/*! 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 12 The flow field for an atmospheri... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 12

The flow field for an atmospheric flow is given by \\[\vec{V}=-\frac{K y}{2 \pi\left(x^{2}+y^{2}\right)} \hat{i}+\frac{K x}{2 \pi\left(x^{2}+y^{2}\right)}\\] where \(K=10^{5} \mathrm{m}^{2} / \mathrm{s},\) and the \(x\) and \(y\) coordinates are parallel to the local latitude and longitude. Plot the velocity magnitude along the \(x\) axis, along the \(y\) axis, and along the line \(y=x,\) and discuss the velocity direction with respect to these three axes. For each plot use a range \(x\) or \(y=-1 \mathrm{km}\) to \(1 \mathrm{km}\), excluding \(\mathrm{L} x\) I or \(|y|<100 \mathrm{m} .\) Find the equation for the streamlines and sketch several of them. What does this flow field model?

Short Answer

Expert verified
The velocity magnitude along all three axes shows singularities at origin due to the division by \(x^{2}+y^{2}\). Velocity directions change with respect to axes. Streamlines, when plotted, illustrate a circular pattern around origin that matches with a vortex flow, suggesting that this flow field could model atmospheric phenomena like storms or whirlwinds.

Step by step solution

01

Calculate the Velocity Magnitudes

First, the velocity magnitudes along the x, y and \(y = x\) axes need to be computed separately by substituting for y or x, from the given equations. For \(x\) axis, put \(y = 0\) and for \(y\) axis, put \(x = 0\). For \(y = x\) line, replace the \(y\) in the equation with \(x\). This will enable us to obtain the velocity magnitudes.
02

Plot the Velocity Magnitudes

Now plot the computed velocity magnitudes. We have to use a range of -1 km to 1 km for \(x\) and \(y\) (excluding \(|x|\) or \(|y|<100 m\)). Plotting software or basic graphing tools can be used to generate these plots.
03

Analyze Velocity Directions

With the velocity magnitude plots along the x, y, and \(y = x\) axes generated, discuss the velocity direction with respect to these three axes. This can be inferred by observing which direction the velocity vector is pointing on each axis.
04

Determine the Equation for Streamlines

The equation of streamlines is obtained by setting the \(dx/V_x = dy/V_y\). On rearranging, we obtain \(y =f(x)\) which is the streamline function. This relationship depicts the trajectory that a particle will follow in the flow field.
05

Sketch and Analyze the Streamlines

Sketch the streamlines represented by plotted functions using the equation we obtained in the previous step. Analyze the plot to understand what the flow field is. Analysis of the streamlines might indicate features like whirls which show where the wind rotates.
06

Identify the Flow Field Model

Finally, based on all calculations, plots, and with the use of physical intuition, identify what this flow field model would represent in real world scenarios. It could for example represent a whirlwind or tornado, or the flow around a pressure high in meteorology.

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Most popular questions from this chapter

A concentric-cylinder viscometer is shown. Viscous torque is produced by the annular gap around the inner cylinder. Additional viscous torque is produced by the flat bottom of the inner cylinder as it rotates above the flat bottom of the stationary outer cylinder. Obtain an algebraic expression for the viscous torque due to flow in the annular gap of width \(a\). Obtain an algebraic expression for the viscous torque due to flow in the bottom clearance gap of height \(b\). Prepare a plot showing the ratio, \(b / a,\) required to hold the bottom torque to 1 percent or less of the annulus torque, versus the other geometric variables. What are the design implications? What modifications to the design can you recommend?

Consider the flow field \(\vec{V}=a x t \hat{i}+b \hat{j},\) where \(a=0.1 \mathrm{s}^{-2}\) and \(b=4 \mathrm{m} / \mathrm{s}\). Coordinates are measured in meters. For the particle that passes through the point \((x, y)=(3,1)\) at the instant \(t=0,\) plot the pathline during the interval from \(t=0\) to 3 s. Compare this pathline with the streamlines plotted through the same point at the instants \(t=1,2,\) and 3 s.

A 73 -mm-diameter aluminum \((\mathrm{SG}=2.64)\) piston of 100 \(\mathrm{mm}\) length resides in a stationary 75 -mm-inner-diameter steel tube lined with SAE \(10 \mathrm{W}-30\) oil at \(25^{\circ} \mathrm{C}\). A mass \(m=2 \mathrm{kg}\) is suspended from the free end of the piston. The piston is set into motion by cutting a support cord. What is the terminal velocity of mass \(m ?\) Assume a linear velocity profile within the oil.

A flow is described by velocity field \(V=a i+b x j\), where \(a=2 \mathrm{m} / \mathrm{s}\) and \(b=1 \mathrm{s}^{-1}\). Coordinates are measured in meters. Obtain the equation for the streamline passing through point \((2,5) .\) At \(t=2 \mathrm{s},\) what are the coordinates of the particle that passed through point (0,4) at \(t=0 ?\) At \(t=3 \mathrm{s},\) what are the coordinates of the particle that passed through point (1,4.25) 2 s earlier? What conclusions can you draw about the pathline, streamline, and streakline for this flow?

The thin outer cylinder (mass \(m_{2}\) and radius \(R\) ) of a small portable concentric cylinder viscometer is driven by a falling mass, \(m_{1},\) attached to a cord. The inner cylinder is stationary. The clearance between the cylinders is \(a\). Neglect bearing friction, air resistance, and the mass of liquid in the viscometer. Obtain an algebraic expression for the torque due to viscous shear that acts on the cylinder at angular speed \(\omega\) Derive and solve a differential equation for the angular speed of the outer cylinder as a function of time. Obtain an expression for the maximum angular speed of the cylinder.
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