Navier-Stokes equations

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The Navier-Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, are a set of equations that describe the motion of fluid substances like liquids and gases. These equations establish that changes in momentum (acceleration) of the particles of a fluid are simply the product of changes in pressure and dissipative viscous forces (similar to friction) acting inside the fluid. These viscous forces originate in molecular interactions and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of the fluid.

They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and economic interest. They are useful to model weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, flow around an airfoil (wing). They are also used in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of the effects of pollution, etc.

The Navier-Stokes equations are differential equations which describe the motion of a fluid. These equations, unlike algebraic equations, do not seek to establish a relation among the variables of interest (e.g. velocity and pressure), rather they establish relations among the rates of change or fluxes of these quantities. In mathematical terms these rates correspond to their derivatives. Thus, the Navier-Stokes for the most simple case of an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solution of the Navier-Stokes for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved in this way and their exact solution is known. These cases often involve non turbulent flow in steady state (flow does not change with time) in which the viscosity of the fluid is large or its velocity is small (small Reynolds number).

For more complex situations, like global weather systems like El Niño or lift in a wing, solution of the Navier-Stokes equations must be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics.

Even though turbulence is an everyday experience it is extremely hard to find solutions for this class of problems. A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whoever makes substantial progress toward a mathematical theory which will help in the understanding of this phenomenon.

Contents 1 Basic assumptions 2 The substantive derivative 3 Conservation laws 3.1 Equation of continuity 3.2 Conservation of momentum 4 The equations 4.1 General form 4.1.1 The form of the equations 5 Special forms 5.1 Newtonian fluids 5.2 Bingham fluids 5.3 Power-law fluid 5.4 Incompressible fluids 6 See also 7 References 8 External links

Basic assumptions

Before going into the details of the Navier-Stokes equations, it is first necessary to make several assumptions about the fluid. The first one is that the fluid is continuous. It signifies that it does not contain voids formed, for example, by bubbles of dissolved gases, or that it does not consist of an aggregate of mist-like particles. Another necessary assumption is that all the fields of interest like pressure, velocity, density, temperature, etc., are differentiable (i.e. no phase transitions).

The equations are derived from the basic principles of conservation of mass, momentum, and energy. For that matter sometimes it is necessary to consider a finite arbitrary volume, called a control volume, over which these principles can be easily applied. This finite volume is denoted by <math>\Omega</math> and its bounding surface <math>\partial \Omega</math>. The control volume can remain fixed in space or can move with the fluid. This leads, however, to special considerations, as we shall see next.

The substantive derivative

Main article substantive derivative.

Changes in properties of a moving fluid can be measured in two different ways. This will be illustrated through the use of the following example: the measurement of changes in wind velocity in the atmosphere. One can measure its changes with the help of an anemometer in a weather station or by mounting it on a weather balloon. Clearly, the anemometer in the first case is measuring the velocity of all the moving particles passing through a fixed point in space, whereas in the second case the instrument is measuring changes in velocity as it moves with the fluid. The same situation arises in the measuring of changes in density, temperature, etc. Therefore when differentiating one must separate out these two cases. The derivative of a field with respect to fixed position in space is called the spatial or Eulerian derivative. The derivative following a moving particle is called the substantive or Lagrangian derivative.

The substantive derivative is defined as the operator:

<math>\frac{D}{Dt}(\cdot) \equiv \frac{\partial(\cdot)}{\partial t} + (\mathbf{v}\cdot\nabla)(\cdot)</math>

Where <math>\mathbf{v}</math> is the velocity of the fluid. The first term on the right-hand side of the equation is the ordinary Eulerian derivative (i.e. the derivative on a fixed reference frame) whereas the second term represents the changes brought about by the moving fluid. This effect is referred to as advection.

Conservation laws

The Navier-Stokes equations are derived from conservation principles of:

• Mass
• Energy
• Momentum
• Angular momentum

Additionally it is necessary to assume a constitutive relation or state law for the fluid.

In its most general form a conservation law states that the rate of change of an extensive property <math>L</math> defined over a control volume must equal what is lost through the boundaries of the volume carried out by the moving fluid plus what is created/consumed by sources and sinks inside the control volume. This is expressed by the following integral equation:

<math>\frac{d}{dt}\int_{\Omega} L \; d\Omega = -\int_{\partial\Omega} L\mathbf{v\cdot n} d\partial\Omega+ \int_{\Omega} Q d\Omega</math>

Where v is the velocity of the fluid and <math>Q</math> represents the sources and sinks in the fluid.

