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Chapter 6: Problem 37

The velocity potential for a given two-dimensional flow field is $$\phi=\left(\frac{5}{3}\right) x^{3}-5 x y^{2}$$ Show that the continuity equation is satisfied and determine the corresponding stream function.

Short Answer

Expert verified
The continuity equation for the given potential \( \left(\frac{5}{3}\right) x^{3}-5 x y^{2} \) is satisfied and the corresponding stream function is \( -\left(\frac{5}{3}\right) x^{3} + 5 x y^{2} \).

Step by step solution

01

Calculating the gradient of the given potential

The given potential is \( \phi=\left(\frac{5}{3}\right) x^{3}-5 x y^{2} \). The velocity field \( \vec{V} \) is equal to the gradient of \( \phi \). Therefore, first differentiate \( \phi \) with respect to \( x \) and \( y \) to find the two components of \( \vec{V} \). The differentiation yields \( \vec{V} = (5x^2-5y^2)\hat{i} - 10xy\hat{j} \).
02

Verifying the continuity equation

Now calculate the divergence of \( \vec{V} \). The divergence \( \nabla \cdot \vec{V} \) is obtained by differentiating the \( i \) and \( j \) components of \( \vec{V} \) with respect to \( x \) and \( y \) respectively, and adding them. The calculation yields \( \nabla \cdot \vec{V} = 10x - 10x = 0 \), which confirms the continuity equation is satisfied.
03

Determining the stream function \(\psi\)

The stream function \( \psi \) for a 2D flow is given by \( \psi = -\phi \). So, substitute the given potential \( \phi \) into this equation to find \( \psi \). This gives \( \psi = -\left(\frac{5}{3}\right) x^{3} + 5 x y^{2} \).

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