/*! 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 114 Consider the flow field represen... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 114

Consider the flow field represented by the velocity potential \(\quad \phi=A x+B x^{2}-B y^{2}, \quad\) where \(\quad A=1 \quad \mathrm{m} \cdot \mathrm{s}^{-1}\) \(B=1 \mathrm{m}^{-1} \cdot \mathrm{s}^{-1},\) and the coordinates are measured in meters. Obtain expressions for the velocity field and the stream function. Calculate the pressure difference between the ori\(\operatorname{gin}\) and point \((x, y)=(1,2)\)

Short Answer

Expert verified
The velocity field is \(V = (A + 2Bx)i - 2Byj\) and the stream function is \(\psi = - Ay + Bxy\). The pressure difference can be calculated using the Bernoulli's principle.

Step by step solution

01

Velocity Field Derivation

Using the definition of velocity potential, recognizing that the velocity of a fluid particle in the field is the gradient of the velocity potential, the velocity field \(V\) can be obtained. The x-component, \(V_x\), and y-component, \(V_y\), are the partial derivatives of the potential with respect to \(x\) and \(y\) respectively.\[V_x = \frac{\partial \phi}{\partial x}\]\[V_y = \frac{\partial \phi}{\partial y}\]Substitute \(\phi = Ax + Bx^2 - By^2\)After differentiation, we get: \[V_x = A + 2Bx\]\[V_y = -2By \]So, the velocity field \(V\) is \(V = (A + 2Bx)i - 2Byj\)
02

Stream Function Derivation

The stream function \( \psi \) can be obtained by using the fact that for a two-dimensional, incompressible fluid flow, the velocity field and stream function are related as follows:\[V_x = \frac{\partial \psi}{\partial y}\]\[V_y = -\frac{\partial \psi}{\partial x}\]Which are partial differential equations. Solving it, we get:\[\psi = -Ay + Bxy + f(x)\]Where \( f(x) \) is an arbitrary function of \(x\). By substiting the value of \(V_y\) into the above equation and further integrating:\[\psi = - Ay + Bxy\]
03

Pressure Difference Calculation

In fluid dynamics, the Bernoulli's equation can be applied under certain conditions, relating pressure, fluid speed, and height. \[P = P_0 + \frac{1}{2} \rho V^2\]where \(P_0\) is the pressure at reference point, \(\rho\) is the density and \(V\) is velocity magnitude at the given point. The pressure difference can be obtained from the Bernoulli's equation by substituting the values of velocity field at the origin and at point (1,2). \[\Delta P = \frac{1}{2} \rho (V_{1,2}^2 - V_{0,0}^2)\]where \(V_{1,2}\) is the magnitude of velocity at point (1,2) and \(V_{0,0}\) is the magnitude at origin. After calculating these values and substiting the correct values, the pressure difference can be obtained.

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

\(\mathrm{A}\) horizontal flow of water is described by the velocity field \(\vec{V}=(-A x+B t) \hat{i}+(A y+B t) \hat{j},\) where \(A=1 \mathrm{s}^{-1}\) and \(B=2 \mathrm{m} / \mathrm{s}^{2}, x\) and \(y\) are in meters, and \(t\) is in seconds. Find expressions for the local acceleration, the convective acceleration, and the total acceleration. Evaluate these at point (1,2) at \(t=5\) seconds. Evaluate \(\nabla p\) at the same point and time.

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 \(50 \mathrm{kPa}(\text { gage })\)

A liquid layer separates two plane surfaces as shown. The lower surface is stationary; the upper surface moves downward at constant speed \(V\). The moving surface has width \(w,\) perpendicular to the plane of the diagram, and \(w \gg L .\) The incompressible liquid layer, of density \(\rho,\) is squeezed from between the surfaces. Assume the flow is uniform at any cross section and neglect viscosity as a first approximation. Use a suitably chosen control volume to show that \(u=V x / b\) within the gap, where \(b=b_{0}-V t .\) Obtain an algebraic expression for the acceleration of a fluid particle located at \(x .\) Determine the pressure gradient, \(\partial p / \partial x,\) in the liquid layer. Find the pressure distribution, \(p(x) .\) Obtain an expression for the net pressure force that acts on the upper (moving) flat surface.

A horizontal axisymmetric jet of air with 0.4 in. diameter strikes a stationary vertical disk of 7.5 in. diameter. The jet speed is \(225 \mathrm{ft} / \mathrm{s}\) at the nozzle exit. A manometer is connected to the center of the disk. Calculate (a) the deflection, if the manometer liquid has \(\mathrm{SG}=1.75,(\mathrm{b})\) the force exerted by the jet on the disk, and (c) the force exerted on the disk if it is assumed that the stagnation pressure acts on the entire forward surface of the disk. Sketch the streamline pattern and plot the distribution of pressure on the face of the disk.

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)\(?\)
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Scientific Method Physics

Read Explanation

Kinematics Physics

Read Explanation

Translational Dynamics

Read Explanation

Magnetism

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.