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Chapter 2: Problem 33

A flow is described by velocity field \(\vec{V}=a x \hat{i}+b \hat{j},\) where \(a=1 / 5 \mathrm{s}^{-1}\) and \(b=1 \mathrm{m} / \mathrm{s}\). Coordinates are measured in meters. Obtain the equation for the streamline passing through point \((1,1) .\) At \(t=5 \mathrm{s}\), what are the coordinates of the particle that initially (at \(t=0\) ) passed through point (1,1)\(?\) What are its coordinates at \(t=10\) s? Plot the streamline and the initial, \(5 \mathrm{s}\), and 10 s positions of the particle. What conclusions can you draw about the pathline, streamline, and streakline for this flow?

Short Answer

Expert verified
The streamline equation passing through (1,1) is \(x=\sqrt{2b/a(y+(a/(2b)))}\). The coordinates of the particle after 5s are \((1+5a, 1+5b)\) and after 10s are \((1+10a, 1+10b)\). The pathline, streamline and streakline all coincide for this steady flow.

Step by step solution

01

Calculate the streamline equation

The streamline equation can be found by integrating the velocity field equations \(dx/V_{x}=dy/V_{y}\). From the given velocity field \(\vec{V}=a x \hat{i}+b \hat{j}\), we get \(dx/(a x) = dy/b\). On integrating both sides of the equation the streamline equation will be \(x=\sqrt{2b/a(y+C)}\) where C is a constant. To find the constant C, use the given point (1,1). After substituting we get the equation of the streamline through the point (1,1) as \(x=\sqrt{2b/a(y+(a/(2b)))}\).
02

Find the coordinates of the particle after specific times

The coordinates of the particle after any time t can be found by integrating the velocity field equations considering that \(dx/dt=V_{x}\) and \(dy/dt=V_{y}\), where dx and dy are the changes in the x and y coordinates of the particle respectively. After integrating and using the initial point (1,1), the coordinates of the particle after any time t would be \((x(t), y(t))=(1+at, 1+bt)\). Using this, we can find the coordinates of the particle after 5s and 10s.
03

Plot the streamline and particle positions

Plot the streamline obtained in step 1 and the positions of particle obtained in step 2 on the same diagram. The particle's initial position, and the positions after 5s and 10s should all be plotted on the streamline.
04

Discuss the relationship between pathline, streamline, and streakline

From the plot and the calculations, observe and describe the relationship between the pathline (the actual path traced by a fluid particle), the streamline (line that is everywhere tangent to the velocity field), and the streakline (line formed by all particles that have passed through a particular spatial point). For this particular exercise, the pathline is the curve traced by the particle (from step 2), and the streamline is the line obtained in step 1. In this case of steady flow, the pathline, streamline, and streakline should all coincide.

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Most popular questions from this chapter

The cone and plate viscometer shown is an instrument used frequently to characterize non-Newtonian fluids. It consists of a flat plate and a rotating cone with a very obtuse angle (typically \(\theta\) is less than 0.5 degrees). The apex of the cone just touches the plate surface and the liquid to be tested fills the narrow gap formed by the cone and plate. Derive an expression for the shear rate in the liquid that fills the gap in terms of the geometry of the system. Evaluate the torque on the driven cone in terms of the shear stress and geometry of the system.

A flow is described by the velocity field \(\vec{V}=(A x+B)\) is \((-A y) j,\) where \(A=10\) ft/s/ft and \(B=20\) ft/s. Plot a few streamlines in the \(x y\) plane, including the one that passes through the point \((x, y)=(1,2)\).

For the velocity fields given below, determine: a. whether the flow field is one-, two-, or three-dimensional, and why. b. whether the flow is steady or unsteady, and why. (The quantities \(a\) and \(b\) are constants.) (1) \(\vec{V}=\left[(a x+t) e^{\log y}\right]\) (2) \(\vec{V}=(a x-b y)\) (3) \(\vec{V}=a x \vec{i}+\left[e^{\ln x}\right]\) (4) \(\vec{V}=a x \hat{i}+b x^{2} j+a x k\) (5) \(\vec{V}=a x \hat{i}+\left[e^{b r}\right] \dot{j}\) (6) \(\vec{V}=a x \hat{i}+b x^{2} \hat{j}+a y k\) (7) \(\vec{V}=a x \hat{i}+\left[e^{\ln }\right] \hat{j}+a y \hat{k}\) (8) \(\vec{V}=a x i+\left[e^{b y}\right] j+a z \hat{k}\)

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