For an irrotational flow, the velocity components are expressed in the scalar form, $φ(x,y,z,t)$. This represents the velocity potential. To further this, the following equations represent the individual components of the velocity potential.
(Eq 1) $u=\frac{∂φ}{∂x}$
(Eq 2) $ν=\frac{∂φ}{∂y}$
(Eq 3) $w=\frac{∂φ}{∂z}$
In turn, the vector form of the above equations are as follows.
(Eq 4) $ v=∇ϕ$
As a result, the velocity gradient is expressed as a scalar function $ϕ$.
The velocity potential is caused by the irrotationality of the flow field. In addition, it can be defined by a three-dimensional flow. On other hand, when we look at the stream function, it is caused by the conservation of mass. Which in turn is restricted to a two-dimension flow. Hence, if we were looking at the conservation of mass of an incompressible fluid it would become the following.
(Eq 5) $∇·v=0$
In turn, when applying equation 4 to equation 5 the following equation would result for an incompressible, irrotational flow.
(Eq 6) $∇^2ϕ=0$
For the above equation $∇^2(~)=∇·∇(~)$ is the Laplacian operator. Finally, equation 6 can be broken down into the following cartesian coordinates.
(Eq 7) $\frac{∂^2ϕ}{∂x^2}+$\frac{∂^2ϕ}{∂y^2}+$\frac{∂^2ϕ}{∂z^2}=0$
Laplace’s Equation
In turn, equations 6 and 7 are a form of Laplace’s equation. Laplace’s equation is a differential equation. Many different areas of engineering use Laplace’s equation. As a result, Laplace’s equation is used to derive inviscid, incompressible, and irrotational flow fields.
In order to use Laplace’s equation, you will have to define boundary conditions. For example, velocity that is specified on the flow field boundaries is an example of a boundary condition. The Laplace’s equation in turn will define the potential flow function of the flow field. This will allow us to determined the velocity at all points in the flow field. In addition, Bernoulli equation will define the pressure at all points in the flow field.
Laplace’s equation is used for both steady and unsteady flows. However, for the remainder of this article I will only be focusing on steady flows.
Next, when we are looking at equations 1-6, these equations will define a potential flow that is irrotational. As a result, the vorticity of the flow is zero throughout the flow field. On the other hand, if the vorticity is not zero than Laplace’s equation cannot be used.
Cylindrical Coordinates
In some cases it is more convenient to use cylindrical coordinates ($r$, $θ$, $z$) instead of cartesian coordinates ($x$, $y$, $z$). As a result, the gradient operator will be the following.
(Eq 8) $∇(~)=\frac{∂(~)}{∂r}\hat{e}_r+\frac{1}{r}\frac{∂(~)}{∂θ}\hat{e}_θ+\frac{∂(~)}{∂z}\hat{e}_z$
or
(Eq 9) $∇(ϕ)=\frac{∂(ϕ)}{∂r}\hat{e}_r+\frac{1}{r}\frac{∂(ϕ)}{∂θ}\hat{e}_θ+\frac{∂(ϕ)}{∂z}\hat{e}_z$
In turn $ϕ=ϕ(r,θ,z)$. Next, the velocity component is represented by the following cylindrical coordinates.
(Eq 10) $v=υ_r\hat{e}_r+ υ_θ\hat{e}_θ+ υ_z\hat{e}_z$
When equation 10 is applied to an irrotational flow, $v=∇v$, the velocity components will become the following.
(Eq 11) $ υ_r=\frac{∂ϕ}{∂r}$
(Eq 12) $ υ_θ=\frac{1}{r}\frac{∂ϕ}{∂θ}$
(Eq 13) $ υ_z=\frac{∂ϕ}{∂z}$
In turn, Laplace’s equation using cylindrical coordinates becomes the following.
(Eq 14) $\frac{1}{r}\frac{∂}{∂r}\left(r\frac{∂ϕ}{∂r}\right) + \frac{1}{r^2}\frac{∂^2ϕ}{∂θ^2} + \frac{∂^2ϕ}{∂z^2} = 0$