} (\rho\mathbf{u} \otimes \mathbf{u} - p\mathbf{I}) \cdot \hat{n}\, ds\, dt \\[1.2ex]


\mathbf e_{m,n+1} &= \mathbf e_{m,n} - \frac{1}{2}\left(u^2_{m,n+1} - u^2_{m,n}\right) - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} \left(\rho e + \frac{1}{2}\rho u^2 + p\right)\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex]


It has been shown that Euler equations are not a complete set of equations, but they require some additional constraints to admit a unique solution: these are the equation of state of the material considered. To be consistent with thermodynamics these equations of state should satisfy the two laws of thermodynamics. On the other hand, by definition non-equilibrium system are described by laws lying outside these laws. In the following we list some very simple equations of state and the corresponding influence on Euler equations.


For an ideal polytropic gas the fundamental equation of state is:


where formula_53 is the specific energy, formula_69 is the specific volume, formula_71 is the specific entropy, formula_193 is the molecular mass, formula_194 here is considered a constant (polytropic process), and can be shown to correspond to the heat capacity ratio. This equation can be shown to be consistent with the usual equations of state employed by thermodynamics.


From this equation one can derive the equation for pressure by its thermodynamic definition:


By inverting it one arrives to the mechanical equation of state:


Then for an ideal gas the compressible Euler equations can be simply expressed in the "mechanical" or "primitive variables" specific volume, flow velocity and pressure, by taking the set of the equations for a thermodynamic system and modifying the energy equation into a pressure equation through this mechanical equation of state. At last, in convective form they result:


and in one-dimensional quasilinear form they results:


In the case of steady flow, it is convenient to choose the Frenet–Serret frame along a streamline as the coordinate system for describing the steady momentum Euler equation:


where formula_1, formula_15 and formula_40 denote the flow velocity, the pressure and the density, respectively.


Let formula_204 be a Frenet–Serret orthonormal basis which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively. Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-hand side of the above equation, the convective derivative of velocity, can be described as follows:


where formula_206 is the radius of curvature of the streamline.


Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form:


For barotropic flow formula_208, Bernoulli's equation is derived from the first equation:


The second equation expresses that, in the case the streamline is curved, there should exist a pressure gradient normal to the streamline because the centripetal acceleration of the fluid parcel is only generated by the normal pressure gradient.


The third equation expresses that pressure is constant along the binormal axis.


Let formula_210 be the distance from the center of curvature of the streamline, then the second equation is written as follows:


In a steady flow of an inviscid fluid without external forces, the center of curvature of the streamline lies in the direction of decreasing radial pressure.


Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature. Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".


This "theorem" explains clearly why there are such low pressures in the centre of vortices, which consist of concentric circles of streamlines.


This also is a way to intuitively explain why airfoils generate lift forces.


All potential flow solutions are also solutions of the Euler equations, and in particular the incompressible Euler equations when the potential is harmonic.


Solutions to the Euler equations with vorticity are:


In fluid dynamics the Milne-Thomson circle theorem or the circle theorem is a statement giving a new stream function for a fluid flow when a cylinder is placed into that flow. It was named after the English mathematician L. M. Milne-Thomson.


Let formula_1 be the complex potential for a fluid flow, where all singularities of formula_1 lie in formula_3. If a circle formula_4 is placed into that flow, the complex potential for the new flow is given by


with same singularities as formula_1 in formula_3 and formula_4 is a streamline. On the circle formula_4, formula_10, therefore


Consider a uniform irrotational flow formula_12 with velocity formula_13 flowing in the positive formula_14 direction and place an infinitely long cylinder of radius formula_15 in the flow with the center of the cylinder at the origin. Then formula_16, hence using circle theorem,


represents the complex potential of uniform flow over a cylinder.


The Prony equation (named after Gaspard de Prony) is a historically important equation in hydraulics, used to calculate the head loss due to friction within a given run of pipe. It is an empirical equation developed by Frenchman Gaspard de Prony in the 19th century:


where "hf" is the head loss due to friction, calculated from: the ratio of the length to diameter of the pipe "L/D", the velocity of the flow "V", and two empirical factors "a" and "b" to account for friction.


This equation has been supplanted in modern hydraulics by the Darcy–Weisbach equation, which used it as a starting point.


In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the Darcy friction factor, a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as well as open-channel flow.


The Darcy friction factor is also known as the "Darcy–Weisbach friction factor", "resistance coefficient" or simply "friction factor"; by definition it is four times larger than the Fanning friction factor.


In this article, the following conventions and definitions are to be understood:


Which friction factor formula may be applicable depends upon the type of flow that exists:


Transition (neither fully laminar nor fully turbulent) flow occurs in the range of Reynolds numbers between 2300 and 4000. The value of the Darcy friction factor is subject to large uncertainties in this flow regime.


