/*! 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 34 A flow is described by velocity ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 34

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?

Short Answer

Expert verified
The streamline passing through (2,5) is given as \(-x^2 / 2 + 2x + 5 = y\). The coordinates of the particle that passed through (0,4) at \(t=0\) at \(t=2s\) are (4,6). For the particle that passed through (1,4.25) 2 seconds earlier, at \(t=3s\) it is at the position (4, 5.5). The streamline, pathline, and streakline are identical in this constant velocity field.

Step by step solution

01

Derive the Streamline Equation

Streamlines denote the exact vector direction flow at a particular point. The equation for the streamline in this velocity field is obtained from the velocity field equation by setting \(dx/dy = V_x/V_y = a / (b * x)\), where \(x\) and \(y\) denote the coordinates, and \(V_x\) and \(V_y\) denote the x and y components of the velocity field respectively. Integrating both sides of this equation gives the streamline equation passing through point \((2,5)\) as \(-x^2 / 2 + 2x + 5 = y\).
02

Calculate the Coordinates of the Particle Passing Through Point (0,4) at \(t=2s\)

To find the expected new coordinates at \(t=2s\), use the streamline equation formed above with initial point as \((0,4)\). After integrating the x and y components of the velocity field equation separately with respect to time, the new coordinates are obtained as \((2*2, 4 + ((1*2^2)/2)) = (4,6)\).
03

Calculate the Coordinates of the Particle that Passed Through Point (1,4.25) 2 seconds Earlier at \(t=3s\)

Utilizing the streamline equation and the initial point \((1,4.25)\) at \(t=1s\), integrate both the x and y components of the velocity field equation with respect to time to locate the particle's new coordinates at \(t=3s\). The results are \((2*(3-1), 4.25 + ((1*(3-1)^2)/2)) = (4, 5.5)\).
04

Conclusion on the Pathline, Streamline, and Streakline

The pathline represents a particle's path in a fluid flow, whereas the streamline shows the fluid flow direction at some instant, and the streakline is the path left by a particle in the fluid. Given the velocity field is constant, the streamline, pathline, and streakline are identical for this flow, as particles are always flowing along these lines.

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

Slowly fill a glass with water to the maximum possible level. Observe the water level closely. Explain how it can be higher than the rim of the glass.

A concentric cylinder viscometer may be formed by rotating the inner member of a pair of closely fitting cylinders. For small clearances, a linear velocity profile may be assumed in the liquid filling the annular clearance gap. A viscometer has an inner cylinder of \(75 \mathrm{mm}\) diameter and 150 mm height, with a clearance gap width of \(0.02 \mathrm{mm}\) A torque of \(0.021 \mathrm{N} \cdot \mathrm{m}\) is required to turn the inner cylinder at \(100 \mathrm{rpm} .\) Determine the viscosity of the liquid in the clearance gap of the viscometer.

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) \(\vec{V}=a x i+\left[e^{b y}\right] j+a z \hat{k}\)

The cone and plate viscometer shown is an instrument used frequently to characterize non-Newtonian fluids. It consists of a flat plate and a rotating cone with a very obtuse angle (typically \(\theta\) is less than 0.5 degrees). The apex of the cone just touches the plate surface and the liquid to be tested fills the narrow gap formed by the cone and plate. Derive an expression for the shear rate in the liquid that fills the gap in terms of the geometry of the system. Evaluate the torque on the driven cone in terms of the shear stress and geometry of the system.

A block weighing 10 lbf and having dimensions 10 in. on cach edge is pulled up an inclined surface on which there is a film of SAE \(10 \mathrm{W}\) oil at \(100^{\circ} \mathrm{F}\). If the speed of the block is \(2 \mathrm{ft} / \mathrm{s}\) and the oil film is 0.001 in. thick, find the force required to pull the block. Assume the velocity distribution in the oil film is linear. The surface is inclined at an angle of \(25^{\circ}\) from the horizontal.
See all solutions

Recommended explanations on Physics Textbooks

Classical Mechanics

Read Explanation

Rotational Dynamics

Read Explanation

Quantum Physics

Read Explanation

Medical Physics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Fundamentals of Physics

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.