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Chapter 6: Problem 110

Consider the flow field presented by the potential function $\phi=x^{5}-10 x^{3} y^{2}+5 x y^{4}-x^{2}+y^{2} .$ Verify that this is an incompressible flow, and obtain the corresponding stream function.

Short Answer

Expert verified
The divergence of the velocity field represented by the potential function is zero, which verifies that the given flow field is incompressible. The corresponding stream function is \(\psi = -x^4y + 5x^2y^3 - y + C\), where \(C\) is a constant.

Step by step solution

01

Compute the Gradient of the Potential Function

The velocity field \(\vec{v}\) is given by the gradient of the potential function \(\phi\). In cartesian coordinates, this is \(\vec{v} = (\frac{d\phi}{dx}, \frac{d\phi}{dy})\). After calculating, you get \(\vec{v} = (5x^4 - 30x^2y^2 + 5y^4 - 2x, -20x^3y + 20xy^3 + 2y)\).
02

Verify the Incompressibility

Next, verify the incompressibility by taking the divergence of the velocity field, for which \(\nabla \cdot \vec{v} = 0\). In this case, the divergence is given by \(\frac{d \vec{v}_x}{dx} + \frac{d \vec{v}_y}{dy}\). After calculating, you get this as \(20x^3 - 60xy^2 + 20xy^3 - 2 + 0 = 0\), confirming that the flow is indeed incompressible.
03

Obtain the Stream Function

We find the stream function \(\psi\) by using the definition of the potential flow. In 2D, we have that \(\frac{d\psi}{dx} = -v_y\) and \(\frac{d\psi}{dy} = v_x\). We solve for \(\psi\) considering one partial derivative at a time. The antiderivative of \(-v_y\) with respect to \(x\) gives \(-x^4y + 5x^2y^3\), and the antiderivative of \(v_x\) with respect to \(y\) provides \(-x^4y + 5x^2y^3 - y\). Comparing these, we obtain \(\psi = -x^4y + 5x^2y^3 - y + C\), where \(C\) is a constant.

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Most popular questions from this chapter

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