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# Introduction

Hydrodynamics  describes the flow properties of viscoelastic  materials. The basic quantities are the density and velocity . Here is the density of the material which at time t happens to be at position . Similarly is the velocity of the material which at time t is at position . A consequence of describing the flow field  this way, is that for example and 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 = = (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 and to the surface dxdy, with constant of proportionality . 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 = = (5.2)

The constant 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 = (5.3)

These forces together will accelerate the mass in our volume element.      Next: Navier-Stokes equations Up: Hydrodynamics Previous: Hydrodynamics
W.J. Briels