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Chapter 4: Problem 20

A constant-density fluid flows in the converging, twodimensional channel shown in Fig. P4.20. The width perpendicular to the paper is quite large compared to the channel height. The velocity in the \(z\) direction is zero. The channel half-height, \(Y\), and the fluid \(x\) velocity, \(u,\) are given by \\[ Y=\frac{Y_{0}}{1+x / \ell} \quad \text { and } \quad u=u_{0}\left(1+\frac{x}{\ell}\right)\left[1-\left(\frac{y}{Y}\right)^{2}\right] \\] Where \(x, y, Y,\) and \(\ell\) are in meters, \(u\) is in \(\mathrm{m} / \mathrm{s}, u_{0}=1.0 \mathrm{m} / \mathrm{s},\) and \(Y_{0}=1.0 \mathrm{m} .\) (a) Is this flow steady or unsteady? Is it one-dimensional, two-dimensional, or three- dimensional? (b) Plot the velocity distribution \(u(y)\) at \(x / \ell=0,0.5,\) and \(1.0 .\) Use \(y / Y\) values of 0 \\[ \pm 0.2,\pm 0.4,\pm 0.6,\pm 0.8, \text { and }\pm 1.0 \\]

Short Answer

Expert verified
The flow is steady and two-dimensional. The velocity distribution can be plotted as per computations using given equations and substitutions of given values of y/Y at given values of x/\( \ell \).

Step by step solution

01

Identify the nature of the fluid flow

From the given equations, check whether any parameter (i.e., \( Y, u \)) is dependent on time. If none, then the flow is steady, otherwise, it's unsteady.
02

Identify the dimensional characteristics

The given variables are checked, and if only one variable (either \( X, Y, \) or \( Z \)) has significant variation relative to the others, it is One-dimensional flow. If two have significant variation, it's Two-dimensional flow. If all three have significant variations, then it is Three-dimensional flow.
03

Compute the velocity distribution for given x/\( \ell \) values at y/Y=0, ±0.2, ±0.4, ±0.6, ±0.8, and ±1.0.

Use the given equation for u and substitute the required values and obtain the corresponding velocity distribution.
04

Plot the velocity distribution

With the given y/Y values and the calculated velocity distributions, prepare a plot for the velocity distribution \( u(y) \) at \( x / \ell = 0, 0.5, and 1.0 \).

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Most popular questions from this chapter

Classify the following flows as one- , two-, or three-dimensional. Sketch a few streamlines for each. (a) Rainwater flow down a wide driveway (b) Flow in a straight horizontal pipe (c) Flow in a straight pipe inclined upward at a \(45^{\circ}\) angle (d) Flow in a long pipe that follows the ground in hilly country (e) Flow over an airplane (f) Wind blowing past a tall telephone (g) Flow in the impeller of a centrifugal pump

A gas flows along the \(x\) axis with a speed of \(V=5 x \mathrm{m} / \mathrm{s}\) and eressure of \(p=10 x^{2} \mathrm{N} / \mathrm{m}^{2},\) where \(x\) is in meters. (a) Determine the time rate of change of pressure at the fixed location \(x=1\) (b) Determine the time rate of change of pressure for a fluid particle flowing past \(x=1 .\) (c) Explain without using any equations why the answers to parts (a) and (b) are different.

The velocity components of \(u\) and \(v\) of a two-dimensional flow are given by \\[ u=a x+\frac{b x}{x^{2} y^{2}} \quad \text { and } \quad v=a y+\frac{b y}{x^{2} y^{2}} \\] where \(a\) and \(b\) are constants. Calculate the acceleration.

Figure \(P 4.67\) illustrates a system and fixed control volume at time \(t\) and the system at a short time \(\delta t\) later. The system temperature is \(T=100^{\circ} \mathrm{F}\) at time \(t\) and \(T=103^{\circ} \mathrm{F}\) at time \(t+\delta t,\) where \(\delta t=0.1 \mathrm{s} .\) The system mass, \(m,\) is 2.0 slugs, and 10 percent of it moves out of the control volume in \(\delta t=0.1\) s. The energy per unit mass \(\tilde{u}\) is \(c_{v} T,\) where \(c_{v}=32.0\) Btu/slug \(\cdot^{\circ} \mathrm{F}\). The energy \(U\) of the system at any time \(t\) is \(m \tilde{u}\). Use the system or Lagrangian approach to evaluate \(D U / D t\) of the system. Compare this result with the \(D U / D t\) evaluated with that of the material time derivative and flux terms.

Water flows through a constant diameter pipe with a uniform velocity given by \(\mathbf{V}=(8 / t+5) \hat{\mathbf{j}} \mathrm{m} / \mathrm{s},\) where \(t\) is in seconds. Determine the acceleration at time \(t=1,2,\) and \(10 \mathrm{s}\)
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