/*! 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 118 The velocity distribution in a t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 118

The velocity distribution in a two-dimensional, steady, inviscid flow field in the \(x y \quad\) plane is \(\vec{V}=(A x+B) \hat{i}+(C-A y) \hat{j}, \quad\) where \(A=3 \quad \mathrm{s}^{-1}, \quad B=6 \mathrm{m} / \mathrm{s}\) \(C=4 \mathrm{m} / \mathrm{s},\) and the coordinates are measured in meters. The body force distribution is \(\vec{B}=-g \hat{k}\) and the density is \(825 \mathrm{kg} / \mathrm{m}^{3} .\) Does this represent a possible incompressible flow field? Plot a few streamlines in the upper half plane. Find the stagnation point(s) of the flow field. Is the flow irrotational? If \(\mathrm{so},\) obtain the potential function. Evaluate the pressure difference between the origin and point \( (x, y, z)=(2,2,2) \)

Short Answer

Expert verified
The given flow field is incompressible, with a stagnation point at (-2, 4/3). However, it is not irrotational. Without further information, the pressure difference between two points cannot be determined.

Step by step solution

01

Test for Incompressibility

Matter is incompressible, if the divergence of its velocity field is zero. Therefore, for the velocity field \(\vec{V} = (Ax + B) \hat{i} + (C - Ay) \hat{j}\) to be incompressible, \(\nabla \cdot \vec{V} = 0\). Calculating, we get \(\partial (Ax + B) / \partial x + \partial (C - Ay) / \partial y = 0\), which simplifies to \(A - A = 0\). So, the flow field is indeed incompressible.
02

Plotting Streamlines

To plot the streamlines, the differential equation for the streamline in 2D is solved. This is dy/dx = Vx/Vy, where Vx and Vy are the x and y components of the velocity. After substitution, we get dy/dx = (3x + 6) / (4 - 3y). This can be solved numerically for the streamlines.
03

Calculating Stagnation points

The stagnation point(s) in a flow field occur where the velocity goes to zero. Thus, we set the x and y components of the velocity \(\vec{V}\) as zero and solve for x and y. This gives (3x + 6) = 0 and (4 - 3y) = 0, that implies x = -2 and y = 4/3. Therefore the stagnation point is (-2, 4/3)
04

Test for Irrotationality

A flow is irrotational if its curl, \(\nabla \times \vec{V}\), is equal to zero. Calculating the curl, we get \(\nabla \times \vec{V} = \mathrm{k}( - \partial (Ax + B) / \partial y + \partial (C - Ay) / \partial x)\) which reduces to \(\mathrm{k}(A + A) = 2A \mathrm{k} \). As the curl is not zero, the flow is not irrotational.
05

Find the pressure difference

Since the velocity potential \(\phi\) cannot be obtained as the flow is not irrotational, Bernoulli's equation cannot be applied directly. Bernoulli's equation includes changes in height and affects both pressure and flow velocities. Since the point is within the fluid, and there is a difference in z coordinates, we need more information about the behavior of the fluid in the z direction apart from its body force distribution (like z component of velocity and z dependence of pressure) in order to find the pressure difference.

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

Imagine a garden hose with a stream of water flowing out through a nozzle. Explain why the end of the hose may be unstable when held a half meter or so from the nozzle end.

A diffuser for an incompressible, inviscid fluid of density \(\rho=1000 \mathrm{kg} / \mathrm{m}^{3}\) consists of a diverging section of pipe. At the inlet the diameter is \(D_{i}=0.25 \mathrm{m},\) and at the outlet the diameter is \(D_{o}=0.75 \mathrm{m} .\) The diffuser length is \(L=1 \mathrm{m},\) and the diameter increases linearly with distance \(x\) along the diffuser. Derive and plot the acceleration of a fluid particle, assuming uniform flow at each section, if the speed at the inlet is \(V_{i}=5 \mathrm{m} / \mathrm{s}\). Plot the pressure gradient through the diffuser, and find its maximum value. If the pressure gradient must be no greater than \(25 \mathrm{kPa} / \mathrm{m}\), how long would the diffuser have to be?

The \(x\) component of velocity in a two-dimensional, incompressible flow field is given by \(u=A x y ;\) the coordinates are measured in meters and \(A=2 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1} .\) There is no velocity component or variation in the \(z\) direction. Calculate the acceleration of a fluid particle at point \((x, y)=(2,1) .\) Estimate the radius of curvature of the streamline passing through this point. Plot the streamline and show both the velocity vector and the acceleration vector on the plot. (Assume the simplest form of the \(y\) component of velocity.)

An incompressible liquid with a density of \(900 \mathrm{kg} / \mathrm{m}^{3}\) and negligible viscosity flows steadily through a horizontal pipe of constant diameter. In a porous section of length \(L=2 \mathrm{m}\) liquid is removed at a variable rate along the length so that the uniform axial velocity in the pipe is \(u(x)=U e^{-x / L},\) where \(U=20 \mathrm{m} / \mathrm{s} .\) Develop expressions for and plot the acceleration of a fluid particle along the centerline of the porous section and the pressure gradient along the centerline. Evaluate the outlet pressure if the pressure at the inlet to the porous section is \(50 \mathrm{kPa}(\text { gage })\)

The stream function of a flow field is \(\psi=A x^{3}+\) \(B\left(x y^{2}+x^{2}-y^{2}\right),\) where \(\psi, x, y, A,\) and \(B\) are all dimensionless. Find the relation between \(A\) and \(B\) for this to be an irrotational flow. Find the velocity potential.
See all solutions

Recommended explanations on Physics Textbooks

Nuclear Physics

Read Explanation

Kinematics Physics

Read Explanation

Waves Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Atoms and Radioactivity

Read Explanation

Physics of Motion

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.