text stringlengths 13 991 |
|---|
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: |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.