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The intent of this article is to highlight the important points of the derivation of the Navier-Stokes equations as well as the application and formulation for different families of fluids.

Contents

Basic assumptions

The convective derivative

Conservation laws

Conservation of mass

Conservation of momentum

General form of the Navier-Stokes equations

Application to different fluids

Newtonian fluid

Bingham fluid

Power-law fluid

Basic assumptions

The Navier-Stokes equations are based on the assumption that the fluid, at the scale of interest, is a continuum, in other words is not made up of discrete particles but rather a continuous substance. Another necessary assumption is that all the of fields of interest like pressure, velocity, density, temperature and so on are differentiable (for example, no phase transitions) and continuous.
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 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.

The convective derivative

Main articles: convective derivative

Changes in properties of a moving fluid can be measured in two different ways. One can measure a given property by either carrying out the measurement on a fixed point in space as particles of the fluid pass by, or by following a parcel of fluid along its streamline. The derivative of a field with respect to a fixed position in space is called the ''spatial'' derivative while the derivative following a moving parcel is called the ''convective'' derivative.
The convective derivative is defined as the operator:
:
abla (star)
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, representing changes at a point with respect to time) whereas the second term represents changes of a quantity with respect to position (advection). This "special" derivative is in reality the ordinary derivative of a function of many variables and can be derived easily through application of the chain rule.
For example, the measurement of changes in wind velocity in the atmosphere can be obtained with the help of an anemometer in a weather station or by mounting it on a weather balloon. 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.

Conservation laws

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

★ Mass

★ Momentum

★ Energy
This is done via the Reynolds transport theorem, an integral relation stating that the changes of some intensive property (call it $L$) defined over a control volume must be equal to what is lost (or gained) through the boundaries of the volume plus what is created/consumed by sources and sinks inside the control volume. This is expressed by the following integral equation:
:
where 'v' is the velocity of the fluid and $Q$ represents the sources and sinks in the fluid. Recall that $Omega$ represents the control volume and $partial\; Omega$ its bounding surface.
The divergence theorem may be applied to the surface integral, changing it into a volume integral:
:
abla cdot ( Lmathbf{v}) dOmega + int_{Omega} Q dOmega
Applying Leibniz's rule to the integral on the left and then combining all of the integrals:
:
abla cdot (Lmathbf{v}) dOmega + int_{Omega} Q dOmega
qquad Rightarrow qquad
int_{Omega} left( rac{partial L}{partial t} +
abla cdot (Lmathbf{v}) + Q
ight) dOmega = 0
The integral must be zero for 'any' control volume; this can only be true if the integrand itself is zero, so that:
:
abla cdot (Lmathbf{v}) + Q = 0
From this valuable relation, three important concepts may be concisely written: conservation of mass, conservation of momentum, and conservation of energy.

Conservation of mass

Mass may be considered with the conservation relation above. Taking Q = 0 (no sources or sinks of mass) and putting in density:
:
ho}{partial t} +
abla cdot (
ho mathbf{v}) = 0
where $$
ho is the mass density (mass per unit volume), and v is the velocity of the fluid. This equation is called the 'continuity equation'.
In the case of an incompressible fluid, $$
ho is a constant and the equation reduces to:
:$$
ablacdotmathbf{v} = 0
which is in fact a statement of the conservation of volume.
The continuity equation isn't generally considered to be one of the Navier-Stokes equations even though it always accompanies them.

