Potential flow pressure on a smooth surface

For an incompressible potential flow around a smooth rigid body, is it true that the pressure on the surface of the body is proportional to $acos^2theta+b$ where $theta$ is the angle the inward unit surface normal vector makes with the velocity of the flow at infinity for some constants $a$ and $b$?

The reason for my conjecture is the following two examples.

1. The incompressible potential flow around a sphere and a cylinder both assume the above relation for the pressure on the surface of the rigid body.

2. Suppose a column of particles with an infinitesimal cross section area $dA$ collide with a facet with its normal vector forming an angle $thetainbig[0,fracpi2big]$ with the particle flow direction vector. The particles bounces off the facet completely elastically. The momentum change is in the normal direction of the facet, and the speed of change is then $2rho v^2costheta dA$, where $rho$ is the density of the air flow and $v$ the speed of it. The area upon which this momentum change occurs is $frac{dA}{costheta}$. Divide the first quantity by the second, we get the pressure $p(theta):=2rho v^2cos^2theta$. Now the early arriving particles bounce off of the surface normally and collide completely elastically with the late arriving particles and bounce back towards the surface again. By symmetry, the average particle velocity near the surface vanishes in the surface normal direction but its component tangent to the surface remains. Macroscopically, the fluid on average as a whole moves along the tangent of the surface. Alternatively we can assume the complete inelastic collision of the air molecule with the surface, so that the momentum normal to the surface completely dissipates only the tangential component is unmolested so the air molecules after the collision move parallel along the surface. In this case, it is clear $p(theta):=rho v^2cos^2theta$ which is half of the previous value as the surface normal momentum transferred is half of that in the elastic case. In the case of fractional elastic collision, the $p(theta):=(1+alpha)rho v^2cos^2theta$ where $alphain[0,1]$ is the coefficient of collision elasticity.