If the control volume is fixed in space then this integral equation can be expressed as

<math>\frac{d}{d t} \int_{\Omega} L d\Omega = -\int_{\Omega} \nabla\cdot ( L\mathbf{v} ) d\Omega + \int_{\Omega} Q d\Omega </math>

Note that Green's theorem was used in the derivation of this last equation in order to express the first term on the right-hand side in the interior of the control volume. Thus:

<math> \frac{d}{dt}\int_{\Omega} L d\Omega = - \int_{\Omega} (\nabla\cdot ( L\mathbf{v} ) - Q) d\Omega </math>

Instead, as this expression is valid for <math>\Omega</math>, which is invariant in time (unlike <math>\partial\Omega</math>), it becomes possible to swap the "<math>\frac{d}{dt}</math>" and "<math> \int_{\Omega}^{} d\Omega</math>" operators. And as the expression is valid for all domains, we can additionally drop the integral.

Introducing the substantive derivative, we get when <math> Q = 0 </math> (no sources or sinks):

<math>\frac{\partial}{\partial t} L + \nabla\cdot\left(L \mathbf{v} \right) = \frac{D}{Dt}L + L \left(\nabla\cdot \mathbf{v}\right) = 0</math>

For a quantity which isn't space-dependent (so that it doesn't have to be integrated over space), D/Dt gives the right comoving time rate of change.

Equation of continuity

Conservation of mass is written:

<math>\frac{\partial \rho}{\partial t} + \nabla\cdot\left(\rho\mathbf{v}\right)=\frac{D\rho}{D t} + \rho\nabla\cdot\mathbf{v}= 0</math>

where

<math>\rho</math> is the density, and v is the velocity of the fluid.

In the case of an incompressible fluid <math>\rho</math> is not a function of time or space; the equation is reduced to:

<math>\nabla\cdot\mathbf{v} = 0</math>

Conservation of momentum

Conservation of momentum is written:

<math>\frac{\partial}{\partial t}\left(\rho\mathbf{v}\right) + \nabla(\rho\mathbf{v}\otimes\mathbf{v}) = \sum\rho\mathbf{f}</math>

Note that <math>\mathbf{v}\otimes\mathbf{v}</math> is a tensor, the <math>\otimes</math> representing the tensor product.

We can simplify it further, using the continuity equation, this becomes:

<math>\rho\frac{D\mathbf{v}}{D t}=\sum\rho\mathbf{f}</math>

In which we recognise the usual F=ma.

The equations

General form

The form of the equations

The general form of the Navier-Stokes equations for the conservation of momentum is:

<math>\rho\frac{D\mathbf{v}}{D t} = \nabla \cdot\mathbb{P} + \rho\mathbf{f}</math>

where <math>\rho</math> is the fluid density, v is the velocity vector, and f is the body force vector. The tensor <math>\mathbb{P}</math> represents the surface forces applied on a fluid particle (the comoving stress tensor). Unless the fluid is made up of spinning degrees of freedom like vortices, <math>\mathbb{P}</math> is a symmetric tensor. In general, we have the form:

<math>\mathbb{P} = \begin{pmatrix}

\sigma_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz} \end{pmatrix} = - \begin{pmatrix} p&0&0\\ 0&p&0\\ 0&0&p \end{pmatrix} + \begin{pmatrix} \sigma_{xx}+p & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_{yy}+p & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_{zz}+p \end{pmatrix} </math>

where the <math>\sigma</math> are normal stresses, <math>\tau</math> tangential stresses (shear stresses), and p is static pressure, associated with the isotropic part of the stress tensor.

The trace <math>\sigma_{xx}+\sigma_{yy}+\sigma_{zz}</math> is always -3p by definition (unless we have bulk viscosity) regardless of whether or not the fluid is in equilibrium.

Finally, we have:

<math>\rho\frac{D\mathbf{v}}{D t} = -\nabla p + \nabla \cdot\mathbb{T} + \rho\mathbf{f}</math>

where <math>\mathbb{T}</math> is the traceless part of <math>\mathbb{P}</math>.

These equations are still incomplete. To complete them, one must make hypotheses on the form of <math>\mathbb{P}</math>, that is, one needs a constitutive law for the stress tensor as shown below.