Conservation of momentum

The most elemental form of the Navier-Stokes equations is obtained when the conservation relation is applied to momentum. Writing momentum as $$
ho v_i gives:
:
ho v_i) +
abla cdot (
ho v_i mathbf{v}) + Q_i = 0
The index $i$ indicates that the above equation is applied to each component of velocity (generally three of them). Noting that a body force (notated $b$) is a source or sink of momentum and expanding the derivatives completely:
:
ho}{partial t} v_i +
ho rac{partial v_i}{partial t} +
abla(
ho v_i) cdot mathbf{v} +
ho v_i
abla cdot mathbf{v} = b_i
:
ho}{partial t} v_i +
ho rac{partial v_i}{partial t} +
abla(
ho) v_i cdot mathbf{v} +
ho
abla(v_i) cdot mathbf{v} + v_i
ho
abla cdot mathbf{v} = b_i
:
ho}{partial t} +
ho rac{partial v_i}{partial t} + v_i mathbf{v} cdot
abla
ho +
ho mathbf{v} cdot
abla v_i +
ho v_i
abla cdot mathbf{v} = b_i
Rearranging and recognizing that $mathbf\{v\}\; cdot$
abla
ho +
ho
abla cdot mathbf{v} =
abla cdot (
ho mathbf{v}):
:
ho}{partial t} + mathbf{v} cdot
abla
ho +
ho
abla cdot mathbf{v}
ight) +
ho left(rac{partial v_i}{partial t} + mathbf{v} cdot
abla v_i
ight) = b_i
:
ho}{partial t} +
abla cdot (
ho mathbf{v})
ight) +
ho left(rac{partial v_i}{partial t} + mathbf{v} cdot
abla v_i
ight) = b_i
The leftmost expression enclosed in parenthesis is, by the continuity equation, equal to zero. Noting that what remains on the left side of the equation is the convective derivative and writing the equation as a vector equation yields:
:$$
ho left(rac{partial mathbf{v}}{partial t} + mathbf{v} cdot
abla mathbf{v}
ight) = mathbf{b}
qquad Rightarrow qquad
horac{D mathbf{v}}{D t} = mathbf{b}
This is simply an expression of Newton's second law ('F' = m'a') in terms of body forces instead of point forces. Each term in any case of the Navier-Stokes equations is a body force. A shorter though less rigorous way to arrive at this result would be the application of the chain rule to acceleration:
:$$
ho rac{d}{d t}(mathbf{v}(x, y, z, t)) = mathbf{b}
qquad Rightarrow qquad
ho left(
rac{partial mathbf{v}}{partial t} +
rac{partial mathbf{v}}{partial x}rac{d x}{d t} +
rac{partial mathbf{v}}{partial y}rac{d y}{d t} +
rac{partial mathbf{v}}{partial z}rac{d z}{d t}
ight) = mathbf{b} qquad Rightarrow
:$$
ho left(
rac{partial mathbf{v}}{partial t} +
u rac{partial mathbf{v}}{partial x} +
v rac{partial mathbf{v}}{partial y} +
w rac{partial mathbf{v}}{partial z}
ight) = mathbf{b}
qquad Rightarrow qquad
ho left(rac{partial mathbf{v}}{partial t} + mathbf{v} cdot
abla mathbf{v}
ight) = mathbf{b}
where $mathbf\{v\}\; =\; (u,\; v,\; w)$.

General form of the Navier-Stokes equations

The generic body force $mathbf\{b\}$ seen previously is made specific first by breaking it up into two new terms, $$
abla cdot sigma_{ij} and 'f'. The tensor $sigma\_\{ij\}$ represents the stresses inside the fluid (its divergence is a vector) while the vector 'f' represents "other" body forces such as gravity.
:$$
horac{Dmathbf{v}}{D t} =
abla cdot sigma_{ij} + mathbf{f}
$sigma\_\{ij\}$ is a symmetric tensor given by:
:
sigma_{xx} & au_{xy} & au_{xz} \
au_{yx} & sigma_{yy} & au_{yz} \
au_{zx} & au_{zy} & sigma_{zz}
end{pmatrix}
where the $sigma$ are normal stresses and $au$ shear stresses. Since pressure is typically a variable of interest, this tensor is split up into terms:
:
sigma_{xx} & au_{xy} & au_{xz} \
au_{yx} & sigma_{yy} & au_{yz} \
au_{zx} & au_{zy} & sigma_{zz}
end{pmatrix}
=
-egin{pmatrix}
p&0&0\
0&p&0\
0&0&p
end{pmatrix}
+
egin{pmatrix}
sigma_{xx}+p & au_{xy} & au_{xz} \
au_{yx} & sigma_{yy}+p & au_{yz} \
au_{zx} & au_{zy} & sigma_{zz}+p
end{pmatrix}

The Navier-Stokes equation may now be written in the most general form:
:$$
horac{Dmathbf{v}}{D t} = -
abla p +
abla cdotmathbb{T} + mathbf{f}
These equations are still incomplete. To complete them, one must make hypotheses on the form of $mathbb\{T\}$, that is, one needs a constitutive law for the stress tensor which can be obtained for specific fluid families; additionally, if the flow is assumed compressible an equation of state will be required, which will likely further require a conservation of energy formulation. In almost all cases the momentum equations are nonlinear.

Application to different fluids

The most general form of the Navier-Stokes equations is not "ready for use", the stress tensors contain too many unkowns so that more information is needed; this information is normally some knowledge of the viscous behavior of the fluid.

