next up previous contents index
Next: Navier-Stokes equations Up: Hydrodynamics Previous: Hydrodynamics

Introduction

Hydrodynamics  describes the flow properties of viscoelastic  materials. The basic quantities are the density $\rho (
\vec{r},t)$ and velocity $\vec{v}(\vec{r},t)$. Here $\rho (
\vec{r},t)$ is the density of the material which at time t happens to be at position $
\vec{r}$. Similarly $\vec{v}(\vec{r},t)$ is the velocity of the material which at time t is at position $
\vec{r}$. A consequence of describing the flow field  this way, is that for example $\rho (
\vec{r},t_{1})$ and $\rho (\vec{r},t_{2})$ are the densities of two different amounts of material. If we want to know the change with time of the density of a given amount of material we shall write

$\displaystyle \frac{D}{Dt}\rho (\vec{r},t)$ = $\displaystyle \lim_{\tau \rightarrow 0}\frac{1}{\tau }
\{\rho (\vec{r}+\vec{v}(\vec{r},t)\tau ,t+\tau )-\rho (\vec{r},t)\}$  
  = $\displaystyle \left\{ \vec{v}(\vec{r},t)\cdot \vec{\nabla}+\frac{\partial }{\partial t}
\right\} \rho (\vec{r},t)$ (5.1)

Similar expressions hold for other quantities. D/Dt is called the total derivative .

A second important concept in hydrodynamics is that of friction . Suppose we have a linear flow field like in Fig. (5.1), where the arrows indicate the velocities. The only nonzero velocity component is in the y-direction, and it only depends on z. Now look at a volume element at (x,y,z). It has a velocity vy(z) which is larger than the velocity vy(z-dz) of the volume element just below it. As a result the element at (x,y,z) will be slowed down and the one at (x,y,z-dz) will be accelerated. Both forces will be proportional to $
\partial v_{y}(z)/\partial z$ and to the surface dxdy, with constant of proportionality $\eta$. Similarly the volume element at (x,y,z) will be accelerated by the one at (x,y,z+dz). In total our volume element will feel a force in the y-direction given by

Fy = $\displaystyle \eta \left\{
\frac{\partial }{\partial z}v_{y}(z+dz)-\frac{\partial }{\partial z}
v_{y}(z)\right\} dxdy$  
  = $\displaystyle \eta \frac{\partial ^{2}v_{y}}{\partial z^{2}}dxdydz.$ (5.2)

The constant $\eta$ is called the viscosity . A second contribution to the force in the y-direction comes from the difference in pressure  on the surfaces at (x,y,z) and (x,y+dy,z):
FyP = P(y)dxdz-P(y+dy)dxdz  
  = $\displaystyle -
\frac{\partial P}{\partial y}dxdydz$ (5.3)

These forces together will accelerate the mass in our volume element.


  
Figure 5.1:
\begin{figure}
\setlength{\unitlength}{1mm}
\begin{picture}
(120,70)(0,10)
%lij...
...{15}}
\put(85,48){\vector(1,0){25}}
\put(75,35){(y,z)}
\end{picture}\end{figure}


next up previous contents index
Next: Navier-Stokes equations Up: Hydrodynamics Previous: Hydrodynamics
W.J. Briels