The Blasius correlation is the simplest equation for computing the Darcy friction


factor. Because the Blasius correlation has no term for pipe roughness, it


is valid only to smooth pipes. However, the Blasius correlation is sometimes


used in rough pipes because of its simplicity. The Blasius correlation is valid


The Darcy friction factor for fully turbulent flow (Reynolds number greater than 4000) in rough conduits can be modeled by the Colebrook–White equation.


The last formula in the "Colebrook equation" section of this article is for free surface flow. The approximations elsewhere in this article are not applicable for this type of flow.


Before choosing a formula it is worth knowing that in the paper on the Moody chart, Moody stated the accuracy is about ±5% for smooth pipes and ±10% for rough pipes. If more than one formula is applicable in the flow regime under consideration, the choice of formula may be influenced by one or more of the following:


The phenomenological Colebrook–White equation (or Colebrook equation) expresses the Darcy friction factor "f" as a function of Reynolds number Re and pipe relative roughness ε / "D"h, fitting the data of experimental studies of turbulent flow in smooth and rough pipes.


The equation can be used to (iteratively) solve for the Darcy–Weisbach friction factor "f".


For a conduit flowing completely full of fluid at Reynolds numbers greater than 4000, it is expressed as:


Note: Some sources use a constant of 3.71 in the denominator for the roughness term in the first equation above.


The Colebrook equation is usually solved numerically due to its implicit nature. Recently, the Lambert W function has been employed to obtain explicit reformulation of the Colebrook equation.


Additional, mathematically equivalent forms of the Colebrook equation are:


The additional equivalent forms above assume that the constants 3.7 and 2.51 in the formula at the top of this section are exact. The constants are probably values which were rounded by Colebrook during his curve fitting; but they are effectively treated as exact when comparing (to several decimal places) results from explicit formulae (such as those found elsewhere in this article) to the friction factor computed via Colebrook's implicit equation.


Equations similar to the additional forms above (with the constants rounded to fewer decimal places, or perhaps shifted slightly to minimize overall rounding errors) may be found in various references. It may be helpful to note that they are essentially the same equation.


Another form of the Colebrook-White equation exists for free surfaces. Such a condition may exist in a pipe that is flowing partially full of fluid. For free surface flow:


The above equation is valid only for turbulent flow. Another approach for estimating "f" in free surface flows, which is valid under all the flow regimes (laminar, transition and turbulent) is the following:


where "Reh" is Reynolds number where "h" is the characteristic hydraulic length (hydraulic radius for 1D flows or water depth for 2D flows) and "Rh" is the hydraulic radius (for 1D flows) or the water depth (for 2D flows). The Lambert W function can be calculated as follows:


The "Haaland equation" was proposed in 1983 by Professor S.E. Haaland of the Norwegian Institute of Technology. It is used to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation, but the discrepancy from experimental data is well within the accuracy of the data.


The Swamee–Jain equation is used to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation.


Serghides's solution is used to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. It was derived using Steffensen's method.


The solution involves calculating three intermediate values and then substituting those values into a final equation.


The equation was found to match the Colebrook–White equation within 0.0023% for a test set with a 70-point matrix consisting of ten relative roughness values (in the range 0.00004 to 0.05) by seven Reynolds numbers (2500 to 108).


Goudar equation is the most accurate approximation to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe. It is an approximation of the implicit Colebrook–White equation. Equation has the following form


Brkić shows one approximation of the Colebrook equation based on the Lambert W-function


The equation was found to match the Colebrook–White equation within 3.15%.


Brkić and Praks show one approximation of the Colebrook equation based on the Wright formula_40-function, a cognate of the Lambert W-function


The equation was found to match the Colebrook–White equation within 0.0497%.


Praks and Brkić show one approximation of the Colebrook equation based on the Wright formula_40-function, a cognate of the Lambert W-function


The equation was found to match the Colebrook–White equation within 0.0012%.


Since Serghides's solution was found to be one of the most accurate approximation of the implicit Colebrook–White equation, Niazkar modified the Serghides's solution to solve directly for the Darcy–Weisbach friction factor "f" for a full-flowing circular pipe.


Niazkar's solution is shown in the following:


Niazkar's solution was found to be the most accurate correlation based on a comparative analysis conducted in the literature among 42 different explicit equations for estimating Colebrook friction factor.


Early approximations for smooth pipes by Paul Richard Heinrich Blasius in terms of the Moody friction factor are given in one article of 1913:


Johann Nikuradse in 1932 proposed that this corresponds to a power law correlation for the fluid velocity profile.


