Laplace equation for irrotational flow

Irrotational flow occurs when the curl of the velocity of the fluid is zero everywhere. That is when

$${\displaystyle \nabla \times {\vec {v}}=0}$$

Similarly, in the case that we assume that our fluid is incompressible, that is

$${\displaystyle \rho (x,y,z,t)=\rho ({\text{a constant}})}$$

Then, starting with the continuity equation:

$${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho {\vec {v}})=0}$$

Our condition of incompressibility means that the time derivative of the density is 0, and that we can pull the density out of the divergence, and divide it out, and thus we have our continuity equation for an incompressible system:

$${\displaystyle \nabla \cdot {\vec {v}}=0}$$

Now, we can use the Helmholtz decomposition to write the velocity as the sum of the gradient of a scalar potential and as the curl of a vector potential. That is we have

$${\displaystyle {\vec {v}}=-\nabla \phi +\nabla \times {\vec {A}}}$$

Note that imposing our condition that ${\displaystyle \nabla \times {\vec {v}}=0}$ implies that

$${\displaystyle \nabla \times (\nabla \times {\vec {A}})=0}$$

Where we have used the fact that the curl of the gradient is always 0. Note that the curl of the curl of a function is only uniformly 0 for the vector potential being 0 itself. So, by our condition of irrotational flow, we have

$${\displaystyle {\vec {v}}=-\nabla \phi }$$

And then using our continuity equation ${\displaystyle \nabla \cdot {\vec {v}}=0}$, we can substitute our scalar potential back in to find Laplace's Equation for irrotational flow:

${\displaystyle \nabla ^{2}\phi =0\,}$

Note that the laplace equation is a well studied linear partial differential equation. It has an infinite amount of solutions, however, we are able to discard most solutions to it when considering physical systems, as boundary conditions completely determine the velocity potential.

Examples of common boundary conditions include the velocity of the fluid, determined by ${\displaystyle {\vec {v}}=-\nabla \phi }$, being 0 on the boundaries of the system.

There is a great amount of overlap with electromagnetism when solving this equation in general, as the laplace equation also models the electrostatic potential in vacuum.

There are many reasons to study Irrotational flow, among them;

• Many real world problems contain large regions of Irrotational flow.
• They can be studied analytically.
• They show us importance of Boundary layers and viscous forces.
• They provide us tools for studying concepts of lift and drag.

References

• Landau, L.D.; Lifshitz, E.M. (1984). Fluid Mechanics (2nd ed.). ISBN 0-7506-2767-0.