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Chapter 6: Problem 111

Consider the flow field presented by the potential function $\phi=x^{6}-15 x^{4} y^{2}+15 x^{2} y^{4}-y^{6} .$ Verify that this is an incompressible flow and obtain the corresponding stream function.

Short Answer

Expert verified
The given potential function represents an incompressible flow. The corresponding stream function is \(6x^5y - 20x^3y^3 + 6x^2y^4 + f(x)\), where \(f(x)\) is an arbitrary function of \(x\), which is not clarified for this problem.

Step by step solution

01

Compute the Velocity Field

The velocity field \(v\) is given by the gradient of the potential function \(\phi\). This is computed as \(v = \nabla \phi = \frac{\partial \phi}{\partial x} i + \frac{\partial \phi}{\partial y} j = (6x^5 - 60x^3y^2 + 30xy^4) i - (30x^4y - 60x^2y^3 + 6y^5) j.\)
02

Compute Divergence of the Velocity Field

Calculate the divergence of the obtained velocity field \(v\). The incompressibility condition requires that the divergence be zero. Divergence of a velocity field \(v\) is given by \(\nabla \cdot v\). Therefore, the divergence is computed as \(\nabla \cdot v = \frac{\partial (\nabla \phi)}{\partial x} + \frac{\partial (\nabla \phi)}{\partial y} = (30x^4 - 180x^2y^2 + 120x^2y^4) - (120x^4y^2 - 180x^2y^4 + 30y^4) = 0.\) Thus the flow is incompressible.
03

Compute the Stream Function

To find the stream function \(\psi\) corresponding to the velocity field, integrate one of its components. Let's integrate the x-component with respect to y (and assume x to be constant during integration). The stream function is then, \(\psi = \int v_x dy = \int (6x^5 - 60x^3y^2 + 30xy^4) dy = 6x^5y - 20x^3y^3 + 6x^2y^4 + f(x) \). Here, \(f(x)\) is the constant of integration, which can be a function of x. There are methods to find \(f(x)\), but for this problem, finding \(f(x)\) is not necessary.

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Most popular questions from this chapter

The velocity field in a two-dimensional, steady, inviscid flow field in the horizontal \(x y\) plane is given by \(\vec{V}=(A x+B) \hat{i}-A y \hat{j},\) where \(A=1 \mathrm{s}^{-1}\) and \(B=2 \mathrm{m} / \mathrm{s} ; x\) and \(y\) are measured in meters. Show that streamlines for this flow are given by \((x+B / A) y=\) constant. Plot streamlines passing through points \((x, y)=(1,1),(1,2),\) and \((2,2) .\) At point \((x, y)=(1,2),\) evaluate and plot the acceleration vector and the velocity vector. Find the component of acceleration along the streamline at the same point; express it as a vector. Evaluate the pressure gradient along the streamline at the same point if the fluid is air. What statement, if any, can you make about the relative value of the pressure at points (1,1) and (2,2)\(?\)

Consider the flow field formed by combining a uniform flow in the positive \(x\) direction and a source located at the origin. Obtain expressions for the stream function, velocity potential, and velocity field for the combined flow. If \(U=25 \mathrm{m} / \mathrm{s}\), determine the source strength if the stagnation point is located at \(x=-1 \mathrm{m} .\) Plot the streamlines and potential lines.

To model the velocity distribution in the curved inlet section of a water channel, the radius of curvature of the streamlines is expressed as \(R=L R_{0} / 2 y .\) As an approximation, assume the water speed along each streamline is \(V=10 \mathrm{m} / \mathrm{s}\) Find an expression for and plot the pressure distribution from \(y=0\) to the tunnel wall at \(y=L / 2,\) if the centerline pressure (gage) is \(50 \mathrm{kPa}, L=75 \mathrm{mm},\) and \(R_{0}=0.2 \mathrm{m} .\) Find the value of \(V\) for which the wall static pressure becomes 35 kPa.

Determine whether the Bernoulli equation can be applied between different radii for the vortex flow fields (a) \(\vec{V}=\omega r \hat{e}_{\theta}\) and (b) \(\vec{V}=\hat{e}_{\theta} K / 2 \pi r\)

The stream function of a flow field is \(\psi=A x^{3}+\) \(B\left(x y^{2}+x^{2}-y^{2}\right),\) where \(\psi, x, y, A,\) and \(B\) are all dimensionless. Find the relation between \(A\) and \(B\) for this to be an irrotational flow. Find the velocity potential.
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