Mishra and Gupta in 1979 proposed a correction for curved or helically coiled tubes, taking into account the equivalent curve radius, Rc:


The following table lists historical approximations to the Colebrook–White relation for pressure-driven flow. Churchill equation (1977) is the only equation that can be evaluated for very slow flow (Reynolds number < 1), but the Cheng (2008), and Bellos et al. (2018) equations also return an approximately correct value for friction factor in the laminar flow region (Reynolds number < 2300). All of the others are for transitional and turbulent flow only.


A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.


Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.


Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point.


Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier–Stokes equations.


Flows governed by continuity equations can be visualized using a Sankey diagram.


A continuity equation is useful when a "flux" can be defined. To define flux, first there must be a quantity which can flow or move, such as mass, energy, electric charge, momentum, number of molecules, etc. Let be the volume density of this quantity, that is, the amount of per unit volume.


The way that this quantity is flowing is described by its flux. The flux of is a vector field, which we denote as j. Here are some examples and properties of flux:


The integral form of the continuity equation states that:


Mathematically, the integral form of the continuity equation expressing the rate of increase of within a volume is:


In a simple example, could be a building, and could be the number of people in the building. The surface would consist of the walls, doors, roof, and foundation of the building. Then the continuity equation states that the number of people in the building increases when people enter the building (an inward flux through the surface), decreases when people exit the building (an outward flux through the surface), increases when someone in the building gives birth (a source, ), and decreases when someone in the building dies (a sink, ).


By the divergence theorem, a general continuity equation can also be written in a "differential form":


This general equation may be used to derive any continuity equation, ranging from as simple as the volume continuity equation to as complicated as the Navier–Stokes equations. This equation also generalizes the advection equation. Other equations in physics, such as Gauss's law of the electric field and Gauss's law for gravity, have a similar mathematical form to the continuity equation, but are not usually referred to by the term "continuity equation", because in those cases does not represent the flow of a real physical quantity.


In the case that is a conserved quantity that cannot be created or destroyed (such as energy), and the equations become:


In electromagnetic theory, the continuity equation is an empirical law expressing (local) charge conservation. Mathematically it is an automatic consequence of Maxwell's equations, although charge conservation is more fundamental than Maxwell's equations. It states that the divergence of the current density (in amperes per square metre) is equal to the negative rate of change of the charge density (in coulombs per cubic metre),


Current is the movement of charge. The continuity equation says that if charge is moving out of a differential volume (i.e. divergence of current density is positive) then the amount of charge within that volume is going to decrease, so the rate of change of charge density is negative. Therefore, the continuity equation amounts to a conservation of charge.


If magnetic monopoles exist, there would be a continuity equation for monopole currents as well, see the monopole article for background and the duality between electric and magnetic currents.


In fluid dynamics, the continuity equation states that the rate at which mass enters a system is equal to the rate at which mass leaves the system plus the accumulation of mass within the system.


The differential form of the continuity equation is:


The time derivative can be understood as the accumulation (or loss) of mass in the system, while the divergence term represents the difference in flow in versus flow out. In this context, this equation is also one of the Euler equations (fluid dynamics). The Navier–Stokes equations form a vector continuity equation describing the conservation of linear momentum.


If the fluid is incompressible (volumetric strain rate is zero), the mass continuity equation simplifies to a volume continuity equation:


which means that the divergence of the velocity field is zero everywhere. Physically, this is equivalent to saying that the local volume dilation rate is zero, hence a flow of water through a converging pipe will adjust solely by increasing its velocity as water is largely incompressible.


In computer vision, optical flow is the pattern of apparent motion of objects in a visual scene. Under the assumption that brightness of the moving object did not change between two image frames, one can derive the optical flow equation as:


Conservation of energy says that energy cannot be created or destroyed. (See below for the nuances associated with general relativity.) Therefore, there is a continuity equation for energy flow:


An important practical example is the flow of heat. When heat flows inside a solid, the continuity equation can be combined with Fourier's law (heat flux is proportional to temperature gradient) to arrive at the heat equation. The equation of heat flow may also have source terms: Although "energy" cannot be created or destroyed, "heat" can be created from other types of energy, for example via friction or joule heating.


If there is a quantity that moves continuously according to a stochastic (random) process, like the location of a single dissolved molecule with Brownian motion, then there is a continuity equation for its probability distribution. The flux in this case is the probability per unit area per unit time that the particle passes through a surface. According to the continuity equation, the negative divergence of this flux equals the rate of change of the probability density. The continuity equation reflects the fact that the molecule is always somewhere—the integral of its probability distribution is always equal to 1—and that it moves by a continuous motion (no teleporting).


