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The Stokes equations represent a considerable simplification of the full Navier–Stokes equations, especially in the incompressible Newtonian case. They are the leading-order simplification of the full Navier–Stokes equations, valid in the distinguished limit formula_10 |
While these properties are true for incompressible Newtonian Stokes flows, the non-linear and sometimes time-dependent nature of non-Newtonian fluids means that they do not hold in the more general case. |
An interesting property of Stokes flow is known as the Stokes' paradox: that there can be no Stokes flow of a fluid around a disk in two dimensions; or, equivalently, the fact there is no non-trivial solution for the Stokes equations around an infinitely long cylinder. |
A Taylor–Couette system can create laminar flows in which concentric cylinders of fluid move past each other in an apparent spiral. A fluid such as corn syrup with high viscosity fills the gap between two cylinders, with colored regions of the fluid visible through the transparent outer cylinder. |
The cylinders are rotated relative to one another at a low speed, which together with the high viscosity of the fluid and thinness of the gap gives a low Reynolds number, so that the apparent mixing of colors is actually laminar and can then be reversed to approximately the initial state. This creates a dramatic demonstration of seemingly mixing a fluid and then unmixing it by reversing the direction of the mixer. |
In the common case of an incompressible Newtonian fluid, the Stokes equations take the (vectorized) form: |
where formula_7 is the velocity of the fluid, formula_13 is the gradient of the pressure, formula_14 is the dynamic viscosity, and formula_4 an applied body force. The resulting equations are linear in velocity and pressure, and therefore can take advantage of a variety of linear differential equation solvers. |
With the velocity vector expanded as formula_16 and similarly the body force vector formula_17, we may write the vector equation explicitly, |
We arrive at these equations by making the assumptions that formula_19 and the density formula_6 is a constant. |
The equation for an incompressible Newtonian Stokes flow can be solved by the stream function method in planar or in 3-D axisymmetric cases |
The linearity of the Stokes equations in the case of an incompressible Newtonian fluid means that a Green's function, formula_21, exists. The Green's function is found by solving the Stokes equations with the forcing term replaced by a point force acting at the origin, and boundary conditions vanishing at infinity: |
where formula_23 is the Dirac delta function, and formula_24 represents a point force acting at the origin. The solution for the pressure "p" and velocity u with |u| and "p" vanishing at infinity is given by |
The terms Stokeslet and point-force solution are used to describe formula_27. Analogous to the point charge in electrostatics, the Stokeslet is force-free everywhere except at the origin, where it contains a force of strength formula_28. |
For a continuous-force distribution (density) formula_29 the solution (again vanishing at infinity) can then be constructed by superposition: |
This integral representation of the velocity can be viewed as a reduction in dimensionality: from the three-dimensional partial differential equation to a two-dimensional integral equation for unknown densities. |
The Papkovich–Neuber solution represents the velocity and pressure fields of an incompressible Newtonian Stokes flow in terms of two harmonic potentials. |
Certain problems, such as the evolution of the shape of a bubble in a Stokes flow, are conducive to numerical solution by the boundary element method. This technique can be applied to both 2- and 3-dimensional flows. |
Hele-Shaw flow is an example of a geometry for which inertia forces are negligible. It is defined by two parallel plates arranged very close together with the space between the plates occupied partly by fluid and partly by obstacles in the form of cylinders with generators normal to the plates. |
Slender-body theory in Stokes flow is a simple approximate method of determining the irrotational flow field around bodies whose length is large compared with their width. The basis of the method is to choose a distribution of flow singularities along a line (since the body is slender) so that their irrotational flow in combination with a uniform stream approximately satisfies the zero normal velocity condition. |
Lamb's general solution arises from the fact that the pressure formula_31 satisfies the Laplace equation, and can be expanded in a series of solid spherical harmonics in spherical coordinates. As a result, the solution to the Stokes equations can be written: |
where formula_33 and formula_34 are solid spherical harmonics of order formula_35: |
and the formula_37 are the associated Legendre polynomials. The Lamb's solution can be used to describe the motion of fluid either inside or outside a sphere. For example, it can be used to describe the motion of fluid around a spherical particle with prescribed surface flow, a so-called squirmer, or to describe the flow inside a spherical drop of fluid. For interior flows, the terms with formula_38 are dropped, while for exterior flows the terms with formula_39 are dropped (often the convention formula_40 is assumed for exterior flows to avoid indexing by negative numbers). |
The drag resistance to a moving sphere, also known as Stokes' solution is here summarised. Given a sphere of radius formula_41, travelling at velocity formula_42, in a Stokes fluid with dynamic viscosity formula_14, the drag force formula_44 is given by: |
The Stokes solution dissipates less energy than any other solenoidal vector field with the same boundary velocities: this is known as the Helmholtz minimum dissipation theorem. |
The Lorentz reciprocal theorem states a relationship between two Stokes flows in the same region. Consider fluid filled region formula_46 bounded by surface formula_47. Let the velocity fields formula_7 and formula_49 solve the Stokes equations in the domain formula_46, each with corresponding stress fields formula_51 and formula_52. Then the following equality holds: |
Where formula_54 is the unit normal on the surface formula_47. The Lorentz reciprocal theorem can be used to show that Stokes flow "transmits" unchanged the total force and torque from an inner closed surface to an outer enclosing surface. The Lorentz reciprocal theorem can also be used to relate the swimming speed of a microorganism, such as cyanobacterium, to the surface velocity which is prescribed by deformations of the body shape via cilia or flagella. |
The Faxén's laws are direct relations that express the multipole moments in terms of the ambient flow and its derivatives. First developed by Hilding Faxén to calculate the force, formula_28, and torque, formula_57 on a sphere, they took the following form: |
where formula_14 is the dynamic viscosity, formula_41 is the particle radius, formula_61 is the ambient flow, formula_62 is the speed of the particle, formula_63 is the angular velocity of the background flow, and formula_64 is the angular velocity of the particle. |
The Faxén's laws can be generalized to describe the moments of other shapes, such as ellipsoids, spheroids, and spherical drops. |
The shallow-water equations are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). The shallow-water equations in unidirectional form are also called Saint-Venant equations, after Adhémar Jean Claude Barré de Saint-Venant (see the related section below). |
The equations are derived from depth-integrating the Navier–Stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity scale of the fluid is small compared to the horizontal velocity scale. It can be shown from the momentum equation that vertical pressure gradients are nearly hydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the horizontal velocity field is constant throughout the depth of the fluid. Vertically integrating allows the vertical velocity to be removed from the equations. The shallow-water equations are thus derived. |
While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. This is an important distinction because, for example, the vertical velocity cannot be zero when the floor changes depth, and thus if it were zero only flat floors would be usable with the shallow-water equations. Once a solution (i.e. the horizontal velocities and free surface displacement) has been found, the vertical velocity can be recovered via the continuity equation. |
Situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are common, so the shallow-water equations are widely applicable. They are used with Coriolis forces in atmospheric and oceanic modeling, as a simplification of the primitive equations of atmospheric flow. |
Shallow-water equation models have only one vertical level, so they cannot directly encompass any factor that varies with height. However, in cases where the mean state is sufficiently simple, the vertical variations can be separated from the horizontal and several sets of shallow-water equations can describe the state. |
The shallow-water equations are derived from equations of conservation of mass and conservation of linear momentum (the Navier–Stokes equations), which hold even when the assumptions of shallow-water break down, such as across a hydraulic jump. In the case of a horizontal bed, with negligible Coriolis forces, frictional and viscous forces, the shallow-water equations are: |
Here "η" is the total fluid column height (instantaneous fluid depth as a function of "x", "y" and "t"), and the 2D vector ("u","v") is the fluid's horizontal flow velocity, averaged across the vertical column. Further "g" is acceleration due to gravity and ρ is the fluid density. The first equation is derived from mass conservation, the second two from momentum conservation. |
Expanding the derivatives in the above using the product rule, the non-conservative form of the shallow-water equations is obtained. Since velocities are not subject to a fundamental conservation equation, the non-conservative forms do not hold across a shock or hydraulic jump. Also included are the appropriate terms for Coriolis, frictional and viscous forces, to obtain (for constant fluid density): |
It is often the case that the terms quadratic in "u" and "v", which represent the effect of bulk advection, are small compared to the other terms. This is called geostrophic balance, and is equivalent to saying that the Rossby number is small. Assuming also that the wave height is very small compared to the mean height (), we have (without lateral viscous forces): |
The one-dimensional (1-D) Saint-Venant equations were derived by Adhémar Jean Claude Barré de Saint-Venant, and are commonly used to model transient open-channel flow and surface runoff. They can be viewed as a contraction of the two-dimensional (2-D) shallow-water equations, which are also known as the two-dimensional Saint-Venant equations. The 1-D Saint-Venant equations contain to a certain extent the main characteristics of the channel cross-sectional shape. |
The 1-D equations are used extensively in computer models such as TUFLOW, Mascaret (EDF), SIC (Irstea), HEC-RAS, SWMM5, ISIS, InfoWorks, Flood Modeller, SOBEK 1DFlow, MIKE 11, and MIKE SHE because they are significantly easier to solve than the full shallow-water equations. Common applications of the 1-D Saint-Venant equations include flood routing along rivers (including evaluation of measures to reduce the risks of flooding), dam break analysis, storm pulses in an open channel, as well as storm runoff in overland flow. |
The system of partial differential equations which describe the 1-D incompressible flow in an open channel of arbitrary cross section – as derived and posed by Saint-Venant in his 1871 paper (equations 19 & 20) – is: |
