Difference between rotational and irrotational flow of fluids
Answers
No real flow is irrotational. It is one of those simplifying conditions we impose on flow fields to make them so we can solve them, especially when learning fluid dynamics. Fluid dynamics is VERY messy and we have to start off with lots of simplifications to make some headway in solving flows and understanding how fluids behave.
Irrotational flow is flow in which all the tiny bits of fluid are moving along and translating and going around obstacles and what have you without every rotating about their own infinitesimal centers of gravity. Irrotational flow can only persist if there is no viscosity and all real fluids have viscosity.
We particularly like to use the simplification of irrotational flow when dealing with two-dimensional potential flow.
Velocity is a vector and if we apply the vector operator called curl to the velocity and set that equal to zero, then that is the mathematical equivalent of making the flow irrotational. So far so good.
Now we use the useful mathematical result that the curl of the gradient of anything will always be zero. (Gradient is another vector operator.) Well if curl of V is zero and curl of gradient of anything is zero, then velocity must be equal to the gradient of something (if the flow is irrotational). Sure enough, we can set velocity equal to the gradient of something. We call that something the potential function. That’s why we end up calling this whole portion of fluid dynamics potential flow.
Next, we notice a couple of other nice and useful math results that coincidentally conspire to have useful interpretation in fluid dynamics.
The divergence of velocity is equal to zero if the fluid has a constant density (is incompressible). The divergence is yet another vector operator courtesy of the mathematicians. We have substituted gradient of the potential function in place of Velocity. So we have divergence of gradient of potential function = 0 for ncompressible and irrotational flow. That divergence of gradient simplifies to the Laplacian.
Well, that is a nice coincidence.
So now our assumptions about irrotational flow AND incompressible flow have led to the equation that the Laplacian of the potential function is zero everywhere in the flow. This is also called Laplaces equation and it comes up all over the place in physics. It describes the flow of heat in a solid. It describes diffusion of a solvent in a solute.
We already knew how to solve Laplaces Equation. So we can go to town with all this potential flow business. We just use known solutions to Laplaces equation. All of these flow fields work as long as we keep the fluid incompressible and irrotational. Sure, we can live with that. That’s how a fluid would behave if it had no viscosity and its density were constant. It’s not real. No real fluid behaves that way. But with the right understanding, we can use this approximation in lots of places. We can learn how fluids behave (with the appropriate restrictions). Very useful.
Turn on viscosity and this all goes out the window. Now we have rotational flow. Things get a whole lot more messy mathematically. All that stuff I just described with curls and gradients and divergences and Laplacians … that’s the easy stuff. Fluid Dynamics is very complicated.