0.6 Solution of the partial differential equations  (Page 8/13)

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$\begin{array}{c}\frac{{\partial }^{2}\varphi }{\partial {x}^{2}}+\frac{{\partial }^{2}\varphi }{\partial {y}^{2}}=0\hfill \\ \frac{{\partial }^{2}\psi }{\partial {x}^{2}}+\frac{{\partial }^{2}\psi }{\partial {y}^{2}}=0\hfill \end{array}$

The Cauchy-Riemann conditions also imply that the gradient of the velocity potential and the stream function are orthogonal.

$\left(\nabla \varphi \right)•\left(\nabla \psi \right)=\frac{\partial \varphi }{\partial x}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \psi }{\partial x}+\frac{\partial \varphi }{\partial y}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \psi }{\partial y}=0$

If the gradients are orthogonal then the equipotential lines and the streamlines are also orthogonal, with the exception of stagnation points where the velocity is zero.

Since the derivative

$\frac{dw}{dz}=\begin{array}{c}lim\\ \left|\delta z\to 0\right|\end{array}\phantom{\rule{0.277778em}{0ex}}\frac{\delta \phantom{\rule{0.166667em}{0ex}}w}{\delta \phantom{\rule{0.166667em}{0ex}}z}$

is independent of the direction of the differential $\delta z$ in the $\left(x,y\right)$ -plane, we may imagine the limit to be taken with $\delta z$ remaining parallel to the x-axis $\left(\delta z=\delta x\right)$ giving

$\frac{dw}{dz}=\frac{\partial \varphi }{\partial x}+i\phantom{\rule{0.166667em}{0ex}}\frac{\partial \psi }{\partial x}={v}_{x}-i\phantom{\rule{0.166667em}{0ex}}{v}_{y}$

Now choosing z to be parallel to the y-axis $\left(\delta z=i\delta y\right)$ ,

$\frac{dw}{dz}=\frac{1}{i}\phantom{\rule{0.166667em}{0ex}}\frac{\partial \varphi }{\partial y}+\frac{\partial \psi }{\partial y}=-i\phantom{\rule{0.166667em}{0ex}}{v}_{y}+{v}_{x}={v}_{x}-i\phantom{\rule{0.166667em}{0ex}}{v}_{y}$

These equations are a restatement that an analytical function has a derivative defined in the complex plane. Moreover, we see that the real part of ${w}^{\text{'}}\left(z\right)$ is equal to ${v}_{x}$ and the imaginary part of ${w}^{\text{'}}\left(z\right)$ is equal to $-{v}_{y}$ . If v is written for the magnitude of $\mathbf{v}$ and for the angle between the direction of $\mathbf{v}$ and the $x$ -axis, the expression for $dw/dz$ becomes

$\begin{array}{c}\frac{dw}{dz}={v}_{x}-i\phantom{\rule{0.166667em}{0ex}}{v}_{y}=v\phantom{\rule{0.166667em}{0ex}}{e}^{-i\theta }\hfill \\ \mathrm{or}\hfill \\ {v}_{x}=\mathrm{real}\left[\frac{dw}{dz}\right]\hfill \\ {v}_{y}=-\mathrm{imaginary}\left[\frac{dw}{dz}\right]\hfill \end{array}$

Flow Fields. The simplest flow field that we can imagine is just a constant translation, $w=\left(U-iV\right)z$ where $U$ and $V$ are real constants. The components of the velocity vector can be determined from the differential.

$\begin{array}{c}w\left(z\right)=\left(U-iV\right)z=\left(U-iV\right)\left(x+i\phantom{\rule{0.166667em}{0ex}}y\right)=Ux+Vy+i\left(-Vx+Uy\right)=\varphi +i\phantom{\rule{0.166667em}{0ex}}\psi \hfill \\ \frac{dw}{dz}=\left(U-iV\right)={v}_{x}-i\phantom{\rule{0.166667em}{0ex}}{v}_{y}\hfill \\ {v}_{x}=U,\phantom{\rule{1.em}{0ex}}{v}_{y}=V\hfill \\ \varphi =Ux+Vy,\phantom{\rule{1.em}{0ex}}\psi =-Vx+Uy\hfill \end{array}$

