Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 10

The velocity field of a flow is given by \(u=-V_{0} y /\left(x^{2}+\right.\) \(\left.y^{2}\right)^{1 / 2}\) and \(v=V_{0} x /\left(x^{2}+y^{2}\right)^{1 / 2},\) where \(V_{0}\) is a constant. Where in the flow field is the speed equal to \(V_{0}\) ? Determine the equation of the streamlines and discuss the various characteristics of this flow.

Short Answer

Expert verified
The speed of the flow is equal to \(V_{0}\) throughout the field as \(V_{0}\) is always positive. The equation of the streamlines is \( x = ± y\), indicating a radial flow that is symmetric with respect to the origin.

Step by step solution

01

Calculate the Speed

The speed of the flow can be calculated by taking the magnitude of the velocity vector: \[\sqrt{u^2 + v^2} = \sqrt{(-V_{0} y /\sqrt{x^{2} + y^{2}} )^2 + ( V_{0} x /\sqrt{x^{2} + y^{2}} )^2}\]
02

Simplify the equation and find when speed equals to \(V_{0}\)

Simplify the above expression to get \[ |V_{0}|\sqrt{(x^{2} + y^{2}) / (x^{2} + y^{2})}\]. This simplifies to \(|V_{0}|\). The magnitude of the velocity is equal to \(V_{0}\) when \(|V_{0}| = V_{0}\). This implies that \(V_{0}\) is always positive.
03

Obtain the equation for streamlines

The streamlines of a fluid are represented by the equation \[\int_{-V_0}^{V_0} \frac{dy}{u} = \int_{-V_0}^{V_0} \frac{dx}{v}\]. Substituting the given values of u and v in the above equation and integrating gives the equation of streamlines as \( x = ± y\).
04

Discuss the Characteristics of the Flow

The streamlines being straight lines passing through the origin indicate that the flow is radial. The fluid particles are moving in straight lines emanating from or converging to the origin. The presence of the ± sign indicates that the flow is symmetric with respect to the origin.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For any steady flow the streamlines and streaklines are the same. For most unsteady flows this is not true. However, there are unsteady flows for which the streamlines and streaklines are the same. Describe a flow field for which this is true.

Assume the temperature of the exhaust in an exhaust pipe can be approximated by \(T=T_{0}\left(1+a e^{-b x}\right)[1+c \cos (\omega t)],\) where \\[ T_{0}=100^{\circ} \mathrm{C}, a=3, b=0.03 \mathrm{m}^{-1}, c=0.05, \text { and } \omega=100 \mathrm{rad} / \mathrm{s} \\] If the exhaust speed is a constant \(3 \mathrm{m} / \mathrm{s}\), determine the time rate of change of temperature of the fluid particles at \(x=0\) and \(x=4 \mathrm{m}\) when \(t=0\)

A constant-density fluid flows through a converging section having an area \(A\) given by \\[ A=\frac{A_{0}}{1+(x / \ell)} \\] where \(A_{0}\) is the area at \(x=0 .\) Determine the velocity and acceleration of the fluid in Eulerian form and then the velocity and acceleration of a fluid particle in Lagrangian form. The velocity is \(V_{0}\) at \(x=0\) when \(t=0\)

The velocity components of \(u\) and \(v\) of a two-dimensional flow are given by \\[ u=a x+\frac{b x}{x^{2} y^{2}} \quad \text { and } \quad v=a y+\frac{b y}{x^{2} y^{2}} \\] where \(a\) and \(b\) are constants. Calculate the acceleration.

Figure \(P 4.67\) illustrates a system and fixed control volume at time \(t\) and the system at a short time \(\delta t\) later. The system temperature is \(T=100^{\circ} \mathrm{F}\) at time \(t\) and \(T=103^{\circ} \mathrm{F}\) at time \(t+\delta t,\) where \(\delta t=0.1 \mathrm{s} .\) The system mass, \(m,\) is 2.0 slugs, and 10 percent of it moves out of the control volume in \(\delta t=0.1\) s. The energy per unit mass \(\tilde{u}\) is \(c_{v} T,\) where \(c_{v}=32.0\) Btu/slug \(\cdot^{\circ} \mathrm{F}\). The energy \(U\) of the system at any time \(t\) is \(m \tilde{u}\). Use the system or Lagrangian approach to evaluate \(D U / D t\) of the system. Compare this result with the \(D U / D t\) evaluated with that of the material time derivative and flux terms.
See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Circular Motion and Gravitation

Read Explanation

Geometrical and Physical Optics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Classical Mechanics

Read Explanation

Atoms and Radioactivity

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free