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Chapter 6: Problem 16

The stream function for an incompressible, two-dimensional flow field is $$\psi=a y-b y^{3}$$ where \(a\) and \(b\) are constants. Is this an irrotational flow? Explain.

Short Answer

Expert verified
The flow is not irrotational because its curl is not equal to zero. However, at y=0, the flow can be considered irrotational since the curl equals zero at this point.

Step by step solution

01

Formulate the necessary equations

Use the given stream function \( \psi = ay - by^3 \). Since this is a two-dimensional problem, we know that there is no variation in the \( z \) direction, thus the stream function in Cartesian coordinates can be written as \( u = \frac{\partial{\psi}}{\partial{y}} \) and \( v = -\frac{\partial{\psi}}{\partial{x}} \). Here \( u \) and \( v \) are the velocity components in the \(x\) and \(y\) directions, respectively.
02

Calculate the derivatives

Calculate the partial derivatives of the stream function w.r.t \( x \) and \( y \). Here \( \frac{\partial{\psi}}{\partial{y}} = a - 3by^2 \) and since there are no \( x \) terms in the stream function, \( \frac{\partial{\psi}}{\partial{x}} = 0 \). Therefore, \( u = a - 3by^2 \) and \( v = 0 \).
03

Calculate the curl

The curl of the velocity field in 2D Cartesian coordinates is given by \( \nabla \times \textbf{v} = \frac{\partial{v}}{\partial{x}} - \frac{\partial{u}}{\partial{y}} \). Substituting \( u \) and \( v \) gives \( \nabla \times \textbf{v} = 0 - \frac{\partial}{\partial{y}}(a - 3by^2) = -6by \).
04

Determine if flow is irrotational

We observe that the curl of the velocity field, \( \nabla \times \textbf{v} \), is not equal to zero, which implies that this is not an irrotational flow. This being said, if the fluid is evaluated at \( y = 0 \), the flow can be considered irrotational at this specific point.

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Most popular questions from this chapter

Air is delivered through a constant-diameter duct by a fan. The air is inviscid, so the fluid velocity profile is "flat" across each cross section. During the fan start-up, the following velocities were measured at the time \(t\) and axial positions \(x\) : $$\begin{array}{llll} & x=0 & x=10 \mathrm{m} & x=20 \mathrm{m} \\ t=0 \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} & V=0 \mathrm{m} / \mathrm{s} \\\t=1.0 \mathrm{s} & V=1.00 \mathrm{m} / \mathrm{s} & V=1.20 \mathrm{m} / \mathrm{s} & V=1.40 \mathrm{m} / \mathrm{s} \\\t=2.0 \mathrm{s} & V=1.70 \mathrm{m} / \mathrm{s} & V=1.80 \mathrm{m} / \mathrm{s} & V=1.90 \mathrm{m} / \mathrm{s} \\\t=3.0 \mathrm{s} & V=2.10 \mathrm{m} / \mathrm{s} & V=2.15 \mathrm{m} / \mathrm{s} & V=2.20 \mathrm{m} / \mathrm{s}\end{array}$$ Estimate the local acceleration, the convective acceleration, and the total acceleration at \(t=1.0 \mathrm{s}\) and \(x=10 \mathrm{m} .\) What is the local acceleration after the fan has reached a steady air flow rate?

An infinitely long, solid, vertical cylinder of radius \(R\) is located in an infinite mass of an incompressible fluid. Start with the Navier-Stokes equation in the \(\theta\) direction and derive an expression for the velocity distribution for the steady-flow case in which the cylinder is rotating about a fixed axis with a constant angular velocity \(\omega\). You need not consider body forces. Assume that the flow is axisymmetric and the fluid is at rest at infinity.

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