Quantum mechanics is another domain where there is a continuity equation related to "conservation of probability". The terms in the equation require the following definitions, and are slightly less obvious than the other examples above, so they are outlined here:


With these definitions the continuity equation reads:


Either form may be quoted. Intuitively, the above quantities indicate this represents the flow of probability. The "chance" of finding the particle at some position and time flows like a fluid; hence the term "probability current", a vector field. The particle itself does "not" flow deterministically in this vector field.


The total current flow in the semiconductor consists of drift current and diffusion current of both the electrons in the conduction band and holes in the valence band.


This section presents a derivation the equation above for electrons. A similar derivation can be found for the equation for holes.


Consider the fact that the number of electrons is conserved across a volume of semiconductor material with cross-sectional area, "A", and length, "dx", along the "x"-axis. More precisely, one can say:


Total electron current is the sum of drift current and diffusion current:


Aplying the product rule reults in the final expression:


The key to solving these equations in real devices is whenever possible to select regions in which most of the mechanisms are negligible so that the equations reduce to a much simpler form.


The notation and tools of special relativity, especially 4-vectors and 4-gradients, offer a convenient way to write any continuity equation.


The density of a quantity and its current can be combined into a 4-vector called a 4-current:


where is the speed of light. The 4-divergence of this current is:


where is the 4-gradient and is an index labelling the spacetime dimension. Then the continuity equation is:


in the usual case where there are no sources or sinks, that is, for perfectly conserved quantities like energy or charge. This continuity equation is manifestly ("obviously") Lorentz invariant.


Examples of continuity equations often written in this form include electric charge conservation


where is the electric 4-current; and energy–momentum conservation


In general relativity, where spacetime is curved, the continuity equation (in differential form) for energy, charge, or other conserved quantities involves the "covariant" divergence instead of the ordinary divergence.


For example, the stress–energy tensor is a second-order tensor field containing energy–momentum densities, energy–momentum fluxes, and shear stresses, of a mass-energy distribution. The differential form of energy–momentum conservation in general relativity states that the "covariant" divergence of the stress-energy tensor is zero:


This is an important constraint on the form the Einstein field equations take in general relativity.


However, the "ordinary" divergence of the stress–energy tensor does "not" necessarily vanish:


The right-hand side strictly vanishes for a flat geometry only.


As a consequence, the "integral" form of the continuity equation is difficult to define and not necessarily valid for a region within which spacetime is significantly curved (e.g. around a black hole, or across the whole universe).


Quarks and gluons have "color charge", which is always conserved like electric charge, and there is a continuity equation for such color charge currents (explicit expressions for currents are given at gluon field strength tensor).


There are many other quantities in particle physics which are often or always conserved: baryon number (proportional to the number of quarks minus the number of antiquarks), electron number, mu number, tau number, isospin, and others. Each of these has a corresponding continuity equation, possibly including source / sink terms.


One reason that conservation equations frequently occur in physics is Noether's theorem. This states that whenever the laws of physics have a continuous symmetry, there is a continuity equation for some conserved physical quantity. The three most famous examples are:


See Noether's theorem for proofs and details.


In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or in short Reynolds theorem, is a three-dimensional generalization of the Leibniz integral rule which is also known as differentiation under the integral sign.


The theorem is named after Osborne Reynolds (1842–1912). It is used to recast derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics.


Consider integrating over the time-dependent region that has boundary , then taking the derivative with respect to time:


If we wish to move the derivative within the integral, there are two issues: the time dependence of , and the introduction of and removal of space from due to its dynamic boundary. Reynolds transport theorem provides the necessary framework.


Reynolds transport theorem can be expressed as follows:


in which is the outward-pointing unit normal vector, is a point in the region and is the variable of integration, and are volume and surface elements at , and is the velocity of the area element ("not" the flow velocity). The function may be tensor-, vector- or scalar-valued. Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used.


In continuum mechanics, this theorem is often used for material elements. These are parcels of fluids or solids which no material enters or leaves. If is a material element then there is a velocity function , and the boundary elements obey


This condition may be substituted to obtain:


If we take to be constant with respect to time, then and the identity reduces to


as expected. (This simplification is not possible if the flow velocity is incorrectly used in place of the velocity of an area element.)


The theorem is the higher-dimensional extension of differentiation under the integral sign and reduces to that expression in some cases. Suppose is independent of and , and that is a unit square in the -plane and has limits and . Then Reynolds transport theorem reduces to


which, up to swapping and , is the standard expression for differentiation under the integral sign.


The Fanning friction factor, named after John Thomas Fanning, is a dimensionless number used as a local parameter in continuum mechanics calculations. It is defined as the ratio between the local shear stress and the local flow kinetic energy density:


In particular the shear stress at the wall can, in turn, be related to the pressure loss by multiplying the wall shear stress by the wall area ( formula_12 for a pipe with circular cross section) and dividing by the cross-sectional flow area ( formula_13 for a pipe with circular cross section). Thus formula_14


This friction factor is one-fourth of the Darcy friction factor, so attention must be paid to note which one of these is meant in the "friction factor" chart or equation consulted. Of the two, the Fanning friction factor is the more commonly used by chemical engineers and those following the British convention.