The flow is assumed to be differentiable and continuous, allowing the conservation laws to be expressed as partial differential equations. In the case of incompressible flow (constant density), the variables to be solved for are the velocity components and the pressure. The three components of the Navier-Stokes equations plus the conservation of mass (continuity equation) conform a closed system of well-posed partial differential equations for these variables, that can be solved , in principle, for suitable boundary conditions. In the case of compressible flow the density becomes another unknown of the system, and can be determined suplementing the system with an equation of state. An equation of state usually involves the temperature of the fluid, so that the equation for conservation of energy must also be solved, coupled with the previous ones. These equations are non-linear, and analytical solutions in closed form are known only for cases with very simple boundary conditions.

The equations can be converted to Wilkinson equations for the secondary variables vorticity and stream function. Solution depends on the fluid properties (such as viscosity, specific heats, and thermal conductivity), and on the boundary conditions of the domain of study.

Special forms

Those are certain usual simplifications of the problem for which sometimes solutions are known.

Newtonian fluids

Main article Newtonian fluids.

In Newtonian fluids the following assumption holds:

<math>p_{ij}=-p\delta_{ij}+\mu\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}-\frac{2}{3}\delta_{ij}\nabla\cdot\mathbf{v}\right)</math>

where

<math>\mu</math> is the viscosity of the fluid.

<math>\delta_{ij}</math> is the Kronecker delta (1 for i=j; 0 for i <math>\ne</math> j).

To see how to "derive" this, we first note that in equilibrium, p_ij=-pδ_ij. For a Newtonian fluid, the deviation of the comoving stress tensor from this equilibrium value is linear in the gradient of the velocity. It obviously can't depend upon the velocity itself because of Galilean covariance. In other words, p_ij+pδ_ij is linear in <math>\partial_i v_j</math>. The fluids that we are considering here are rotationally invariant (i.e., they are not liquid crystals). p_ij+pδ_ij decomposes into a traceless symmetric tensor and a trace. Similarly, <math>\partial_i v_j</math> decomposes into a traceless symmetric tensor, a trace and an antisymmetric tensor. Any linear map from the latter to the former has to map the antisymmetric part to zero (Schur's lemma) and has two coefficients corresponding to the traceless symmetric part and the trace part. The traceless symmetric part of <math>\partial_i v_j</math> is <math>\partial_i v_j +\partial_j v_i - \frac{2}{d} \delta_{ij}\partial_k v_k</math> where d is the number of spatial dimensions and the trace part is <math>\delta_{ij} \partial_k v_k</math>. Therefore, the most general rotationally invariant linear map is given by

<math>p_{ij}+p\delta_{ij}=\mu\left(\partial_i v_j+\partial_j v_i -\frac{2}{d}\delta_{ij}\nabla\cdot\mathbf{v}\right)+\mu_B \delta_{ij} \nabla\cdot \mathbf{v}</math>

for some coefficients μ and μ_B. μ is called the shear viscosity and μ_B is called the bulk viscosity. It is an empirical observation that the bulk viscosity is negligible for most fluids of interest, which is why it is often dropped. This explains the factor of −2/3 appearing in this equation. This factor has to be modified in 1 or 2 spatial dimensions.

<math>\rho \left(\frac{\partial \mathbf{v}}{\partial t}+\nabla_{\mathbf{v}}\mathbf{v}\right)=\rho \mathbf{f}-\nabla p +\mu\left(\Delta \mathbf{v}+\frac{1}{3}\nabla\left(\nabla\cdot \mathbf{v}\right)\right)</math>

<math>\rho \left(\frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j}\right)=\rho f_i-\frac{\partial p}{\partial x_i}+\mu\left(\frac{\partial ^2 v_i}{\partial x_j \partial x_j}+\frac{1}{3}\frac{\partial ^2 v_j}{\partial x_i \partial x_j}\right)</math>

where we have used the Einstein summation convention.

