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Chapter 6: Problem 107

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.

Short Answer

Expert verified
The relation between the constants A and B for the flow to be irrotational is given by \( A = - \frac{B}{3} \) and the computed velocity potential is \( \phi = Ax^{3} + Bxy^{2} + C \), where \( C \) is the constant of integration.

Step by step solution

01

Computing the Velocity Components

The velocity components \( u \) and \( v \) in 2D flow can be determined from the stream function \( \psi \) as: \( u = \frac{\partial \psi}{\partial y} \) and \( v = - \frac{\partial \psi}{\partial x} \) . Apply these formulas to obtain \( u=2Bxy \) and \( v=-3Ax^2-Bx^2+2By^2 \).
02

Verifying the Irrotational Flow

A flow field is irrotational if its vorticity or the curl of the velocity vector is zero everywhere. This requires \( \frac{\partial v}{\partial x} = \frac{\partial u}{\partial y} \) . Apply this formula on the computed velocity components to get the relation between \( A \) and \( B \). This yields: \( -6Ax - 2Bx = 2Bx \). Simplifying the equation, you get \( A = - \frac{B}{3} \).
03

Calculating the Velocity Potential

The velocity potential \( \phi \) can be found by solving the Cauchy-Riemann conditions which state, \( \frac{\partial \phi}{\partial x} = u \) and \( \frac{\partial \phi}{\partial y} = -v \). Integrating these equations yields the velocity potential function. Integrating \( \phi_x \) and \( \phi_y \) separately, we get two equations which when compared gives us: \( \phi = Ax^{3} + Bxy^{2} + C \).

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Most popular questions from this chapter

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 \((x, y)=(2,2)\)

A velocity field is given by \(\vec{V}=\left[A x^{3}+B x y^{2}\right] \hat{i}+\) \(\left[A y^{3}+B x^{2} y\right] \hat{j} ; A=0.2 \mathrm{m}^{-2} \cdot \mathrm{s}^{-1}, B\) is a constant, and the coordinates are measured in meters. Determine the value and units for \(B\) if this velocity field is to represent an incompressible flow. Calculate the acceleration of a fluid particle at point \((x, y)=(2,1) .\) Evaluate the component of particle acceleration normal to the velocity vector at this point.

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

Consider the flow field with velocity given by \(\vec{V}=\left[A\left(x^{2}-y^{2}\right)-3 B x\right] \hat{i}-[2 A x y-3 B y] \hat{j},\) where \(A=1 \mathrm{ft}^{-1}\) \(\mathrm{s}^{-1}, B=1 \mathrm{s}^{-1},\) and the coordinates are measured in feet. The density is 2 slug/ft \(^{3}\) and gravity acts in the negative \(y\) direction. Determine the acceleration of a fluid particle and the pressure gradient at point \((x, y)=(1,1)\)

For the flow of Problem 4.123 show that the uniform radial velocity is \(V_{r}=Q / 2 \pi r h .\) Obtain expressions for the \(r\) component of acceleration of a fluid particle in the gap and for the pressure variation as a function of radial distance from the central holes.
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