The formulas below may be used to obtain the Fanning friction factor for common applications.


The Darcy friction factor can also be expressed as


For laminar flow in a round tube.


From the chart, it is evident that the friction factor is never zero, even for smooth pipes because of some roughness at the microscopic level.


The friction factor for laminar flow of Newtonian fluids in round tubes is often taken to be:


where Re is the Reynolds number of the flow.


For a square channel the value used is:


For turbulent flow in a round tube.


Blasius developed an expression of friction factor in 1913 for the flow in the regime formula_21.


Koo introduced another explicit formula in 1933 for a turbulent flow in region of formula_23


When the pipes have certain roughness formula_25, this factor must be taken in account when the Fanning friction factor is calculated. The relationship between pipe roughness and Fanning friction factor was developed by Haaland (1983) under flow conditions of formula_26


As the roughness extends into turbulent core, the Fanning friction factor becomes independent of fluid viscosity at large Reynolds numbers, as illustrated by Nikuradse and Reichert (1943) for the flow in region of formula_30. The equation below has been modified from the original format which was developed for Darcy friction factor by a factor of formula_31


For the turbulent flow regime, the relationship between the Fanning friction factor and the Reynolds number is more complex and is governed by the Colebrook equation which is implicit in formula_2:


Various explicit approximations of the related Darcy friction factor have been developed for turbulent flow.


Stuart W. Churchill developed a formula that covers the friction factor for both laminar and turbulent flow. This was originally produced to describe the Moody chart, which plots the Darcy-Weisbach Friction factor against Reynolds number. The Darcy Weisbach Formula formula_35, also called Moody friction factor, is 4 times the Fanning friction factor formula_36 and so a factor of formula_37 has been applied to produce the formula given below.


Due to geometry of non-circular conduits, the Fanning friction factor can be estimated from algebraic expressions above by using hydraulic radius formula_41 when calculating for Reynolds number formula_42


The friction head can be related to the pressure loss due to friction by dividing the pressure loss by the product of the acceleration due to gravity and the density of the fluid. Accordingly, the relationship between the friction head and the Fanning friction factor is:


In fluid mechanics and astrophysics, the relativistic Euler equations are a generalization of the Euler equations that account for the effects of general relativity.They have applications in high-energy astrophysics and numerical relativity, where they are commonly used for describing phenomena such as gamma-ray bursts, accretion phenomena, and neutron stars, often with the addition of a magnetic field. "Note: for consistency with the literature, this article makes use of natural units, namely the speed of light" formula_1 "and the Einstein summation convention."


For most fluids observable on Earth, traditional fluid mechanics based on Newtonian mechanics is sufficient. However, as the fluid velocity approaches the speed of light or moves through strong gravitational fields, or the pressure approaches the energy density (formula_2), these equations are no longer valid. Such situations occur frequently in astrophysical applications. For example, gamma-ray bursts often feature speeds only formula_3 less than the speed of light, and neutron stars feature gravitational fields that are more than formula_4 times stronger than the Earth's. Under these extreme circumstances, only a relativistic treatment of fluids will suffice.


The equations of motion are contained in the continuity equation of the stress–energy tensor formula_5:


where formula_7 is the covariant derivative. For a perfect fluid,


Here formula_9 is the total mass-energy density (including both rest mass and internal energy density) of the fluid, formula_10 is the fluid pressure, formula_11 is the four-velocity of the fluid, and formula_12 is the metric tensor. To the above equations, a statement of conservation is usually added, usually conservation of baryon number. If formula_13 is the number density of baryons this may be stated


These equations reduce to the classical Euler equations if the fluid three-velocity is much less than the speed of light, the pressure is much less than the energy density, and the latter is dominated by the rest mass density.To close this system, an equation of state, such as an ideal gas or a Fermi gas, is also added.


In the case of flat space, that is formula_15 and using a metric signature of formula_16, the equations of motion are,


Where formula_18 is the energy density of the system, with formula_10 being the pressure, and formula_20 being the four-velocity of the system.


Expanding out the sums and equations, we have, (using formula_21 as the material derivative)


Then, picking formula_23 to observe the behavior of the velocity itself, we see that the equations of motion become


Note that taking the non-relativistic limit, we have formula_25. This says that the of the system is dominated by the rest energy of the fluid in question.


In this limit, we have formula_26 and formula_27, and can see that we return the Euler Equation of formula_28.


