/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 The velocity field \(\vec{V}=A x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 5

The velocity field \(\vec{V}=A x \hat{i}-A y \tilde{j},\) where \(A=2 s^{-1},\) can be interpreted to represent flow in a corner. Find an equation for the flow streamlines. Explain the relevance of \(A .\) Plot several streamlines in the first quadrant, including the one that passes through the point \((x, y)=(0,0)\).

Short Answer

Expert verified
The streamline equation for the given velocity field \(\vec{V} = A x \hat{i} - A y \tilde{j}\) is \(xy=k\), where \(k\) is a constant obtained from integration. The value of \(A\) does not influence the slope or location of the streamlines, rather it affects the velocity along these lines. Any streamline can be plotted by choosing some value of \(k\). The streamline passing through the origin has \(k = 0\).

Step by step solution

01

Understanding the Streamline Concept

A streamline in fluid dynamics is a line that is tangential to the velocity vector at each point along it in the flow field. Essentially, describing flow patterns. Given \(\vec{V}=A x \hat{i}-A y \tilde{j}\), a streamline is found by setting \(dx/V_{x} = dy/V_{y}\). Or, equating the ratios of the differentials of \(x\) and \(y\) to the corresponding velocity field values.
02

Derivation of Streamline Equation

Plugging \(\vec{V} = A x \hat{i}-A y \tilde{j}\) into \(dx/V_{x} = dy/V_{y}\) gives \(dx/(A x) = -dy/(A y)\). This simplifies to \(dx/x = -dy/y\), which upon integration leads to \(\ln|x| = -\ln|y| + c\), where \(c\) is the constant of integration. By manipulating the logarithmic equation to exponential form, the streamline equation \(xy = e^c\) is obtained, where \(e^c\) is another constant often denoted by \(k\) yielding the final equation as \(xy = k\).
03

The Role of \(A\)

\(A\), the constant in velocity field equation, acts as a scale factor governing the field's rate of change. Its value does not affect the shape or location of the streamlines, but it does affect the speed of the flow along these lines: higher \(A\) equates to faster flow.
04

Plotting the Streamlines

The equation of the streamline is \(xy = k\). Since the streamlines do not depend on \(A\), any set of streamlines should look the same, just scaled differently. The stream line which passes through the origin has \(k = 0\), because any line passing through origin in the \(xy\) plane is either on the \(x\) or \(y\) axis itself. As the value of \(x\) or \(y\) decreases, respectively, the streamlines extend towards the respective \(y\) and \(x\) axis. The shape of the streamlines remains the same for all cases in the first quadrant irrespective of the value of \(A\) or \(k\).

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

A viscous liquid is sheared between two parallel disks; the upper disk rotates and the lower one is fixed. The velocity field between the disks is given by \(\vec{V}=\hat{e}_{0} r \omega z / h\). (The origin of coordinates is located at the center of the lower disk; the upper disk is located at \(z=h .\) ) What are the dimensions of this velocity field? Does this velocity field satisfy appropriate physical boundary conditions? What are they?

A proposal has been made to use a pair of parallel disks to measure the viscosity of a liquid sample. The upper disk rotates at height \(h\) above the lower disk. The viscosity of the liquid in the gap is to be calculated from measurements of the torque needed to turn the upper disk steadily. Obtain an algebraic expression for the torque needed to turn the disk. Could we use this device to measure the viscosity of a nonNewtonian fluid? Explain.

A block of mass \(10 \mathrm{kg}\) and measuring \(250 \mathrm{mm}\) on each edge is pulled up an inclined surface on which there is a film of SAE \(10 \mathrm{W}-30\) oil at \(30^{\circ} \mathrm{F}\) (the oil film is \(0.025 \mathrm{mm}\) thick). Find the steady speed of the block if it is released. If a force of \(75 \mathrm{N}\) is applied to pull the block up the incline, find the steady speed of the block. If the force is now applied to push the block down the incline, find the steady speed of the block. Assume the velocity distribution in the oil film is linear. The surface is inclined at an angle of \(30^{\circ}\) from the horizontal.

A flow is described by velocity field \(V=a i+b x j\), where \(a=2 \mathrm{m} / \mathrm{s}\) and \(b=1 \mathrm{s}^{-1}\). Coordinates are measured in meters. Obtain the equation for the streamline passing through point \((2,5) .\) At \(t=2 \mathrm{s},\) what are the coordinates of the particle that passed through point (0,4) at \(t=0 ?\) At \(t=3 \mathrm{s},\) what are the coordinates of the particle that passed through point (1,4.25) 2 s earlier? What conclusions can you draw about the pathline, streamline, and streakline for this flow?

For the velocity fields given below, determine: a. whether the flow field is one-, two-, or three-dimensional, and why. b. whether the flow is steady or unsteady, and why. (The quantities \(a\) and \(b\) are constants.) (1) \(\vec{V}=\left[(a x+t) e^{\log y}\right]\) (2) \(\vec{V}=(a x-b y)\) (3) \(\vec{V}=a x \vec{i}+\left[e^{\ln x}\right]\) (4) \(\vec{V}=a x \hat{i}+b x^{2} j+a x k\) (5) \(\vec{V}=a x \hat{i}+\left[e^{b r}\right] \dot{j}\) (6) \(\vec{V}=a x \hat{i}+b x^{2} \hat{j}+a y k\) (7) \(\vec{V}=a x \hat{i}+\left[e^{\ln }\right] \hat{j}+a y \hat{k}\) (8) \(\vec{V}=a x i+\left[e^{b y}\right] j+a z \hat{k}\)
See all solutions

Recommended explanations on Physics Textbooks

Fluids

Read Explanation

Electromagnetism

Read Explanation

Oscillations

Read Explanation

Quantum Physics

Read Explanation

Medical Physics

Read Explanation

Work Energy and Power

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.