Navier-Stokes equations

In this section we shall present two important equations governing the dynamics of fluid flow. First we obtain the equation describing the conservation of mass. Next we present the equation of motion. In order to do this we have to generalize the concept of viscosity as introduced in the previous section.

Consider a volume element with edges (dx,0,0), (0,dy,0) and (0,0,dz). If we follow the mass in our volume element along the flow during a short time $\tau$, the line element (dx,0,0) will transform into

$$dx\left( 1+\frac{\partial v_{x}}{\partial x}\tau ,\frac{\part... ...artial x}\tau ,\frac{\partial v_{z}}{\partial x}\tau \right) .$$ (5.4)

Analogous equations hold for the other line elements. The volume after time $\tau$ will be

$${dxdydz\left( 1+\frac{\partial v_{x}}{\partial x}\tau ,\frac{ \pa... ... v_{y}}{\partial y}\tau ,\frac{\partial v_{z}}{\partial y}\tau \right) \times }$$

$$\left( \frac{\partial v_{x}}{\partial z}\tau ,\frac{\partial v_{y}}{ \partial z}\tau ,1+\frac{\partial v_{z}}{\partial z}\tau \right)$$

$$dxdydz\left( 1+\frac{\partial v_{x}}{\partial x}\tau ,\frac{\part... ...\partial v_{x}}{\partial y}\tau ,-\frac{\partial v_{x}}{\partial z}\tau \right)$$ =

$$dxdydz\left\{ 1+\left( \frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{\partial v_{z}}{\partial z}\right) \tau \right\}$$ = (5.5)

As a result we have

$$\frac{DV}{Dt}=V\vec{\nabla}\cdot \vec{v}$$ (5.6)

or using $\rho =mN/V$

$$\frac{D\rho }{Dt}=-\rho \vec{\nabla}\cdot \vec{v}.$$ (5.7)

This equation describes the conservation of mass .

We shall now generalize the concept of viscosity as it was given in section 1. Consider a surface element of size dA, and normal $\vec{t}$. Let $\vec{F}$ be the force exerted by the fluid below the surface element on the fluid above the surface element. Then we define the stress tensor  S by

$$F_{i}=-\sum_{j}S_{ij}t_{j}dA=-( \bar{S}\cdot \vec{t})_{i}dA$$ (5.8)

The total force in i-direction on a given volume element at (x,y,z) then is

$$\{S_{ix}(x+dx,y,z)-S_{ix}(x,y,z)\}dydz$$ F_i =

$$+\{S_{iy}(x,y+dy,z)-S_{iy}(x,y,z)\}dxdz$$

$$+\{S_{iz}(x,y,z+dz)-S_{iz}(x,y,z)\}dxdy$$

$$\sum_{j}\frac{\partial }{\partial x_{j}}S_{ij}dxdydz$$ = (5.9)

where x₁=x, x₂=y and x₃=z. In vector notation

$$\vec{F}=V\vec{\nabla}\cdot \bar{S}.$$ (5.10)

Since S will turn out to be symmetric, there is no ambiguity in this notation. This force accelerates the mass in our volume element, which is $\rho V$. Newton's second law then reads

$$\frac{D}{Dt}(\rho V\vec{v})=V\vec{\nabla}\cdot \bar{S}$$ (5.11)

Writing $D/Dt(\rho V\vec{v})=\rho VD/Dt(\vec{v})$, because $\rho V$ is constant along the flow, and dividing by V we get

$$\rho \frac{D}{Dt}\vec{v}=\vec{\nabla}\cdot \bar{S}.$$ (5.12)

This is the Navier-Stokes equation .

Many fluids may be described by assuming that the stress tensor S consists of a part, which is independent of the velocity, and a part which depends linearly on the derivatives $\partial v_{i}/\partial x_{j}$. In hydrodynamics it is shown that the most general stress tensor  having these properties then reads

$$S_{ij}=\eta \left\{ \frac{\partial v_{i}}{\partial x_{j}}+\f... ...ac{2}{3}\eta -\kappa )\vec{\nabla}\cdot \vec{v}\}\delta _{ij}.$$ (5.13)

Here $\eta$ is the shear viscosity , $\kappa$ is the bulk viscosity , and P the pressure .

Many flow fields of interest may be described assuming that the fluid is incompressible , i.e. that the density along the flow is constant. In that case $\vec{\nabla}\cdot \vec{v}=0$, as follows from Eq. (5.7). Assuming moreover that the velocities are small, and that the second order non-linear term $\vec{v}\cdot \vec{\nabla}\vec{v}$ may be neglected we obtain Stokes equations  for incompressible flow

$$\rho \frac{\partial \vec{v}}{\partial t} \eta \nabla ^{2}\vec{v}-\vec{\nabla}P$$ = (5.14)

$$\vec{\nabla}\cdot \vec{v}$$ = 0. (5.15)

These are the equations that we are going to use in the next two sections.