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

$$\frac{D}{Dt}\rho (\vec{r},t) \lim_{\tau \rightarrow 0}\frac{1}{\tau } \{\rho (\vec{r}+\vec{v}(\vec{r},t)\tau ,t+\tau )-\rho (\vec{r},t)\}$$ =

$$\left\{ \vec{v}(\vec{r},t)\cdot \vec{\nabla}+\frac{\partial }{\partial t} \right\} \rho (\vec{r},t)$$ = (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 v_y(z) which is larger than the velocity v_y(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

$$\eta \left\{ \frac{\partial }{\partial z}v_{y}(z+dz)-\frac{\partial }{\partial z} v_{y}(z)\right\} dxdy$$ F_y =

$$\eta \frac{\partial ^{2}v_{y}}{\partial z^{2}}dxdydz.$$ = (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):

F_y^P = P(y)dxdz-P(y+dy)dxdz

$$- \frac{\partial P}{\partial y}dxdydz$$ = (5.3)

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

  

Figure 5.1: