/*! 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 36 Incompressible fluid flows stead... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 36

Incompressible fluid flows steadily through a plane diverging channel, At the inlet, of height \(H,\) the flow is uniform with magnitude \(V_{1}\). At the outlet, of height \(2 H\), the velocity profile is \\[ V_{2}=V_{m} \cos \left(\frac{\pi y}{2 H}\right) \\] where \(y\) is measured from the channel centerline. Express \(V_{m}\) in terms of \(V_{1}\)

Short Answer

Expert verified
To express \(V_{m}\) in terms of \(V_{1}\), we use the principle of flow conservation for incompressible steady flow, which tells us that the mass flow rate at the inlet equals the mass flow rate at the outlet. By integrating the velocity profile at the outlet and equating it to the flow at the inlet, we solve for \(V_{m}\) in terms of \(V_{1}\).

Step by step solution

01

Identify key principles

Recognize that the flow conservation principle applies in this exercise. For incompressible steady flow, the mass flow rate at the inlet equals the mass flow rate at the outlet.
02

Calculate mass flow rate at the inlet

Calculate the mass flow rate at the inlet. Mass flow rate is given by the equation \(\rho AV\), where \(\rho\) is the fluid density, \(A\) is the cross-sectional area, and \(V\) is the velocity. Here, the velocity is \(V_{1}\) and area of the inlet is \(H\). Since the fluid is incompressible, \(\rho\) remains constant, thus the mass flow rate is proportional to \(HV_{1}\)
03

Determine expression for velocity at the outlet

At the outlet, the velocity profile \(V_{2}\) is given as \(V_{m} \cos \left(\frac{\pi y}{2 H}\right)\). The mass flow rate at the outlet must also be proportional to \(V_{1}\), so we'll integrate the velocity profile over the height of the outlet (0 to \(2H\)) to determine \(V_{m}\) in terms of \(V_{1}\).
04

Set up the integral for the outlet's velocity profile

Set up the integral from 0 to \(2H\) for the velocity profile \(V_{2}\).
05

Solve the integral and equate to the inlet flow

Evaluate the integral and get the result in terms of \(V_{m}\). Next, set up the expression from step 2 equal to the result of the integral to isolate \(V_{m}\) in terms of \(V_{1}\).
06

Solve for \(V_{m}\)

Solve the equation for \(V_{m}\) in terms of \(V_{1}\). This will give us the relationship between the maximum velocity at the outlet and the uniform velocity at the inlet of the channel.

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

\(\mathrm{A}\) rocket sled traveling on a horizontal track is slowed by a retro- rocket fired in the direction of travel. The initial speed of the sled is \(U_{0}=500 \mathrm{m} / \mathrm{s}\). The initial mass of the sled is \(M_{0}=1500 \mathrm{kg}\). The retro-rocket consumes fuel at the rate of 7.75 kgls, and the exhaust gases leave the nozzle at atmospheric pressure and a speed of \(2500 \mathrm{m} / \mathrm{s}\) relative to the rocket. The retro- rocket fires for 20 s. Neglect acrodynamic drag and rolling resistance. Obtain and plot an algebraic expression for sled speed \(U\) as a function of firing time. Calculate the sled speed at the end of retro-rocket firing.

A rocket sled with initial mass of 3 metric tons, including 1 ton of fuel, rests on a level section of track. At \(t=\) \(0,\) the solid fuel of the rocket is ignited and the rocket burns fuel at the rate of \(75 \mathrm{kg} / \mathrm{s}\). The exit speed of the exhaust gas relative to the rocket is \(2500 \mathrm{m} / \mathrm{s}\), and the pressure is atmospheric. Neglecting friction and air resistance, calculate the acceleration and speed of the sled at \(t=10\) s.

\( \mathrm{A}\) disk, of mass \(M,\) is constrained horizontally but is free to move vertically. A jet of water strikes the disk from below. The jet leaves the nozzle at initial speed \(V_{0}\), Obtain a differential equation for the disk height, \(h(t),\) above the jet exit plane if the disk is released from large height, \(H .(\mathrm{You}\) will not be able to solve this ODE, as it is highly nonlinear!) Assume that when the disk reaches equilibrium, its height above the jet exit plane is \(h_{0}\) (a) Sketch \(h(t)\) for the disk released at \(t=0\) from \(H>h_{0}\) (b) Explain why the sketch is as you show it.

A Venturi meter installed along a water pipe consists of a convergent section, a constant-area throat, and a divergent section. The pipe diameter is \(D=100 \mathrm{mm},\) and the throat diameter is \(d=50 \mathrm{mm}\). Find the net fluid force acting on the convergent section if the water pressure in the pipe is \(200 \mathrm{kPa}(\text { gage })\) and the flow rate is \(1000 \mathrm{L} / \mathrm{min}\). For this analysis, neglect viscous effects.

A centrifugal water pump with a 0.1-m-diameter inlet and a 0.1 -m-diameter discharge pipe has a flow rate of \(0.02 \mathrm{m}^{3} / \mathrm{s}\). The inlet pressure is \(0.2 \mathrm{m}\) Hg vacuum and the exit pressure is \(240 \mathrm{kPa}\). The inlet and outlet sections are located at the same elevation. The measured power input is \(6.75 \mathrm{kW}\). Determine the pump efficiency.
See all solutions

Recommended explanations on Physics Textbooks

Magnetism

Read Explanation

Fields in Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Classical Mechanics

Read Explanation

Electrostatics

Read Explanation

Scientific Method 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.