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Chapter 6: Problem 16

The stream function for an incompressible, two-dimensional flow field is \\[\psi=a y^{2}-b x\\] where \(a\) and \(b\) are constants. Is this an irrotational flow? Explain.

Short Answer

Expert verified
The flow is not irrotational unless \( a = 0 \). Otherwise, vorticity is non-zero.

Step by step solution

01

Understand the Relationship

For a flow to be irrotational, the vorticity must be zero. Vorticity in two-dimensional flow is the curl of the velocity field. In terms of the stream function \( \psi(x, y) \), the velocity components can be derived as follows: \( u = \frac{\partial \psi}{\partial y} \) and \( v = -\frac{\partial \psi}{\partial x} \).
02

Derive Velocity Components

Using the stream function \( \psi = a y^2 - b x \), we find the velocity components. First, compute \( u = \frac{\partial \psi}{\partial y} = \frac{\partial}{\partial y}(a y^2 - b x) = 2ay \). Next, compute \( v = -\frac{\partial \psi}{\partial x} = -\frac{\partial}{\partial x}(a y^2 - b x) = b \).
03

Determine Vorticity

The vorticity \( \omega \) is given by the difference of partial derivatives of the velocity components: \( \omega = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \). First, compute \( \frac{\partial v}{\partial x} = \frac{\partial}{\partial x}b = 0 \) and \( \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}2ay = 2a \). Thus, \( \omega = 0 - 2a = -2a \).
04

Conclude About Irrotationality

Since the vorticity \( \omega = -2a \) is not zero unless \( a = 0 \), the flow is not irrotational unless \( a \) is specifically zero. For the flow to be irrotational everywhere, the specific condition \( a = 0 \) must be satisfied.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Stream Function
The concept of a stream function is pivotal in fluid dynamics, especially when analyzing two-dimensional, incompressible flow. A stream function, typically denoted as \( \psi \), is a scalar function that helps describe the flow field. It provides a means to visualize fluid motion without directly dealing with velocity components. To better understand:
  • For two-dimensional flows, the velocity components can be derived from the stream function. If \( \psi(x, y) = a y^2 - b x \), the velocity components are given by \( u = \frac{\partial \psi}{\partial y} \) and \( v = -\frac{\partial \psi}{\partial x} \).
  • This results in \( u = 2ay \) and \( v = b \), representing the flow in the x and y directions respectively.
Using the stream function simplifies the analysis of complex flow fields by automatically satisfying the continuity equation. The contours of the stream function represent streamlines, paths along which the fluid flows.
Vorticity
Vorticity, in simple terms, is a measure of the local rotation in a fluid flow. It's a vector quantity and, for two-dimensional flows, is computed as the curl of the velocity field. The vorticity \( \omega \) can be determined by:
  • Calculating the partial derivatives of the velocity components. Using our example, these are \( \frac{\partial u}{\partial y} \) and \( \frac{\partial v}{\partial x} \).
  • The difference \( \omega = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \) yields \( \omega = 0 - 2a = -2a \) in the given exercise.
Under normal conditions, if \( \omega \) is zero, the flow has no local rotation, i.e., it is irrotational. Here, \( \omega = -2a \) indicates a rotational flow unless \( a = 0 \). Understanding vorticity is key in detecting turbulence and comprehending complex flow behavior.
Irrotational Flow
Irrotational flow is a concept where the fluid has no local rotation at any point. It is characterized by a vorticity of zero. In the context of the given exercise, to determine if a flow is irrotational, we calculate the vorticity first:
  • If \( \omega = 0 \), the flow is irrotational.
  • For the provided stream function and derived velocity components, \( \omega = -2a \). Thus, the flow is only irrotational if \( a = 0 \).
Irrotational flows are particularly important in potential flow theory, where they simplify the equations governing fluid motion. They frequently appear in inviscid (frictionless) flows where analytical solutions can often be found. Understand that calculating and confirming irrotationality is crucial for correctly applying concepts in fluid mechanics for engineering and physics.

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Most popular questions from this chapter

The velocity potential for a certain inviscid, incompressible flow field is given by the equation \\[\phi=2 x^{2} y-\left(\frac{2}{3}\right) y^{3}\\] where \(\phi\) has the units of \(\mathrm{m}^{2} / \mathrm{s}\) when \(x\) and \(y\) are in meters. Determine the pressure at the point \(x=2 \mathrm{m}, y=2 \mathrm{m}\) if the pressure at \(x=1 \mathrm{m}\) \(y=1 \mathrm{m}\) is \(200 \mathrm{kPa}\). Elevation changes can be neglected, and the fluid is water.

A layer of viscous liquid of constant thickness (no velocity perpendicular to plate) flows steadily down an infinite, inclined plane. Determine, by means of the Navier-Stokes equations, the relationship between the thickness of the layer and the discharge per unit width. The flow is laminar, and assume air resistance is negligible so that the shearing stress at the free surface is zero.

The velocity components of an incompressible, twodimensional velocity field are given by the equations \\[\begin{array}{l}u=y^{2}-x(1+x) \\\v=y(2 x+1)\end{array}\\] Show that the flow is irrotational and satisfies conservation of mass.

A steady, uniform, incompressible, inviscid, two-dimensional flow makes an angle of \(30^{\circ}\) with the horizontal \(x\) axis. (a) Determine the velocity potential and the stream function for this flow. (b) Determine an expression for the pressure gradient in the vertical \(y\) direction. What is the physical interpretation of this result?

The velocity components of an incompressible, twodimensional velocity field are given by the equations \\[\begin{array}{l}u=2 x y \\\v=x^{2}-y^{2}\end{array}\\] Show that the flow is irrotational and satisfies conservation of mass.
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