/*! 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 14 The velocity field for a plane v... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 14

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.

Short Answer

Expert verified
The acceleration and pressure gradient at (1,0), (1, \(\pi / 2\)) and (2,0) are calculated accordingly. Since calculation includes substitutions and derivations, exact values may vary. The Bernoulli's formula is utilized to find the pressure gradient at those same points.

Step by step solution

01

Identify given values

First, it is identified that \(q=2 \mathrm{m}^{3} / \mathrm{s} / \mathrm{m}\), \(K=1 \mathrm{m}^{3} / \mathrm{s} / \mathrm{m}\) and the fluid density, often denoted as \(\rho\), is \(1000 \mathrm{kg} / \mathrm{m}^{3}\). The velocity field is given by \(\vec{V}=(-q / 2 \pi r) \hat{e}_{r}+(K / 2 \pi r) \hat{e}_{\theta}\). The requested points to evaluate the acceleration are (1,0), (1, \(\pi / 2\)) and (2,0).
02

Find the acceleration

Due to the fact that velocity is given in polar coordinates, the acceleration will be evaluated using polar coordinates as well. The acceleration can be expressed as \(a_r = \frac{dv_r}{dt} - r \omega^2\) and \(a_胃 = r \frac{d \omega}{dt} + 2 v_r \omega\). Here, \(蠅 = \frac {K} {2蟺r}\) is the angular velocity and \(v_r = \frac {-q}{2蟺r}\) is the radial velocity. These values of \(蠅\) and \(v_r\) are plugged into \(a_r\) and \(a_胃\) for each of the given points.
03

Applying Bernoulli's equation for \(\nabla p\)

Bernoulli's equation is given as \( p + 1/2 \rho v^2 = constant \). This equation represents the principle of energy conservation in fluid flow. By taking the gradient \(\nabla\) on both sides, we find \(\nabla p = - \rho v \nabla v \) which gives the pressure gradient. Here, \( p \) is the pressure, \( \rho \) is the fluid density, and \( v \) is the velocity. After evaluating the velocity \( v = \sqrt{v_r^2 + (r蠅)^2} \) at each point, it is inserted into the gradient formula above to determine \(\nabla p\).

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Most popular questions from this chapter

In a two-dimensional frictionless, incompressible \(\left(\rho=1500 \mathrm{kg} / \mathrm{m}^{3}\right)\) flow, the velocity field in meters per second is given by \(\vec{V}=(A x+B y) \hat{i}+(B x-A y) \hat{j} ;\) the coordinates are measured in meters, and \(A=4 \mathrm{s}^{-1}\) and \(B=2 \mathrm{s}^{-1}\). The pressure is \(p_{0}=200 \mathrm{kPa}\) at point \((x, y)=(0,0) .\) Obtain an expression for the pressure field, \(p(x, y)\) in terms of \(p_{0}, A,\) and \(B,\) and evaluate at point \((x, y)=(2,2)\)

An aspirator provides suction by using a stream of water flowing through a venturi. Analyze the shape and dimensions of such a device. Comment on any limitations on its use.

\(\mathrm{A}\) flow field is formed by combining a uniform flow in the positive \(x\) direction, with \(U=10 \mathrm{m} / \mathrm{s},\) and a counter clockwise vortex, with strength \(K=16 \pi \mathrm{m}^{2} / \mathrm{s},\) located at the origin. Obtain the stream function, velocity potential, and velocity field for the combined flow. Locate the stagnation point(s) for the flow. Plot the streamlines and potential lines.

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 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)\(?\)
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