Newtonian fluid

Main articles: Newtonian fluid

The formulation for Newtonian fluids stems from an observation made by Newton that, for most fluids,
:
In order to apply this to the Navier-Stokes equations, three assumptions were made by Stokes:
:
★ The stress tensor is a linear function of the strain rates.
:
★ The fluid is isotropic.
:
★ For a fluid at rest, $$
abla cdot mathbb{T} must be zero (so that hydrostatic pressure results).
Applying these assumptions will lead to:
:
ight) + delta_{ij} lambda
abla cdot mathbf{v}
$delta\_\{ij\}$ is the Kronecker delta and $lambda$ is the second coefficient of viscosity (related to bulk viscosity), a viscous effect associated with volume change. The value of this parameter is very difficult to determine, not even its sign is known. Even in compressible flows, the term involving $lambda$ is often negligible; however it can occasionally be important even in nearly incompressible flows and is a matter of controversy. When taken nonzero, the most common approximation is .
A straightforward substitution of $mathbb\{T\}\_\{ij\}$ into the momentum conservation equation will yield the 'Navier-Stokes equations for a compressible Newtonian fluid':
:$$
ho left(rac{partial u}{partial t} + u rac{partial u}{partial x} + v rac{partial u}{partial y} + w rac{partial u}{partial z}
ight) = -rac{partial p}{partial x} +
rac{partial}{partial x}left(2 mu rac{partial u}{partial x} + lambda
abla cdot mathbf{v}
ight) +
rac{partial}{partial y}left(muleft(rac{partial u}{partial y} + rac{partial v}{partial x}
ight)
ight) +
rac{partial}{partial z}left(muleft(rac{partial u}{partial z} + rac{partial w}{partial x}
ight)
ight) +
ho g_x
:$$
ho left(rac{partial v}{partial t} + u rac{partial v}{partial x} + v rac{partial v}{partial y}+ w rac{partial v}{partial z}
ight) = -rac{partial p}{partial y} +
rac{partial}{partial x}left(muleft(rac{partial v}{partial x} + rac{partial u}{partial y}
ight)
ight) +
rac{partial}{partial y}left(2 mu rac{partial v}{partial y} + lambda
abla cdot mathbf{v}
ight) +
rac{partial}{partial z}left(muleft(rac{partial v}{partial z} + rac{partial w}{partial y}
ight)
ight) +
ho g_y
:$$
ho left(rac{partial w}{partial t} + u rac{partial w}{partial x} + v rac{partial w}{partial y}+ w rac{partial w}{partial z}
ight) = -rac{partial p}{partial z} +
rac{partial}{partial x}left(muleft(rac{partial w}{partial x} + rac{partial u}{partial z}
ight)
ight) +
rac{partial}{partial y}left(muleft(rac{partial w}{partial y} + rac{partial v}{partial z}
ight)
ight) +
rac{partial}{partial z}left(2 mu rac{partial w}{partial z} + lambda
abla cdot mathbf{v}
ight) +
ho g_z
Gravity has been accounted for as "the" body force, ie $mathbf\{f\}\; =$
ho mathbf{g}. The associated continuity equation is:
:
ho}{partial t} +
abla cdot (
ho mathbf{v}) = 0
In addition to the continuity equation, an equation of state and an equation for the conservation of energy is needed. The equation of state to use depends on context (often the ideal gas law), the conservation of energy will read:
:$$
ho rac{D h}{D t} = rac{D p}{D t} +
abla cdot (k
abla T) + Phi
Here, $h$ is the enthalpy, $T$ is the temperature, and $Phi$ is a function representing the dissipation of energy due to viscous effects:
:
ight)^2 + 2left(rac{partial v}{partial y}
ight) + 2left(rac{partial w}{partial z}
ight)^2 + left(rac{partial v}{partial x} + rac{partial u}{partial y}
ight)^2 + left(rac{partial w}{partial y} + rac{partial v}{partial z}
ight)^2 + left(rac{partial u}{partial z} + rac{partial w}{partial x}
ight)^2
ight) + lambda (
abla cdot mathbf{v})^2
With a good equation of state and good functions for the dependence of parameters (such as viscosity) on the variables, this system of equations seems to properly model the dynamics of all known gases and most liquids.
For the special but very common case of incompressible flow, the momentum equations simplify significantly. For example, looking at the viscous terms of the ''x'' momentum equation (note that viscosity will now be a constant and the second viscosity effect will be zero):
:
&rac{partial}{partial x}left(2 mu rac{partial u}{partial x} + lambda
abla cdot mathbf{v}
ight) +
rac{partial}{partial y}left(muleft(rac{partial u}{partial y} + rac{partial v}{partial x}
ight)
ight) +
rac{partial}{partial z}left(muleft(rac{partial u}{partial z} + rac{partial w}{partial x}
ight)
ight) \ \
& =
2 mu rac{partial^2 u}{partial x^2} +
mu rac{partial^2 u}{partial y^2} + mu rac{partial^2 v}{partial y , partial x} +
mu rac{partial^2 u}{partial z^2} + mu rac{partial^2 w}{partial z , partial x} \ \
& =
mu rac{partial^2 u}{partial x^2} +
mu rac{partial^2 u}{partial y^2} +
mu rac{partial^2 u}{partial z^2} +
mu rac{partial^2 u}{partial x^2} + mu rac{partial^2 v}{partial y , partial x} + mu rac{partial^2 w}{partial z , partial x} \ \
& = mu
abla^2 u + mu rac{partial}{partial x} left(rac{partial u}{partial x} + rac{partial v}{partial y} + rac{partial w}{partial z}
ight) = mu
abla^2 u
end{align}

Bingham fluid

Main articles: Bingham plastic

In Bingham fluids, the situation is slightly different:
:$$
rac {partial u} {partial y} = left{
egin{matrix}
0 &, quad au < au_0 \
( au - au_0)/ {mu} &, quad au ge au_0
end{matrix}
ight.
These are fluids capable of bearing some shear before they start flowing. Some common examples are toothpaste and clay.

Power-law fluid

Main articles: Power-law fluid

A power law fluid is an idealised fluid for which the shear stress, $au$, is given by
:
ight)^n
This form is useful for approximating all sorts of general fluids, including shear thinning (such as latex paint) and shear thickening (such as corn starch water mixture).