where "x" is the space coordinate along the channel axis, "t" denotes time, "A"("x","t") is the cross-sectional area of the flow at location "x", "u"("x","t") is the flow velocity, "ζ"("x","t") is the free surface elevation and τ("x","t") is the wall shear stress along the wetted perimeter "P"("x","t") of the cross section at "x". Further ρ is the (constant) fluid density and "g" is the gravitational acceleration. |
Closure of the hyperbolic system of equations ()–() is obtained from the geometry of cross sections – by providing a functional relationship between the cross-sectional area "A" and the surface elevation ζ at each position "x". For example, for a rectangular cross section, with constant channel width "B" and channel bed elevation "z"b, the cross sectional area is: . The instantaneous water depth is with "z"b("x") the bed level (i.e. elevation of the lowest point in the bed above datum, see the cross-section figure). For non-moving channel walls the cross-sectional area "A" in equation () can be written as: |
with "b"("x","h") the effective width of the channel cross section at location "x" when the fluid depth is "h" – so for rectangular channels. |
The wall shear stress τ is dependent on the flow velocity "u", they can be related by using e.g. the Darcy–Weisbach equation, Manning formula or Chézy formula. |
Further, equation () is the continuity equation, expressing conservation of water volume for this incompressible homogeneous fluid. Equation () is the momentum equation, giving the balance between forces and momentum change rates. |
The bed slope "S"("x"), friction slope "S"f("x","t") and hydraulic radius "R"("x","t") are defined as: |
Consequently, the momentum equation () can be written as: |
The momentum equation () can also be cast in the so-called conservation form, through some algebraic manipulations on the Saint-Venant equations, () and (). In terms of the discharge : |
where "A", "I"1 and "I"2 are functions of the channel geometry, described in the terms of the channel width "B"(σ,"x"). Here σ is the height above the lowest point in the cross section at location "x", see the cross-section figure. So σ is the height above the bed level "z"b("x") (of the lowest point in the cross section): |
Above – in the momentum equation () in conservation form – "A", "I"1 and "I"2 are evaluated at . The term describes the hydrostatic force in a certain cross section. And, for a non-prismatic channel, gives the effects of geometry variations along the channel axis "x". |
In applications, depending on the problem at hand, there often is a preference for using either the momentum equation in non-conservation form, () or (), or the conservation form (). For instance in case of the description of hydraulic jumps, the conservation form is preferred since the momentum flux is continuous across the jump. |
The Saint-Venant equations ()–() can be analysed using the method of characteristics. The two celerities d"x"/d"t" on the characteristic curves are: |
The Froude number determines whether the flow is subcritical () or supercritical (). |
For a rectangular and prismatic channel of constant width "B", i.e. with and , the Riemann invariants are: |
so the equations in characteristic form are: |
The Riemann invariants and method of characteristics for a prismatic channel of arbitrary cross-section are described by Didenkulova & Pelinovsky (2011). |
The characteristics and Riemann invariants provide important information on the behavior of the flow, as well as that they may be used in the process of obtaining (analytical or numerical) solutions. |
The dynamic wave is the full one-dimensional Saint-Venant equation. It is numerically challenging to solve, but is valid for all channel flow scenarios. The dynamic wave is used for modeling transient storms in modeling programs including Mascaret (EDF), SIC (Irstea), HEC-RAS, InfoWorks_ICM, MIKE 11, Wash 123d and SWMM5. |
In the order of increasing simplifications, by removing some terms of the full 1D Saint-Venant equations (aka Dynamic wave equation), we get the also classical Diffusive wave equation and Kinematic wave equation. |
For the diffusive wave it is assumed that the inertial terms are less than the gravity, friction, and pressure terms. The diffusive wave can therefore be more accurately described as a non-inertia wave, and is written as: |
The diffusive wave is valid when the inertial acceleration is much smaller than all other forms of acceleration, or in other words when there is primarily subcritical flow, with low Froude values. Models that use the diffusive wave assumption include MIKE SHE and LISFLOOD-FP. In the SIC (Irstea) software this options is also available, since the 2 inertia terms (or any of them) can be removed in option from the interface. |
For the kinematic wave it is assumed that the flow is uniform, and that the friction slope is approximately equal to the slope of the channel. This simplifies the full Saint-Venant equation to the kinematic wave: |
The kinematic wave is valid when the change in wave height over distance and velocity over distance and time is negligible relative to the bed slope, e.g. for shallow flows over steep slopes. The kinematic wave is used in HEC-HMS. |
The 1-D Saint-Venant momentum equation can be derived from the Navier–Stokes equations that describe fluid motion. The "x"-component of the Navier–Stokes equations – when expressed in Cartesian coordinates in the "x"-direction – can be written as: |
where "u" is the velocity in the "x"-direction, "v" is the velocity in the "y"-direction, "w" is the velocity in the "z"-direction, "t" is time, "p" is the pressure, ρ is the density of water, ν is the kinematic viscosity, and "f"x is the body force in the "x"-direction. |
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. |
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