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Chapter 6: Problem 112

Show by expanding and collecting real and imaginary terms that \(\left.f=z^{6} \text { (where } z \text { is the complex number } z=x+i y\right)\) leads to a valid velocity potential (the real part of \(f\) ) and a corresponding stream function (the negative of the imaginary part of \(f\), an irrotational and incompressible flow. Then show that the real and imaginary parts of \(d f / d z\) yield \(-u\) and \(v,\) respectively.

Short Answer

Expert verified
Upon expanding and simplifying the function, the real part can be shown as the velocity potential and the negative of the imaginary part as the stream function. By differentiating \(f\) with respect to \(z\) and breaking it into real and imaginary components, we can identify \(u\) and \(v\) as the respective components.

Step by step solution

01

Expand the complex number

Start by expanding the function \(f=z^6\). Given \(z = x + iy\), replacing \(z\) in \(f\) and expanding will yield \(f = (x+iy)^6\).
02

Divide the equation into real and imaginary parts

Write \(f\) as a sum of two terms, one containing all the real parts, and the other the imaginary parts. This requires simplifying \(f = (x+iy)^6\). You must expand the above equation using the Newton binomial theorem and collect like terms.
03

Identify the velocity potential and the stream function

The velocity potential is the real part of \(f\), and the stream function is the negative of the imaginary. Check that the Laplacians of the two parts are zero, to prove that the flow is incompressible and irrotational respectively.
04

Derive \(f\) with respect to \(z\)

Find \(df/dz\) using the Chain Rule. This derivative should also be divided into real and imaginary parts.
05

Identify components of velocity

Show that the real part of the derivative is \(u\) and the imaginary part is \(v\). These are the speeds in the x and y directions respectively. Show that these speeds make physical sense in this context.

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Most popular questions from this chapter

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