Introduction

Two-Dimensional Fluid Flow

Suppose that a fluid flows over the complex plane and that the velocity at the point z = x + iy is given by the vector

We also require that the velocity does not depend on time and that the components p(x,y) and q(x,y) have continuous partial derivatives. The divergence of the vector field in this equation is given by

and is a measure of the extent to which the velocity field diverges near the point. We will consider only fluid flows for which the divergence is zero. This is more precisely characterized by requiring that the net flow through any simple closed contour be identically zero.

If we consider the flow out of the small rectangle with edges and , then the rate of outward flow equals the line integral of the exterior normal component of taken over the sides of the rectangle. The exterior normal component is given by -q(x,y) on the bottom edge, p(x,y) on the right edge, q(x,y) on the top edge, and -p(x,y) on the left edge. Integrating and setting the resulting net flow equal to zero yields

Since p(x,y) and q(x,y) are continuously differentiable, the mean value theorem cn be used to show that

Substitution of the expressions into the previous equations and subsequently dividing through by results in

The mean value theorem for integrals can be used in this equation to show that

The last equation is called the equation of continuity.

Theorem 11.5. (Invariance of Flow) Let denote the complex potential for a fluid flow in a domain G in the w plane where the velocity is . If the function is a one-to-one conformal mapping from a domain D in the z plane onto G, then the composite function

is the complex potential for a fluid flow in D where the velocity is .

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(c) John Mathews, 1998, 2006