Flow between plates set at a small angle when one moves
We need some relatively simple physics to help us to understand bearings. The model is for two plane surfaces, one fixed and the other moving at some constant speed , that make a small angle . There must be some physical restraint for the fixed surface because it cannot extend to the point where the surfaces meet. We also need a constraint at the open end. The result looks like a block with a tapering gap making an angle to a second moving surface in figure 16-12. The diagram is grossly out of scale because the angle is too large. A practical angle will be in the region of one second of arc or a gap of less than one tenth of a millimetre.
I have let the fixed surface have a length and be inclined at . This surface can be at any height above the moving surface and, if we extend the fixed surface to cut the moving surface at point the intersection will be at some distance from the leading edge of the fixed surface. We now have the two surfaces located relative to each other by three dimensions, .
What we need to find out is how the pressure varies with distance along the fixed surface and we see immediately that, if the surface is of finite width, there will be leakage of lubricant from the sides. We cannot handle that so we must simplify and consider only two-dimensional flow that might well prevail along the centre line. If we accept that, we have to decide what to do about the velocity distribution between the two surfaces. We have already done this for parallel plates and we found that the velocity distribution can be found from :-
Now we have converging surfaces and we have to deal with a pressure that will rise from zero at the leading edge of the block and fall to zero again at the trailing edge. We are not looking for the velocity distribution but the variation of pressure with distance .
The best solution comes from defining the distance between the plates at distance from the leading edge of the fixed plate and supposing that the velocity distribution is given by the equation for parallel plates. Then if we imagine the pressure changes to take place in increments it turns out that it is best to use not the equation for velocity but the equation for volume flow:-
to find the incremental change in pressure.
We can write to give flow per unit width and then :-
which can be changed to :- where is the pressure drop in distance at distance from the leading edge. Then :-
In this case is a function of and given the geometry in figure 15- we can say that :-
. Then :-
.
This can be rearranged for integration to give :-
and .
So
Now we have to decide what to do with this equation. We know that the pressures at are both equal to the absolute pressure of the surroundings. So we have two conditions that will let us find a value for . However we must also recognize that is fixed if are fixed. This means that we could use the two pressure conditions to find and then eliminate . This is hideously long but it yields :-
We need a better mental picture of the behaviour of the lubricant between the surfaces and all we have is this equation. What we are talking about is lubricant being dragged by viscous forces between two plates that might be less than 0.1 mm apart at the entry and closer at exit. This is a thin film with a thickness about equal to that of 80 gram copying paper and certainly thicker than the films that I found in my simple testing.
It seems to me that the important thing to find out is how the pressure varies between inlet to the plates and exit. We can find that from a plot of pressure against . The expression for pressure shows that the pressures throughout will be proportional to the value of the coefficient of viscosity and that the pressures will increase directly with the speed of the moving plate. There will be no such simple relationship for the remaining variables where is the gap at inlet. These three are linked by the geometry and we must think about how the geometry can be exploited in preparation for an exploration using Mathcad.
We
must have some way to tackle this because we shall want to know how the
pressure distribution varies with the angle for a given value of at inlet and how it varies with for a given value of angle .
I have said that the geometry of this arrangement of a surface at a fixed angle
and a second moving surface can be defined from a knowledge first of the angle ,
then of the gap at the leading edge and then of the length of the fixed surface.
If distance is defined from figure 16-12 it follows that or the expression for pressure will become
invalid if .
So we need combinations of and that give .
It is usual an engineering to explore relationships between the quantities in equations by plotting graphs of the relationship between any two with all the other quantities constant. Here all we can do is find the pressure distribution for a fixed value of with varying and vice versa.
Figure 16-13 gives the programme for drawing the graphs in graph 16-6 and 16-7. I simply chose likely values for viscosity, velocity and length and fiddled out suitable values for the angle and the gap at inlet. They are given in figure 16-13 where I also give the expression reworked in terms of and instead of .
The information for graphs 16-6 and 16-7 is given in the legends on the graphs. I have used the same vertical scale in both graphs just to show that changing the inlet gap for a given angle gives much greater pressures than changing the angle for a given inlet gap. It is also evident that in both graphs the peak values move towards the exit as the pressures rise.
The gaps and angles are so small that in any practical application it is most unlikely they would be designed quantitatively. For the important applications of this physics we must exploit the fact that one surface is free to move relative to the other to find a geometry that gives equilibrium operation.