When written-out in full it becomes clear how complex these equations really are (but only if we insist on writing every single component out explicitly):

Conservation of momentum:

<math>\rho \cdot \left({\partial u \over \partial t}+ u {\partial u \over \partial x}+ v {\partial u \over \partial y}+ w {\partial u \over \partial z}\right) = k_x -{\partial p \over \partial x} + {\partial \over \partial x} \left[ \mu \cdot \left(2 \cdot {\partial u \over \partial x}-\frac{2}{3}\cdot (\nabla \cdot \mathbf{v}) \right) \right] + {\partial \over \partial y}\left[\mu \cdot \left({\partial u \over \partial y} + {\partial v \over \partial x} \right) \right] + {\partial \over \partial z}\left[\mu \cdot \left({\partial w \over \partial x} + {\partial u \over \partial z} \right) \right]</math>

<math> \rho \cdot \left({\partial v \over \partial t}+ u {\partial v \over \partial x}+ v {\partial v \over \partial y}+ w {\partial v \over \partial z}\right) = k_y -{\partial p \over \partial y} + {\partial \over \partial y} \left[ \mu \cdot \left(2 \cdot {\partial v \over \partial y}-\frac{2}{3}\cdot (\nabla \cdot \mathbf{v}) \right) \right] + {\partial \over \partial z}\left[\mu \cdot \left({\partial v \over \partial z} + {\partial w \over \partial y} \right) \right] + {\partial \over \partial x}\left[\mu \cdot \left({\partial u \over \partial y} + {\partial v \over \partial x} \right) \right]</math>

<math> \rho \cdot \left({\partial w \over \partial t}+ u {\partial w \over \partial x}+ v {\partial w \over \partial y}+ w {\partial w \over \partial z}\right) = k_z -{\partial p \over \partial z} + {\partial \over \partial z} \left[ \mu \cdot \left(2 \cdot {\partial w \over \partial z}-\frac{2}{3}\cdot (\nabla \cdot \mathbf{v}) \right) \right] + {\partial \over \partial x}\left[\mu \cdot \left({\partial w \over \partial x} + {\partial u \over \partial z} \right) \right] + {\partial \over \partial y}\left[\mu \cdot \left({\partial v \over \partial z} + {\partial w \over \partial y} \right) \right]</math>

Conservation of mass:

<math> {\partial \rho \over \partial t} + {\partial (\rho \cdot u) \over \partial x}+{\partial (\rho \cdot v) \over \partial y}+{\partial (\rho \cdot w) \over \partial z}=0 </math>

Since density is an unknown another equation is required.

Conservation of energy:

<math> \rho \left({\partial e \over \partial t}+ u {\partial e \over \partial x}+ v {\partial e \over \partial y}+ w {\partial e \over \partial z}\right) = \left( {\partial \over \partial x} \left(\lambda \cdot {\partial T \over \partial x} \right) + {\partial \over \partial y} \left(\lambda \cdot {\partial T \over \partial y} \right) + {\partial \over \partial z} \left(\lambda \cdot {\partial T \over \partial z} \right) \right) - p \cdot \left( \nabla \cdot \mathbf{v} \right) + \mathbf{k} \cdot \mathbf{v} + \rho \cdot \dot{q}_s + \mu \cdot \Phi</math>

Where:

<math>\Phi = 2 \cdot \left[ \left({\partial u \over \partial x} \right)^2+\left({\partial v \over \partial y}\right)^2+\left({\partial w \over \partial z}\right)^2 \right]

+ \left({\partial v \over \partial x}+{\partial u \over \partial y} \right)^2 + \left({\partial w \over \partial y}+{\partial v \over \partial z} \right)^2 + \left({\partial u \over \partial z}+{\partial w \over \partial x} \right)^2 -\frac{2}{3} \cdot \left({\partial u \over \partial x}+{\partial v \over \partial y}+{\partial w \over \partial z} \right)^2</math>

<math>\Phi</math> is sometimes referred to as "viscous dissipation". <math>\Phi</math> can often be neglected unless dealing with extreme flows such as high supersonic and hypersonic flight (e.g., hypersonic planes and atmospheric reentry).

Assuming an ideal gas:

<math>e = c_p \cdot T - \frac{p}{\rho}</math>

The above is a system of six equations and six unknowns (u, v, w, T, e and <math>\rho</math>).

Bingham fluids

Main article Bingham plastic.

In Bingham fluids, we have something slightly different:

<math>\tau_{ij}=\tau_0 + \mu\frac{\partial v_i}{\partial x_j},\;\frac{\partial v_i}{\partial x_j}>0 </math>

Those are fluids capable of bearing some shear before they start flowing. Some common examples are toothpaste and Silly Putty.

Power-law fluid

Main article Power-law fluid.

It is an idealised fluid for which the shear stress, <math>\tau</math>, is given by

<math>\tau = K \left( \frac {\partial u} {\partial y} \right)^n </math>

This form is useful for approximating all sorts of general fluids.

