Application of Bernoulli’s equation begins by drawing a streamline connecting the two points we wish to analyze. One point of our analysis a certain volume of the air along the radial direction in terms of r. However, where does our streamline start?

It would be a mistake to draw a streamline from a point before the pump. Remember Bernoulli’s equation is a representation of the conservation of mechanical energy and the streamline going through the pump disobeys this as the pump adds mechanical energy. Instead, we will look at the streamline from the point after the pump and to the volume of air at a distance r,as shown below.

The Bernoulli equation is given as

Let O be the location shortly after the pump. The velocity at O is related to the volumetric flow rate by

since the diameter is given by c. Hence applying Bernoulli’s equation to point O gives us

Now let’s look at any point in the radial location r  along the gap, in particular the terms for pressure and velocity. As seen from the diagram drawn previously, you could be quick to conclude the pressure as  as the layer of air is in contact with the free surface. This is NOT the case. We are looking at a cross-section area of whole layer of air in the gap. The air that flows out forms a parabolic velocity profile which is constantly flowing. Moreover, if the pressure is  for whatever value of r, the velocity of the air is the same. This is not the case as shown from our previous lesson. Instead, we label this pressue as  as it changes as r changes.

How about the velocity term. By recalling that volume of air enters the control volume with a volumetric flow rate of Q. However, this volume of air is exiting the control volume through the control surface at the same rate. Therefore, our velocity is given by

Again, noticing that the velocity is dependent on r and d. Now we can apply the equation.

At any radial location r along the gap,

Equating the two and rearranging, we get,

With this pressure function, we can calculuate the required force to suspend the hovercraft. The pressure function is dependent on r, thus we need to consider a ring of radius r where the pressure is given by .

Recalling that force is equal to pressure multiplied by area, in this case is  for a small area of constant r, the required force is given by

which we would have to refer to a table of integrals to find the final result.

And there we go, using Bernoulli’s equation to derive the pressure function and then using it to calculate the required force to suspend the hovercraft. This problem is a personal favourite of mine.