/*! 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 63 Steady, frictionless, and incomp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 63

Steady, frictionless, and incompressible flow from left to right over a stationary circular cylinder, of radius \(a,\) is represented by the velocity field \\[ \vec{V}=U\left[1-\left(\frac{a}{r}\right)^{2}\right] \cos \theta \hat{e}_{r}-U\left[1+\left(\frac{a}{r}\right)^{2}\right] \sin \theta \hat{e}_{\theta} \\] Obtain an expression for the pressure distribution along the streamline forming the cylinder surface, \(r=a\). Determine the locations where the static pressure on the cylinder is equal to the freestream static pressure.

Short Answer

Expert verified
The pressure distribution along the streamline forming the cylinder surface, \(r=a\), is given by \( P=P_{\infty}+ a \times \frac{U^{2}}{2} \times\left(2 \cos ^{2} \theta-1\right)\). The static pressure on the cylinder is equal to the freestream static pressure when \(\theta = \pi/4\) or \(\theta = 3\pi/4\) respectively.

Step by step solution

01

Use Bernoulli's Equation

Let's begin with using Bernoulli鈥檚 principle which states that for an inviscid flow, the sum of pressure, kinetic energy and potential energy per unit volume is constant. If we ignore gravitational effects, it simplifies to the formula \(P_{0}=P+\frac{1}{2} \rho V^{2}\), where \(P_{0}\) is the constant total pressure, P is the variable pressure, \(\rho\) is the fluid density, and V is the speed derived from the velocity field.
02

Find the fluid speed

We first need to find the speed of the fluid at the cylinder surface, which is \(r=a\). Plug this values into the given velocity field expression to get the speed on the cylinder surface to be \( V = U \sqrt{2 (1-\cos^2\theta)} \).
03

Solve for Pressure Variation

Let's substitute the given fluid speed into Bernoulli's equation: \[ P=a\times \frac{U^{2}}{2} \times\left(2 \cos ^{2} \theta-1\right) \].
04

Find the Locations of Equal Pressure

We now equate this with the freestream pressure \(P_{\infty}\) and solve to get the locations: This occurs when the value \(\cos^{2}\theta = 1/2\). The solutions to this equation are where \(\theta = \pi/4\) or \(\theta = 3\pi/4\).

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

An incompressible liquid with negligible viscosity and density \(\rho=1250 \mathrm{kg} / \mathrm{m}^{3}\) flows steadily through a \(5-\mathrm{m}\) -long convergent-divergent section of pipe for which the area varies as \\[ A(x)=A_{0}\left(1+e^{-x / a}-e^{-x / 2 a}\right) \\] where \(A_{0}=0.25 \mathrm{m}^{2}\) and \(a=1.5 \mathrm{m} .\) Plot the area for the first \(5 \mathrm{m} .\) Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, for the first \(5 \mathrm{m}\) if the inlet centerline velocity is \(10 \mathrm{m} / \mathrm{s}\) and inlet pressure is 300 kPa. Hint: Use relation \\[ u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right) \\]

An incompressible liquid with negligible viscosity and density \(\rho=1.75\) slug/ft \(^{3}\) flows steadily through a horizontal pipe. The pipe cross- section area linearly varies from 15 in \(^{2}\) to 2.5 in \(^{2}\) over a length of 10 feet. Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, if the inlet centerline velocity is \(5 \mathrm{ft} / \mathrm{s}\) and inlet pressure is 35 psi. What is the exit pressure? Hint: Use relation \\[ u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right) \\]

The velocity field in a two-dimensional, steady, inviscid flow field in the horizontal \(x y\) plane is given by \(\vec{V}=(A x+B) \hat{i}-A y \hat{j},\) where \(A=1 \mathrm{s}^{-1}\) and \(B=2 \mathrm{m} / \mathrm{s} ; x\) and \(y\) are measured in meters. Show that streamlines for this flow are given by \((x+B / A) y=\) constant. Plot streamlines passing through points \((x, y)=(1,1),(1,2),\) and \((2,2) .\) At point \((x, y)=(1,2),\) evaluate and plot the acceleration vector and the velocity vector. Find the component of acceleration along the streamline at the same point; express it as a vector. Evaluate the pressure gradient along the streamline at the same point if the fluid is air. What statement, if any, can you make about the relative value of the pressure at points (1,1) and (2,2)\(?\)

The velocity distribution in a two-dimensional steady flow field in the \(x y\) plane is \(\vec{V}=(A x-B) \hat{i}+(C-A y) \hat{j}\) where \(A=2 \mathrm{s}^{-1}, B=5 \mathrm{m} \cdot \mathrm{s}^{-1},\) and \(\mathrm{C}=3 \mathrm{m} \cdot \mathrm{s}^{-1} ;\) the coordinates are measured in meters, and the body force distribution is \(\vec{g}=-g \hat{k} .\) Does the velocity field represent the flow of an incompressible fluid? Find the stagnation point of the flow field. Obtain an expression for the pressure gradient in the flow field. Evaluate the difference in pressure between point \((x, y)=(1,3)\) and the origin, if the density is \(1.2 \mathrm{kg} / \mathrm{m}^{3}\)

The velocity field for a plane vortex sink is given by \(\vec{V}=(-q / 2 \pi r) \hat{e}_{r}+(K / 2 \pi r) \hat{e}_{\theta}, \quad\) where \(\quad q=2 \quad \mathrm{m}^{3} / \mathrm{s} / \mathrm{m} \quad\) and \(K=1 \mathrm{m}^{3} / \mathrm{s} / \mathrm{m} .\) The fluid density is \(1000 \mathrm{kg} / \mathrm{m}^{3} .\) Find the acceleration at \((1,0),(1, \pi / 2),\) and \((2,0) .\) Evaluate \(\nabla p\) under the same conditions.
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Turning Points in Physics

Read Explanation

Fields in Physics

Read Explanation

Torque and Rotational Motion

Read Explanation

Electrostatics

Read Explanation

Radiation

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.