In order to determine the equations of motion, we take advantage of the following spatial projection tensor condition:


We prove this by looking at formula_30 and then multiplying each side by formula_31. Upon doing this, and noting that formula_32, we have formula_33. Relabeling the indices formula_34 as formula_35 shows that the two completely cancel. This cancellation is the expected result of contracting a temporal tensor with a spatial tensor.


Where we have implicitly defined that formula_37.


Then, let's note the fact that formula_40 and formula_41. Note that the second identity follows from the first. Under these simplifications, we find that


We have two cancellations, and are thus left with


The Starling equation describes the net flow of fluid across a semipermeable membrane. It is named after Ernest Starling. It describes the balance between capillary pressure, interstitial pressure, and osmotic pressure. The classic Starling equation has in recent years been revised. The Starling principle of fluid exchange is key to understanding how plasma fluid (solvent) within the bloodstream (intravascular fluid) moves to the space outside the bloodstream (extravascular space).


Transendothelial fluid exchange occurs predominantly in the capillaries, and is a process of plasma ultrafiltration across a semi-permeable membrane. It is now appreciated that the ultrafilter is the glycocalyx of the plasma membrane of the endothelium, whose interpolymer spaces function as a system of small pores, radius circa 5 nm. Where the endothelial glycocalyx overlies an inter endothelial cell cleft, the plasma ultrafiltrate may pass to the interstitial space. Some continuous capillaries may feature fenestrations that provide an additional subglycocalyx pathway for solvent and small solutes. Discontinuous capillaries as found in sinusoidal tissues of bone marrow, liver and spleen have little or no filter function.


The rate at which fluid is filtered across vascular endothelium (transendothelial filtration) is determined by the sum of two outward forces, capillary pressure (formula_1) and interstitial protein osmotic pressure (formula_2), and two absorptive forces, plasma protein osmotic pressure (formula_3) and interstitial pressure (formula_4). The Starling equation describes these forces in mathematical terms. It is one of the Kedem–Katchalski equations which bring nonsteady state thermodynamics to the theory of osmotic pressure across membranes that are at least partly permeable to the solute responsible for the osmotic pressure difference. The second Kedem–Katchalsky equation explains the trans endothelial transport of solutes, formula_5.


The classic Starling equation reads as follows:


By convention, outward force is defined as positive, and inward force is defined as negative. If Jv is positive, solvent is leaving the capillary (filtration). If negative, solvent is entering the capillary (absorption).


Applying the classic Starling equation, it had long been taught that continuous capillaries filter out fluid in their arteriolar section and reabsorb most of it in their venular section, as shown by the diagram.


However, empirical evidence shows that, in most tissues, the flux of the intraluminal fluid of capillaries is continuous and, primarily, effluent. Efflux occurs along the whole length of a capillary. Fluid filtered to the space outside a capillary is mostly returned to the circulation via lymph nodes and the thoracic duct.


A mechanism for this phenomenon is the Michel-Weinbaum model, in honour of two scientists who, independently, described the filtration function of the glycocalyx. Briefly, the colloid osmotic pressure πi of the interstitial fluid has been found to have no effect on Jv and the colloid osmotic pressure difference that opposes filtration is now known to be π'p minus the subglycocalyx π, which is close to zero while there is adequate filtration to flush interstitial proteins out of the interendothelial cleft. Consequently, Jv is much less than previously calculated, and the unopposed diffusion of interstitial proteins to the subglycocalyx space if and when filtration falls wipes out the colloid osmotic pressure difference necessary for reabsorption of fluid to the capillary.


The revised Starling equation is compatible with the steady-state Starling principle:


Pressures are often measured in millimetres of mercury (mmHg), and the filtration coefficient in millilitres per minute per millimetre of mercury (ml·min−1·mmHg−1).


In some texts the product of hydraulic conductivity and surface area is called the filtration co-efficient Kfc.


Staverman's reflection coefficient, "σ", is a unitless constant that is specific to the permeability of a membrane to a given solute.


The Starling equation, written without "σ", describes the flow of a solvent across a membrane that is impermeable to the solutes contained within the solution.


"σn" corrects for the partial permeability of a semipermeable membrane to a solute "n".


Where "σ" is close to 1, the plasma membrane is less permeable to the denotated species (for example, larger molecules such as albumin and other plasma proteins), which may flow across the endothelial lining, from higher to lower concentrations, more slowly, while allowing water and smaller solutes through the glycocalyx filter to the extravascular space.


Following are typically quoted values for the variables in the classic Starling equation:


It is reasoned that some albumin escapes from the capillaries and enters the interstitial fluid where it would produce a flow of water equivalent to that produced by a hydrostatic pressure of +3 mmHg. Thus, the difference in protein concentration would produce a flow of fluid into the vessel at the venous end equivalent to 28 − 3 = 25 mmHg of hydrostatic pressure. The total oncotic pressure present at the venous end could be considered as +25 mmHg.