Another simple function that is analytical with the exception at the origin is

$\begin{array}{ccc}\hfill w\left(z\right)& =& A\phantom{\rule{0.166667em}{0ex}}{z}^{n}=A\phantom{\rule{0.166667em}{0ex}}{r}^{n}{e}^{i\phantom{\rule{0.166667em}{0ex}}n\phantom{\rule{0.166667em}{0ex}}\theta }\hfill \\ & =& A\phantom{\rule{0.166667em}{0ex}}{r}^{n}cos\phantom{\rule{0.166667em}{0ex}}n\theta +i\phantom{\rule{0.166667em}{0ex}}A\phantom{\rule{0.166667em}{0ex}}{r}^{n}sin\phantom{\rule{0.166667em}{0ex}}n\theta \hfill \\ & =& \varphi +i\phantom{\rule{0.166667em}{0ex}}\psi \hfill \\ & & \mathrm{thus}\hfill \\ \hfill \varphi & =& A\phantom{\rule{0.166667em}{0ex}}{r}^{n}cos\phantom{\rule{0.166667em}{0ex}}n\theta \hfill \\ \hfill \psi & =& A\phantom{\rule{0.166667em}{0ex}}{r}^{n}sin\phantom{\rule{0.166667em}{0ex}}n\theta \hfill \\ \hfill \frac{dw}{dz}& =& n\phantom{\rule{0.166667em}{0ex}}A\phantom{\rule{0.166667em}{0ex}}{z}^{n-1}={v}_{x}-i\phantom{\rule{0.166667em}{0ex}}{v}_{y}\hfill \end{array}$

where $A$ and $n$ are real constants. The boundary condition at stationary solid surfaces for irrotational flow is that the normal component of velocity is zero or the surface coincides with a streamline. The expression above for the stream function is zero for all $r$ when $\theta =0$ and when $\theta =\pi /n$ . Thus these equations describe the flow between these boundaries are illustrated below.

Earlier we discussed the Green's function solution of a line source in two dimensions. The same solution can be found in the complex domain. A function that is analytical everywhere except the singularity at ${z}_{o}$ is the function for a line source of strength $m$ .

$\begin{array}{c}w\left(z\right)=\frac{m}{2\pi }ln\left(z-{z}_{o}\right),\phantom{\rule{1.em}{0ex}}\mathrm{line}\phantom{\rule{0.277778em}{0ex}}\mathrm{source}\hfill \\ \frac{\mathrm{dw}}{\mathrm{dz}}=\frac{m}{2\pi }\frac{1}{\left(z-{z}_{o}\right)}={v}_{x}-i\phantom{\rule{0.166667em}{0ex}}{v}_{y}\hfill \end{array}$

This results can be generalized to multiple line sources or sinks by superposition of solutions. A special case is that of a source and sink of the same magnitude.

$\begin{array}{c}w\left(z\right)=\sum _{i}\frac{{m}_{i}}{2\pi }ln\left(z-{z}_{i}\right),\phantom{\rule{1.em}{0ex}}\mathrm{multiple}\phantom{\rule{0.277778em}{0ex}}\mathrm{line}\phantom{\rule{0.277778em}{0ex}}\mathrm{sources}\hfill \\ w\left(z\right)=\frac{m}{2\pi }ln\left(\frac{z-{z}_{o}}{z+{z}_{o}}\right),\phantom{\rule{1.em}{0ex}}\mathrm{source}-\mathrm{sink}\phantom{\rule{0.277778em}{0ex}}\mathrm{pair}\hfill \end{array}$

The above flow fields can be viewed with the MATLAB code corner.m, linesource.m, and multiple.m in the complex subdirectory.

Assignment 7.5

Line Source Solution For ${z}_{o}$ at the origin, derive expressions for the flow potential, stream function, components of velocity, and magnitude of velocity for the solution to the line source in terms of $r$ and $\theta$ . Plot the flow potentials and stream functions. Compute and plot the flow potentials and stream function for the superposition of multiple line sources corresponding to the zero flux boundary conditions at $y=+1$ and $-1$ of the earlier assignment.

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