Flows in the Complex Plane

Abstract

One of the primary goals in this book is to obtain analytical solutions for the equations of motion of fluids. In order that such solutions are found, fluid motion is often restricted to be stationary and two dimensional (2-D) in a plane. All fluid fields depend then on the x- and y-coordinates only, and the fluid velocity has no z-component, \(\vec {u}= u_x(x,y)\;\hat{x}+ u_y(x,y)\;\hat{y}\). The motion of a fluid parcel is restricted to a planar layer, and the whole body of fluid is made up of layers stacked one upon the other. With the usual identification of the vector space \(\mathbb {R}^2\) and the field \(\mathbb {C}\) of complex numbers, complex function theory becomes available for 2-D fluid dynamics. In this chapter, stationary fluid dynamics in 2-D is framed as a problem of complex function theory, by introducing a complex potential Open image in new window for the complex speed field w. We discuss some rather elementary topics from potential theory (Laplace equation, boundary conditions, and maximum property). An Appendix gives an overview of theorems from complex function theory on analytic and meromorphic functions without proofs. We discuss the Schwarz–Christoffel theorem on conformal mappings, which is applied on Chap. 4 to flows with jets, wakes, and cavities. Finally, a brief introduction to Riemann surfaces is given, which will find applications in later chapters.