/*! 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 33 The velocity field in a two-dime... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 33

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)\(?\)

Short Answer

Expert verified
The key steps in solving this exercise are to derive the equation of streamlines, plot these streamlines for select points, compute velocity and acceleration at a given point, and then find the pressure gradient at that point. The pressure at two given points can be compared.

Step by step solution

01

Derive the Streamline Equation

From the given velocity field \(\vec{V}=(A x+B) \hat{i}-A y \hat{j}\), where A and B are constants, we can equate the two components of the velocity field to zero to get the equations of streamline, which are \(dx / (Ax + B) = dy / (-Ay)\). Using the technique of separation of variables and integrating both sides, we get \(ln |x + B/A| = - ln |y| + constant\), or after taking exponent of both sides, \(|x + B/A| = 1 / |y|\), implying that the streamlines are given by \((x + B/A) y = constant\).
02

Plot Streamlines

For the given points (1,1), (1,2) and (2,2), when we substitute into the streamline equation \((x + B/A) y = constant\), the streamlines passing through these points can be plotted in the XY plane.
03

Compute Velocity and Acceleration Vector

The velocity vector at a given point is simply the given velocity field evaluated at that point. So at the point (1,2), the velocity vector is \((A(1)+B) \hat{i}-A(2) \hat{j}\). The acceleration vector can be obtained by taking the derivative of the velocity vector with respect to time.
04

Evaluate Component of Acceleration

The component of the acceleration along the streamline at the point (1,2) can be obtained by taking the dot product of the acceleration vector with the unit tangent vector of the streamline at that point.
05

Evaluate Pressure Gradient

The pressure gradient at the point (1,2) can be obtained by applying Bernoulli鈥檚 equation along the streamline.
06

Discuss Relative Pressure

Based on the computed pressure gradients and the assumption that the fluid is air (which is incompressible at low speeds), we can make a statement about the relative value of the pressure at points (1,1) and (2,2).

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Most popular questions from this chapter

A velocity field is given by \(\vec{V}=\left[A x^{3}+B x y^{2}\right] \hat{i}+\) \(\left[A y^{3}+B x^{2} y\right] \hat{j} ; A=0.2 \mathrm{m}^{-2} \cdot \mathrm{s}^{-1}, B\) is a constant, and the coordinates are measured in meters. Determine the value and units for \(B\) if this velocity field is to represent an incompressible flow. Calculate the acceleration of a fluid particle at point \((x, y)=(2,1) .\) Evaluate the component of particle acceleration normal to the velocity vector at this point.

Determine whether the Bernoulli equation can be applied between different radii for the vortex flow fields (a) \(\vec{V}=\omega r \hat{e}_{\theta}\) and (b) \(\vec{V}=\hat{e}_{\theta} K / 2 \pi r\)

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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 \(1.2 \mathrm{kg} / \mathrm{m}^{3}\)

Show by expanding and collecting real and imaginary terms that \(\left.f=z^{6} \text { (where } z \text { is the complex number } z=x+i y\right)\) leads to a valid velocity potential (the real part of \(f\) ) and a corresponding stream function (the negative of the imaginary part of \(f\), an irrotational and incompressible flow. Then show that the real and imaginary parts of \(d f / d z\) yield \(-u\) and \(v,\) respectively.
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