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Chapter 6: Problem 109

Consider the flow field represented by the potential function $\phi=A x^{2}+B x y-A y^{2} .$ Verify that this is an incompressible flow and determine the corresponding stream function.

Short Answer

Expert verified
By calculating the velocity vectors from the given potential function and substituting those in the continuity equation, it can be verified that the given flow is incompressible as the equation equals to zero. The stream function can then be determined by integrating the relations between the velocity vectors and the stream function.

Step by step solution

01

Verify the Incompressibility

The continuity equation for an incompressible flow in two dimensions is obtained when the divergence of the velocity vector is zero i.e \( \nabla \cdot \vec{v} = 0 \). The velocity vectors can be derived from the potential function as \( u = \frac{\partial \phi}{\partial x} \) and \( v = -\frac{\partial \phi}{\partial y} \). Substitute these expressions in the continuity equation and check if it equals zero.
02

Calculate Velocity Vectors

Calculate \( u = \frac{\partial \phi}{\partial x} \) and \( v = -\frac{\partial \phi}{\partial y} \). For \( u = \frac{\partial \phi}{\partial x} \), take the derivative of \( \phi \) with respect to \( x \) and likewise for \( v = -\frac{\partial \phi}{\partial y} \), take the derivative of \( \phi \) with respect to \( y \). This will give the expressions for \( u \) and \( v \).
03

Apply Continuity Equation

Substitute the expressions derived for \( u \) and \( v \) in the continuity equation \( \nabla \cdot \vec{v} = 0 \) which gives \( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \). If this equation equals to zero, this verifies the flow is incompressible.
04

Determine the Stream Function

Once the flow has been verified as incompressible, determine the stream function \( \psi \). The stream function \( \psi \) is related to the velocity vectors as \( u = \frac{\partial \psi}{\partial y} \) and \( v = -\frac{\partial \psi}{\partial x} \). The stream function \( \psi \) can be obtained by integrating these expressions.

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Most popular questions from this chapter

Show that any differentiable function \(f(z)\) of the complex number \(z=x+i y\) leads to a valid potential (the real part of \(f\), and a corresponding stream function (the negative of the imaginary part of \(f\) ) of an incompressible, irrotational flow. To do so, prove using the chain rule that \(f(z)\) automatically satisfies the Laplace equation. Then show that \(d f / d z=-u+i v\)

In a two-dimensional frictionless, incompressible \(\left(\rho=1500 \mathrm{kg} / \mathrm{m}^{3}\right)\) flow, the velocity field in meters per second is given by \(\vec{V}=(A x+B y) \hat{i}+(B x-A y) \hat{j} ;\) the coordinates are measured in meters, and \(A=4 \mathrm{s}^{-1}\) and \(B=2 \mathrm{s}^{-1}\). The pressure is \(p_{0}=200 \mathrm{kPa}\) at point \((x, y)=(0,0) .\) Obtain an expression for the pressure field, \(p(x, y)\) in terms of \(p_{0}, A,\) and \(B,\) and evaluate at point \((x, y)=(2,2)\)

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.

\(\mathrm{A}\) flow field is formed by combining a uniform flow in the positive \(x\) direction, with \(U=10 \mathrm{m} / \mathrm{s},\) and a counter clockwise vortex, with strength \(K=16 \pi \mathrm{m}^{2} / \mathrm{s},\) located at the origin. Obtain the stream function, velocity potential, and velocity field for the combined flow. Locate the stagnation point(s) for the flow. Plot the streamlines and potential lines.

Calculate the dynamic pressure that corresponds to a speed of $100 \mathrm{km} / \mathrm{hr}$ in standard air. Express your answer in millimeters of water.
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