Illustrative example for complex analysis in biofluid dynamics applications? I am looking for an application of complex analysis to solve problems related to biofluid dynamics.
We know that complex analysis can be used for solving 2 dimensional, inviscid and irrotational flow problems. There are many well-known examples like the Kutta-Joukowski Theorem, Joukowski transformation for airfoils and the Schwarz-Christoffel mapping.
In order to demonstrate the usefulness of complex analysis for biofluid dynamics I want to give a nice but not too complicated use of complex analysis in this field. I have flipped through the Milne-Thompsons book on theoretical hydrodynamics but I was not able to detect something illustrative (maybe you know something from this book?).
It would be nice if someone could provide me with an example of the application of complex analysis to the field of biofluid dynamics (related papers are also welcome).
 A: Perhaps the simplest case of applying complex analysis in fluid dynamics (and not only biofluids) is the following:
For irrational fluids, one can define a scalar velocity potential:
$$\text{u}=\nabla\phi$$
and if the fluid is also incompressible $(\nabla\cdot\text u=0)$, this velocity potential would satisfy Laplace equation:
$$\nabla^2\phi=0$$
The incompressibility also implies the existence of a scalar stream-function $\psi$, such that in 2D fluids:
$$\text u(x,y)=-\nabla\times(\hat{\textbf z}\psi)$$
This leads to the fact that $\nabla\phi\cdot\nabla\psi=0$, i.e. equipotential lines are orthogonal to streamlines. So for an incompressible and irrational 2D flow, one can obtain these equations:
$$u_x\equiv-\frac{\partial\phi}{\partial x},\quad u_y\equiv-\frac{\partial\phi}{\partial y}$$
which somehow resembles a Cauchy-Riemann like condition. Thus if we consider $u_x$ and $u_y$ as the components of a single complex variable, one can therefore apply powerful techniques from complex analysis to solve for the flow field.
As a side-note, the Cauchy-Riemann like conditions can also be directly derived from the orthogonality condition $(\nabla\phi\cdot\nabla\psi=0)$.
