Franklin's Notes

Complex potential

Given a (possibly multivalued) holomorphic function $F$, we may interpret $F$ as encoding information about a potential fluid flow in the two-dimensional plane and refer to it as a complex potential. We may consider the real and imaginary parts of $F$ separately: where $\varphi$ is called the velocity potential and $\psi$ is called the stream function. The derivative $F'(z)=f(z)$ with respect to $z$ yields the velocity field of the fluid flow, if the value of $f=u-iv$ at each point $z\in\mathbb C$ is interpreted as a vector $(u,v)$ encoding the velocity of the fluid flow at that point. Fluid flow occurs along the streamlines, or lines of constant $\psi$, and at right angles to the equipotential lines, or lines of constant $\varphi$.

To understand geometrically why this analogy works, let's consider the intuitive interpretation of the derivative in terms of infinitesimals. If $F'=f$, we have that for small $\delta z$. If $(u,v)$ defines the velocity field on a region $\Omega$, we have that the velocity field $f$ is defined by $f(z)=\overline{u+iv}=u-iv$, so that we have Note that the real part of this difference is equal to the dot product $\langle \delta x,\delta y\rangle \cdot \langle u,v\rangle$, which intuitively measures the extent to which the change in $z$ and the velocity field are pointing in the same direction. Hence, when $\delta z$ moves "along" the velocity field, the real part of $F$ increases. On the other hand, the imaginary part of this difference is the dot product $\langle \delta x,\delta y\rangle \cdot \langle -v, u\rangle$, where $\langle -v,u\rangle$ is the vector $\langle u,v\rangle$ rotated by 90 degrees, so that this quantity measures how perpendicular to the flow the movement $\delta z$ is. This means that when $\delta z$ moves at a 90 degree counterclockwise angle perpendicular to the flow, the imaginary part of $F$ increases, and it decreases when $\delta z$ moves at a 90 degree clockwise angle perpendicular to the flow.

More generally, when we encode a velocity field $(u,v)$ with the function $f=u-iv$, we have that the line integral along a curve $\gamma$ equals a complex number whose real part measures how much a particle moving along $\gamma$ moves "with the flow", and whose imaginary part measures how much the particle moves "across the flow". If $\gamma$ is a closed curve, the real part measures circulation around $\gamma$ while the imaginary part measures flux through $\gamma$.



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