This is interesting because we have, in effect, a combination of Newton’s conceptual model with which he defined his coefficient of viscosity and steady viscous flow. In figures 6.5a, b, and c, I bring the diagram from Chapter 6 that shows this model. It is not practical as I said in Chapter 6 but it shows us that the flow between parallel plates where one is moving can either be regarded as a case of Newton’s model with a flow superimposed on it or as a flow with Newton’s model superimposed on it.
In this treatment of the system we already have an expression for the flow between fixed plates.
The derivation above for stationary parallel plates can be adapted for the case of parallel plates when one moves at a constant speed.
We can go back to this equation:- and note that now when , and, when , where is the speed of the moving plate and is supposed to be in the same direction as . Then as before but now :-
Substituting gives from which:-
and finally :-
This is clearly the first term is the former equation for a parabolic distribution and the second term is a new term that adds or subtracts a proportion of the velocity in the ratio as in Newton’s model. It gives a skewed velocity profile. In graph 16-2 I have shown the velocity distribution for the same figures as I used for the previous graph. The water behaves as if the new velocity is the sum of the velocity for fixed plates with a linear proportion of the speed of the upper plate added.
It is possible for the upper plate to move in the direction of the flow or against the flow and a graph can be drawn to show the result It is graph 16-3 where I have let take values from by increments of 0.25 m/s. The skewing is progressive and the maximum velocity moves towards the moving plate.
It is clear that, for the same pressure gradient, the flow will be reduced as the moving plate changes from moving with the flow to moving against it.
It is possible to find the flow.
This is an interesting expression because can be negative. This means that despite the pressure difference acting on the flow the net flow can be zero or indeed be in the opposite direction to the pressure difference.
We can find the condition for the flow to be zero. We can write :-
If this is evaluated for water flowing between plates, as in my example above, zero flow will occur when and the velocity distribution for can be plotted as in graph 16-4.
The outcome is intriguing because we now have, in a distance of 1 mm, flow in one direction near to the stationary plate and a flow in the opposite direction near to the moving plate. The cross-hatched areas must be equal.
However there is no reason why the value of could not be much greater so that the flow in the direction of the pressure difference becomes very small.
My figures are for water and a very wide gap by the standards of lubrication between the plates but the viscosity of air is lower than that of water by a factor of abut 50 and the viscosity of oil can be several thousand times that of water. All sorts of combinations of velocity and gap will be needed in practice for various applications. The worked examples at the end of this chapter will give some idea of magnitudes.