Question

Which of these equations is satisfied by the velocity potential equation?

(a) Laplace equation

(b) Fano’s equation

(c) Bernoulli’s equation

(d) Rayleigh equation

This question was posed to me in class test.

The doubt is from The Velocity Potential Equation topic in portion Velocity Potential Equation of Aerodynamics

Answer 1

Right choice is (a) Laplace equation

The best explanation: The velocity component is the negative derivative of the velocity potential in that direction. According to this,

u = –\(\frac {∂ϕ}{∂x}\), v = –\(\frac {∂ϕ}{∂y}\), w = –\(\frac {∂ϕ}{∂z}\)

The continuity equation for three – dimensional flow is given by:

\(\frac {∂u}{∂x} + \frac {∂v}{∂y} + \frac {∂w}{∂z}\) = 0

Substituting the velocity components in the continuity equation, we get

\(\frac {∂}{∂x} \big ( – \frac {∂ϕ}{∂x} \big ) + \frac {∂}{∂y} \big ( – \frac {∂ϕ}{∂x} \big ) + \frac {∂}{∂z} \big ( – \frac {∂ϕ}{∂x} \big )\) = 0

\(\frac {∂^2 ϕ}{∂x} + \frac {∂^2 ϕ}{∂y} + \frac {∂^2 ϕ}{∂z}\) = 0

The above final equation is known as Laplace equation, thus velocity potential satisfies the Laplace equation.