Dynamic
response from DPZTFFC device

by
Navier-Stokes equation exact solution

Boundary layer theory was first developed by Prandtl. He
showed that for a moving fluid all friction losses occur within a thin layer
adjacent to a solid boundary (called the boundary layer) and that flow outside
this layer can be considered frictionless. The velocity near the boundary is
affected by boundary shear. In general, the boundary layer is very thin at the
upstream boundaries of an immersed object but increases in thickness due to the
continual action of shear stress.

At low Reynolds numbers, the entire boundary layer is governed
by viscous forces, and laminar flow occurs therein. For high Reynolds numbers,
the entire boundary layer is turbulent.

Flow described by Navier-Stokes equations. In general the
problem of finding exact solutions of the NS equations presents insurmountable
mathematical difficulties. This is primarily a consequence of their being
non-linear.

Nevertheless, it is possible to find exact solutions in
certain particular cases.

In case of small viscosity many of the exact solutions have a
boundary-layer structure which means that the influence of viscosity is confined
to thin layer near the wall.

It can be seen that the thickness of the boundary layer (d
) will increase as the square root of character dimension x increases and also as the
square root of the kinematic viscosity increases, while d
will decrease as the square root of the velocity increases. Similarly, the
boundary shear t0 will increase as r and m increase,
will decrease as the square root of x
increases, and will increase as the square of V increases.

Let's discuss the parallel flows, constitute a particularly
simple class of motions. A flow is called parallel if only one velocity
component is a different from zero and all fluid particles are moving in the one
direction.

For irrotational, incompressible flow with F=0 , the Navier-Stokes
equation then simplifies to

For low Reynolds number, t he inertia term is smaller than the
viscous term and can therefore be ignored, leaving the equation of creeping
motion:

In creeping motion regime, viscous interactions have an
influence over large distances from an obstacle.

For low Reynolds number, flow at low pressure, the
Navier-Stokes equation becomes a diffusion equation:

This type of flow would be prevail flow inside channels of
DPZTFFC device and we must discuss it first.

Equation (3) with the boundary conditions u=0 for y=+/-b
describe steady flow in channel between two parallel plates.

And solution for this case is parabola.

Another simple solution of equation (3) is obtained for
so-called Couette flow between two parallel flat walls, one of which is a rest
the another moving in its own plane with constant velocity.

We will investigate another application for equation (3) - the
flow near oscillating flat plate - the Stokes's second
problem.

Case
1.

Consider the laminar flow of incompressible liquid near an
oscillatory plate.

This is a speed of plate oscillation (at the wall /plate y=0), with U - amplitude of speed and
w -
frequency.

The second B.C. is u(oo,t)=0

The solution is:

where

and

- is kinematic viscosity

Let's compute this equation for two different frequencies 5
and 50 rad/sec:

The velocity profile has the form of a damped harmonic
oscillation in which a fluid layer at a distance y has a phase lag

ky
with respect to the motion of the wall.

The layer which is a carried by the wall has a thickness
proportional to: and decreases for decreasing kinematic viscosity and increasing
frequency.

Case
2

Consider now more complicate problem: a plane laminar flow of
incompressible viscous fluid between two parallel plates separated by
distance 2*b. The
bottom plate moves, oscillates with velocity u(0,t)=Ucos(w.t) , while the upper plate
is held stationary, so that B.C. is u(2b,t)=0 . Initial
condition: u(y,0)=Uo*(1-y/2b).

Will solve the same diffusion equation (3) with different B.C.
and I.C.

In this case the solution is:

This is an attempt to solve same problem,

same case but different way.

Developed by
Roman N Tunkel,

Research
Engineer, Ph.D.,

tunkelrn@utrc.utc.com