In the beginning (arteriolar end) of a capillary, there is a net driving force (formula_28) outwards from the capillary of +9 mmHg. In the end (venular end), on the other hand, there is a net driving force of −8 mmHg.


Assuming that the net driving force declines linearly, then there is a mean net driving force outwards from the capillary as a whole, which also results in that more fluid exits a capillary than re-enters it. The lymphatic system drains this excess.


J. Rodney Levick argues in his textbook that the interstitial force is often underestimated, and measurements used to populate the revised Starling equation show the absorbing forces to be consistently less than capillary or venular pressures.


Glomerular capillaries have a continuous glycocalyx layer in health and the total transendothelial filtration rate of solvent (formula_7) to the renal tubules is normally around 125 ml/ min (about 180 litres/ day). Glomerular capillary formula_7 is more familiarly known as the glomerular filtration rate (GFR). In the rest of the body's capillaries, formula_7 is typically 5 ml/ min (around 8 litres/ day), and the fluid is returned to the circulation "via" afferent and efferent lymphatics.


The Starling equation can describe the movement of fluid from pulmonary capillaries to the alveolar air space.


The principles behind the equation are useful for explaining physiological phenomena in capillaries, such as the formation of edema.


Woodcock and Woodcock showed in 2012 that the revised Starling equation (steady-state Starling principle) provides scientific explanations for clinical observations concerning intravenous fluid therapy.


The Starling equation is named for the British physiologist Ernest Starling, who is also recognised for the Frank–Starling law of the heart. Starling can be credited with identifying that the "absorption of isotonic salt solutions (from the extravascular space) by the blood vessels is determined by this osmotic pressure of the serum proteins" in 1896.


In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.


The mild-slope equation models the propagation and transformation of water waves, as they travel through waters of varying depth and interact with lateral boundaries such as cliffs, beaches, seawalls and breakwaters. As a result, it describes the variations in wave amplitude, or equivalently wave height. From the wave amplitude, the amplitude of the flow velocity oscillations underneath the water surface can also be computed. These quantities—wave amplitude and flow-velocity amplitude—may subsequently be used to determine the wave effects on coastal and offshore structures, ships and other floating objects, sediment transport and resulting bathymetric changes of the sea bed and coastline, mean flow fields and mass transfer of dissolved and floating materials. Most often, the mild-slope equation is solved by computer using methods from numerical analysis.


A first form of the mild-slope equation was developed by Eckart in 1952, and an improved version—the mild-slope equation in its classical formulation—has been derived independently by Juri Berkhoff in 1972. Thereafter, many modified and extended forms have been proposed, to include the effects of, for instance: wave–current interaction, wave nonlinearity, steeper sea-bed slopes, bed friction and wave breaking. Also parabolic approximations to the mild-slope equation are often used, in order to reduce the computational cost.


In case of a constant depth, the mild-slope equation reduces to the Helmholtz equation for wave diffraction.


For monochromatic waves according to linear theory—with the free surface elevation given as formula_1 and the waves propagating on a fluid layer of mean water depth formula_2—the mild-slope equation is:


The phase and group speed depend on the dispersion relation, and are derived from Airy wave theory as:


For a given angular frequency formula_7, the wavenumber formula_12 has to be solved from the dispersion equation, which relates these two quantities to the water depth formula_20.


the mild slope equation can be cast in the form of an inhomogeneous Helmholtz equation:


In spatially coherent fields of propagating waves, it is useful to split the complex amplitude formula_4 in its amplitude and phase, both real valued:


This transforms the mild-slope equation in the following set of equations (apart from locations for which formula_30 is singular):


The last equation shows that wave energy is conserved in the mild-slope equation, and that the wave energy formula_32 is transported in the formula_33-direction normal to the wave crests (in this case of pure wave motion without mean currents). The effective group speed formula_40 is different from the group speed formula_41


The first equation states that the effective wavenumber formula_33 is irrotational, a direct consequence of the fact it is the derivative of the wave phase formula_43, a scalar field. The second equation is the eikonal equation. It shows the effects of diffraction on the effective wavenumber: only for more-or-less progressive waves, with formula_44 the splitting into amplitude formula_45 and phase formula_43 leads to consistent-varying and meaningful fields of formula_45 and formula_33. Otherwise, "κ"2 can even become negative. When the diffraction effects are totally neglected, the effective wavenumber "κ" is equal to formula_12, and the geometric optics approximation for wave refraction can be used.


