Flow between parallel plates

Figure 16-4 is a drawing of a fitting that could be made and fitted to the side of a tank so that liquid could flow through the slit formed between two flat plates and two spacers. The resulting duct has a width of  and a length of . The rate of flow will depend on the pressure drop between inlet and outlet. We know enough about flow of this sort to know that if  is large compared to  the effects of the sides on the flow is likely to be very small and, if  is long when compared with  the effect of the entry to the duct will be small.

 

Further, even for water with its low viscosity, the flow will be laminar if  is of the order of 1 mm.

 

So, if we treat the flow through this slit as steady and one dimensional the outcome is likely to be quite accurate.

 

Figure 16-5 represents an element of the flow moving from left to right though a slit made up of the two parallel plates of width  and length  set at a distance  apart. As always the velocity of the fluid in contact with the plates is zero and, as there is a flow and it is laminar, the velocity must increase in some orderly way to a maximum in the middle. Within this flow we can consider an element of thickness , of width  and at a distance  from the lower face.

 

As the flow is steady this element must be in equilibrium under the forces caused by the pressure difference and the shearing forces due to viscosity. Over the distance  the pressure changes from  to . Of course the pressure will fall but this gives a direction to the mathematics.

 

Then the force on the element in the direction of motion is given by :-

                                    .

The shearing force on one face of the element is given by Newton’s expression :-

 where  is the coefficient of viscosity of the fluid,  is the effective area and  is the velocity gradient.

 

Later, I want to consider the possibility that the flow is between two surfaces when one of the surfaces is moving as might be the case in a dashpot. As a result I need to deal with the velocity gradient as a general case. In figure 16-3 I have shown the lower face and a curve that is entirely arbitrary representing the velocity distribution through the flow between the plates. On the diagram I have shown the element of thickness  at distance  from the plate and two tangents to the velocity curve at  and at . Then the velocity gradient at  is  and at  is .

This gives two velocity gradients on equal areas and then the net resisting force on the element is given by :-

                                    .

Equating forces:-

                     , from which :-

                                                  .

As we are looking for the velocity distribution we must integrate twice. Rearranging:-

              and then .

Integrating again gives .-

                                .

We know that   at  and at  so . Then :-   from which  and

In this equation  will be constant if  is the same throughout the flow so  can be replaced by  where  is the pressure drop between inlet and outlet.

A good example of this simple system is the Hele-Shaw apparatus. Figure 16-7 shows my home built version. The working section is made up of two glass plates stuck to 1 mm thick ply at the sides to form a duct with a cross section of a slit. The combination of the two plates is attached to the water reservoir and water is fed into the reservoir at the same rate as it flows out. It is used to produce flow patterns, in this case round the profile of a sail. The pattern is produced in dye injected through 20 or so holes in the feed box.

 

I have used typical dimensions for such a piece of apparatus to find the velocity profile.

 

I let the depth in the feed box be 300 mm, the distance between the plates be 1 mm, the length of the plates be 500 mm and the viscosity of the water be 0.001 kg/ms.

In graph 16- I have plotted the two terms separately and the sum of the terms. The velocity distribution is in black and is parabolic with a maximum value of about 0.75 m/s.

 

Obviously it would be valuable to have an expression for the mean velocity as in the Poiseuille expression for flow in pipes. This involves another integral.

 

                                       

                                               

Then the flow                     

                                               

                                               

 

I have already pointed out that  if  is constant. This means that we can either write                    as required.

 

This can be compared with the Poiseuille expression for laminar flow in a pipe :-