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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 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, such as 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.

Basic assumptionsEdit

Before going into the details of the Navier-Stokes equations, first, it is 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 \Omega and its bounding surface \partial \Omega. 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 derivativeEdit

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:

\frac{D}{Dt}(\cdot) = \frac{\partial(\cdot)}{\partial t} + (\mathbf{v}\cdot\nabla)(\cdot)

Where \mathbf{v} 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 is effect is referred as advection.

Conservation lawsEdit

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

Aditionally it is necesary 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 L 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:

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

Where v is the velocity of the fluid and Q represents the sources and sinks in the fluid.

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

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

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:

 \frac{d}{dt}\int_{\Omega} L d\Omega = - \int_{\Omega} (\nabla\cdot ( L\mathbf{v} ) + Q d\Omega )

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

Introducing the substantive derivative, we get:

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

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

Equation of continuityEdit

Conservation of mass is written:

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

where

\rho is the density of the fluid.

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

\nabla\cdot\mathbf{v} = 0

Conservation of momentumEdit

Conservation of momentum is written:

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

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

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

\rho\frac{D\mathbf{v}}{D t}=\sum\rho\mathbf{f}

In which we recognise the usual F=ma.

The equationsEdit

General formEdit

The form of the equationsEdit

The general form of the Navier-Stokes equations is:

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

For the conservation of momentum. The tensor \mathbb{P} 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, \mathbb{P} is a symmetric tensor. In general, we have the form:

\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}

Where the \sigma are normal constraints and \tau tangential constraints.

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


To which we add the continuity equation:

\frac{D\rho}{Dt} + \rho\nabla\cdot\mathbf{v} = 0

where

p is the pressure

finally, we have:

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

where \mathbb{T} is the traceless part of \mathbb{P}.

The closure problemEdit

These equations are incomplete. To complete them, one must make hypotheses on the form of \mathbb{P}. In the case of a perfect fluid \tau components are nil, for example. Those equations used to complete the set are equations of state.

For example, the pressure can be function of, notably, density and temperature.

The variables to be solved for are the velocity components, the fluid density, static pressure, and temperature. The flow is assumed to be differentiable and continuous, allowing these balances to be expressed as partial differential equations. 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.

The components of \mathbb{P} are the constraints on an infinitesimal element of fluid. They represent the normal and shear constraints. \mathbb{P} is symmetric unless there is a nonzero spin density.

So-called non-Newtonian fluids are simply fluids where this tensor has no special properties allowing for special solutions of the equations.

Special formsEdit

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

Newtonian fluidsEdit

Main article Newtonian fluids.

In Newtonian fluids the following assumption holds:

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)

where

\mu is the viscosity of the fluid.

To see how to "derive" this, we first note that in equilibrium, pij=-pδij. 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, pij+pδij is linear in \partial_i v_j. The fluids that we are considering here are rotationally invariant (i.e., they are not liquid crystals). pij+pδij decomposes into a traceless symmetric tensor and a trace. Similarly, \partial_i v_j 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 \partial_i v_j is \partial_i v_j +\partial_j v_i - \frac{2}{d} \delta_{ij}\partial_k v_k where d is the number of spatial dimensions and the trace part is \delta_{ij} \partial_k v_k. Therefore, the most general rotationally invariant linear map is given by

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}

for some coefficients μ and μB. μ is called the shear viscosity and μB 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.

\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)
\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)

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:

\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]
 \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]
 \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]

Conservation of mass:

 {\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

Since density is an unknown another equation is required.

Conservation of energy:

 \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

Where:

\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

\Phi is sometimes referred to as "viscous dissipation". \Phi 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:

e = c_p \cdot T - \frac{p}{\rho}

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

Bingham fluidsEdit

Main article Bingham plastic.

In Bingham fluids, we have something slightly different:

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

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

Power-law fluidEdit

Main article Power-law fluid.

It is an idealised fluid for which the shear stress, \tau, is given by

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

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

Incompressible fluidsEdit

Main article Incompressible fluids.

The Navier-Stokes equations are


\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]

for momentum conservation and

\nabla\cdot\mathbf{v}=0

for conservation of mass.

where

\rho is the density,
u_i (i=1,2,3) the three components of velocity,
f_i body forces (such as gravity),
p the pressure,
\mu the dynamic viscosity, of the fluid at a point;
e_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right);
\Delta=e_{ii} is the divergence,
\delta_{ij} is the Kronecker delta.

If \mu is uniform over the fluid, the momentum equation above simplifies to


\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)

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

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

 \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
 \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
 \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

Continuity equation (assuming incompressibility):

 {\partial v_x \over \partial x}+{\partial v_y \over \partial y}+{\partial v_z \over \partial z}=0
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 Edit

References Edit

  • 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 Edit


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