When formula_50 is used in the mild-slope equation, the result is, apart from a factor formula_51:


Now both the real part and the imaginary part of this equation have to be equal to zero:


The effective wavenumber vector formula_33 is "defined" as the gradient of the wave phase:


Note that formula_33 is an irrotational field, since the curl of the gradient is zero:


Now the real and imaginary parts of the transformed mild-slope equation become, first multiplying the imaginary part by formula_45:


The first equation directly leads to the eikonal equation above for formula_61, while the second gives:


which—by noting that formula_63 in which the angular frequency formula_7 is a constant for time-harmonic motion—leads to the wave-energy conservation equation.


The mild-slope equation can be derived by the use of several methods. Here, we will use a variational approach. The fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational. These assumptions are valid ones for surface gravity waves, since the effects of vorticity and viscosity are only significant in the Stokes boundary layers (for the oscillatory part of the flow). Because the flow is irrotational, the wave motion can be described using potential flow theory.


Luke's Lagrangian formulation gives a variational formulation for non-linear surface gravity waves.


For the case of a horizontally unbounded domain with a constant density formula_36, a free fluid surface at formula_66 and a fixed sea bed at formula_67 Luke's variational principle formula_68 uses the Lagrangian


where formula_70 is the horizontal Lagrangian density, given by:


where formula_72 is the velocity potential, with the flow velocity components being formula_73 formula_74 and formula_75 in the formula_76, formula_77 and formula_78 directions, respectively.


Luke's Lagrangian formulation can also be recast into a Hamiltonian formulation in terms of the surface elevation and velocity potential at the free surface.


Taking the variations of formula_79 with respect to the potential formula_72 and surface elevation formula_81 leads to the Laplace equation for formula_82 in the fluid interior, as well as all the boundary conditions both on the free surface formula_66 as at the bed at formula_84


In case of linear wave theory, the vertical integral in the Lagrangian density formula_70 is split into a part from the bed formula_86 to the mean surface at formula_87 and a second part from formula_88 to the free surface formula_89. Using a Taylor series expansion for the second integral around the mean free-surface elevation formula_87 and only retaining quadratic terms in formula_82 and formula_92 the Lagrangian density formula_93 for linear wave motion becomes


The term formula_95 in the vertical integral is dropped since it has become dynamically uninteresting: it gives a zero contribution to the Euler–Lagrange equations, with the upper integration limit now fixed. The same is true for the neglected bottom term proportional to formula_96 in the potential energy.


The waves propagate in the horizontal formula_6 plane, while the structure of the potential formula_82 is not wave-like in the vertical formula_78-direction. This suggests the use of the following assumption on the form of the potential formula_100


Here formula_104 is the velocity potential at the mean free-surface level formula_103 Next, the mild-slope assumption is made, in that the vertical shape function formula_106 changes slowly in the formula_6-plane, and horizontal derivatives of formula_106 can be neglected in the flow velocity. So:


The Euler–Lagrange equations for this Lagrangian density formula_93 are, with formula_114 representing either formula_115 or formula_116


Now formula_118 is first taken equal to formula_115 and then to formula_120


As a result, the evolution equations for the wave motion become:


with ∇ the horizontal gradient operator: ∇ ≡(∂/∂"x" ∂/∂"y")T where T denotes the transpose.


The next step is to choose the shape function formula_106 and to determine formula_123 and formula_124


Vertical shape function from Airy wave theory.


Since the objective is the description of waves over mildly sloping beds, the shape function formula_125 is chosen according to Airy wave theory. This is the linear theory of waves propagating in constant depth formula_126 The form of the shape function is:


with formula_128 now in general not a constant, but chosen to vary with formula_76 and formula_77 according to the local depth formula_2 and the linear dispersion relation:


Here formula_133 a constant angular frequency, chosen in accordance with the characteristics of the wave field under study. Consequently, the integrals formula_123 and formula_135 become:


The following time-dependent equations give the evolution of the free-surface elevation formula_81 and free-surface potential formula_138


From the two evolution equations, one of the variables formula_115 or formula_141 can be eliminated, to obtain the time-dependent form of the mild-slope equation:


and the corresponding equation for the free-surface potential is identical, with formula_141 replaced by formula_144 The time-dependent mild-slope equation can be used to model waves in a narrow band of frequencies around formula_145


Consider monochromatic waves with complex amplitude formula_4 and angular frequency formula_147


with formula_7 and formula_133 chosen equal to each other, formula_151 Using this in the time-dependent form of the mild-slope equation, recovers the classical mild-slope equation for time-harmonic wave motion:


Applicability and validity of the mild-slope equation.


The standard mild slope equation, without extra terms for bed slope and bed curvature, provides accurate results for the wave field over bed slopes ranging from 0 to about 1/3. However, some subtle aspects, like the amplitude of reflected waves, can be completely wrong, even for slopes going to zero. This mathematical curiosity has little practical importance in general since this reflection becomes vanishingly small for small bottom slopes.