Demonstration #4: Pressure and Force

At the outset, we noted that
these vector fields are the results of the flow of an inviscid,
incompressible fluid. Such fluids satisfy Euler's equations; these
equations have two variables- the fluid velocity and the fluid pressure . If we write , then the equations are

,

.

In addition to these auditions, the velocity field must (and does) satisfy appropriate boundary conditions. If is any boundary of the flow, then on , where is the outward normal to .

If the velocity field is also conservative, meaning ,
then the pressure can be determined directly from the velocity. Indeed,
taking the first component of the first equation, we find

so that

.

A similar argument for the second component shows that

,

and hence we have

.

This fact is called Bernoulli's Equation.
Because only the gradient of the pressure appears in Euler's equations,
the pressure is only determined up to an additive constant .

All of the flows we have considered are
conservative, as they all can (locally) be written as the gradient of
some potential.

Because we are using fluid flows that neglect
the effects of friction and viscosity, the only way that the fluid
exerts a force on an object is through the pressure. In particular, if is an obstacle in the path of a fluid with boundary curve , then the force that the fluid exerts on the obstacle is

where is the outward normal to the obstacle.

Flow Around a Cylinder

Point Vortex Flow Around a Cylinder

Flow Around a Cylinder with Circulation

Flow in a Corner

Flow in a Closed Channel

Flow Around a Rankine Half-Body