Incompressible fluids

Main article Incompressible fluids.

The Navier-Stokes equations are

<math>

\rho\frac{Du_i}{Dt}=\rho f_i-\frac{\partial p}{\partial x_i}+\frac{\partial}{\partial x_j}\left[ 2\mu\left(e_{ij}-\frac{\Delta\delta_{ij}}{3}\right)\right]</math> for momentum conservation and

<math>\nabla\cdot\mathbf{v}=0</math>

for conservation of mass.

where

<math>\rho</math> is the density,

<math>u_i</math> (<math>i=1,2,3</math>) the three components of velocity,

<math>f_i</math> body forces (such as gravity),

<math>p</math> the pressure,

<math>\mu</math> the dynamic viscosity, of the fluid at a point;

<math>e_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)</math>;

<math>\Delta=e_{ii}</math> is the divergence,

<math>\delta_{ij}</math> is the Kronecker delta.

If <math>\mu</math> is uniform over the fluid, the momentum equation above simplifies to

<math>

\rho\frac{Du_i}{Dt}=\rho f_i-\frac{\partial p}{\partial x_i} +\mu \left(

 \frac{\partial^2u_i}{\partial x_j\partial x_j}+                                                                                                                                         \frac{1}{3}\frac{\partial\Delta}{\partial x_i}\right)

</math>

(if <math>\mu=0</math> but the fluid is compressible, the resulting equations are known as the Euler equations; there, the emphasis is on compressible flow and shock waves).

If now in addition <math>\rho</math> is assumed to be constant we obtain the following system:

<math> \rho \left({\partial v_x \over \partial t}+ v_x {\partial v_x \over \partial x}+ v_y {\partial v_x \over \partial y}+ v_z {\partial v_x \over \partial z}\right)= \mu \left[{\partial^2 v_x \over \partial x^2}+{\partial^2 v_x \over \partial y^2}+{\partial^2 v_x \over \partial z^2}\right]-{\partial p \over \partial x} +\rho g_x</math>

<math> \rho \left({\partial v_y \over \partial t}+ v_x {\partial v_y \over \partial x}+ v_y {\partial v_y \over \partial y}+ v_z {\partial v_y \over \partial z}\right)= \mu \left[{\partial^2 v_y \over \partial x^2}+{\partial^2 v_y \over \partial y^2}+{\partial^2 v_y \over \partial z^2}\right]-{\partial p \over \partial y} +\rho g_y</math>

<math> \rho \left({\partial v_z \over \partial t}+ v_x {\partial v_z \over \partial x}+ v_y {\partial v_z \over \partial y}+ v_z {\partial v_z \over \partial z}\right)= \mu \left[{\partial^2 v_z \over \partial x^2}+{\partial^2 v_z \over \partial y^2}+{\partial^2 v_z \over \partial z^2}\right]-{\partial p \over \partial z} +\rho g_z</math>

Continuity equation (assuming incompressibility):

<math> {\partial v_x \over \partial x}+{\partial v_y \over \partial y}+{\partial v_z \over \partial z}=0 </math>

Simplified version of the N-S equations. Adapted from Incompressible Flow, second edition by Ronald Panton

Note that the Navier-Stokes equations can only describe fluid flow approximately and that, at very small scales or under extreme conditions, real fluids made out of mixtures of discrete molecules and other material, such as suspended particles and dissolved gases, will produce different results from the continuous and homogeneous fluids modelled by the Navier-Stokes equations. Depending on the Knudsen number of the problem, statistical mechanics may be a more appropriate approach. However, the Navier-Stokes equations are useful for a wide range of practical problems, providing their limitations are borne in mind.

See also

• Reynolds transport theorem
• Reynolds number
• Mach number
• Reynolds-averaged Navier-Stokes equations

References

• Inge L. Rhyming Dynamique des fluides, 1991 PPUR
• A.D. Polyanin, A.M. Kutepov, A.V. Vyazmin, and D.A. Kazenin, Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, 2002. ISBN 0-415-27237-8

External links

• Derivation and detail discussion of Navier-Stokes equation
• The Clay Mathematics Institute's Navier-Stokes equation prize
  • Official statement of the problem
• A good derivation of the Navier-Stokes equations
• Derivation of the Navier-Stokes equations
• NASA page on the Navier-Stokes equations
• Navier-Stokes equations (exact solutions) at EqWorld: The World of Mathematical Equationsar:معادلات نافير-ستوك

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