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Chapter 6: Problem 104

The stream function of a flow field is \(\psi=A x^{2} y-B y^{3},\) where \(A=1 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1}, B=\frac{1}{3} \mathrm{m}^{-1} \cdot \mathrm{s}^{-1},\) and the coordinates are measured in meters. Find an expression for the velocity potential.

Short Answer

Expert verified
The velocity potential function from the given stream function is \(- A x^{2} y+B y^{3}+\frac{1}{3} A x^{3} + C\)

Step by step solution

01

Derive Velocities from Stream Function

First, you have to obtain the velocity components from the stream function \(\psi\). The velocities in x and y direction, denoted as \(u\) and \(v\), are '-partial derivative of \(\psi\) with respect to y' and 'partial derivative of \(\psi\) with respect to x'. The result will be \[u=-\frac{\partial \psi}{\partial y}\] and \[v=\frac{\partial \psi}{\partial x}\] By plugging the given expression of \(\psi\) into these equations, they become \[u=-2 A x y+3 B y^{2}\] and \[v=A x^{2}\]
02

Integrate velocities to obtain Potential Function

The potential function \(\phi\) is obtained by integrating the x-velocity with respect to x and adding it to the integral of the y-velocity with respect to y plus an arbitray constant \(C\). Thus the expression for \(\phi\) becomes \[\phi=\int u d x+\int v d y + C.\] Substituting for \(u\) and \(v\) gives the integral expression that needs to be solved \[\phi=\int(-2 A x y+3 B y^{2}) dx+\int(A x^{2}) dy + C\]. Solve these integrals to find the potential function.
03

Simplify the Velocity Potential Function

On integrating, \[\phi=- A x^{2} y+B y^{3}+\frac{1}{3} A x^{3} + C\]. The potential function comes to \[\phi=- A x^{2} y+B y^{3}+\frac{1}{3} A x^{3} + C\]. This is the final result.

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Most popular questions from this chapter

An incompressible, inviscid fluid flows into a horizontal round tube through its porous wall. The tube is closed at the left end and the flow discharges from the tube to the atmosphere at the right end. For simplicity, consider the \(x\) component of velocity in the tube uniform across any cross section. The density of the fluid is \(\rho,\) the tube diameter and length are \(D\) and \(L,\) respectively, and the uniform inflow velocity is \(v_{0}\) The flow is steady. Obtain an algebraic expression for the \(x\) component of acceleration of a fluid particle located at position \(x,\) in terms of \(v_{0}, x,\) and \(D .\) Find an expression for the pressure gradient, \(\partial p / \partial x,\) at position \(x .\) Integrate to obtain an expression for the gage pressure at \(x=0\)

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