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Chapter 8: Problem 30

When soup is stirred in a bowl, there is considerable turbulence in the resulting motion (see Video \(\mathrm{V} 8.7\) ). From a very simplistic standpoint, this turbulerce consists of numerous intertwined swirls, each involving a characteristic diameter and velocity. As time goes by, the smaller swirls (the fine scale structure) die out relatively quickly, leaving the large swirls that continue for quite some time. Explain why this is to be expected.

Short Answer

Expert verified
The smaller swirls dissipate faster due to their higher surface area to volume ratio, which leads to higher energy loss through friction. Larger swirls have more energy and lose it more slowly due to their smaller surface area to volume ratio, allowing them to persist longer. This is consistent with the principles of energy conservation and fluid dynamics.

Step by step solution

01

Understanding Turbulence

Turbulence is chaotic or irregular fluid flow characterized by the presence of eddies and swirls. It occurs in various forms - from tiny, irregular fluctuations to large turbulent eddies. When stirring soup, the turbulence created involves numerous interlinked swirls, each having a unique diameter and velocity.
02

Differing Life-Spans of Swirls

During the stirring, smaller swirls or 'fine scale structure' appear, which die out relatively quickly. On the contrary, larger swirls continue for a more extended period. The question is why this happens. One must consider the energy involved in these swirls to understand this.
03

Energy Dissipation in Swirls

Each swirl (or eddy) contains energy. Smaller swirls have less energy compared to larger ones due to their smaller mass and size. Moreover, due to their higher surface area to volume ratio, they have larger interface with the surrounding fluid. This leads to a higher rate of energy dissipation through friction (viscous drag), causing them to 'die out' faster.
04

Persistence of Larger Swirls

Conversely, larger swirls have more energy and a smaller surface area to volume ratio. This means they lose energy at a slower rate through viscous drag. Consequently, larger swirls persist much longer compared to the smaller ones.

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Most popular questions from this chapter

A fluid flows through a horizontal 0.1-in.-diameter pipe. When the Reynolds number is \(1500,\) the head loss over a \(20-f t\) lenzth of the pipe is \(6.4 \mathrm{ft}\). Determine the fluid velocity.

Asphalt at \(120^{\circ} \mathrm{F}\), considered to be a Newtonian fluid with a viscosity 80,000 times that of water and a specific gravity of 1.09 flows through a pipe of diameter 2.0 in. If the pressure gradient is 1.6 psi/ft determine the flowrate assuming the pipe is (a) horizontal; (b) vertical with flow up.

The Churchill formula for the friction factor is $$f=8\left[\left(\frac{8}{\mathrm{Re}}\right)^{12}+\frac{1}{(A+B)^{15}}\right]^{1 / 12}$$ where $$\begin{array}{l} A=\left\\{-2.457 \ln \left[\left(\frac{7}{\mathrm{Re}}\right)^{0.9}+\frac{\varepsilon}{3.7 D}\right]\right\\}^{16} \\ B=\left(\frac{37.530}{\mathrm{Re}}\right)^{16} \end{array}$$ Compare this equation for \(f\) for both the laminar and turbulent regimes for \(\varepsilon / D=0.00001,0.0001,0.001,\) and 0.01 and Reynolds numbers of \(10,10^{2}, 10^{3}, 10^{4}, 10^{5}, 10^{6},\) and \(10^{7}\) with the Moody chart and decide whether it is an acceptable replacement for the Colebrook formula.

For fully developed laminar pipe flow in a circular pipe, the velocity profile is given by \(u(r)=2\left(1-r^{2} / R^{2}\right)\) in \(\mathrm{m} / \mathrm{s},\) where \(R\) is the inner radius of the pipe. Assuming that the pipe diameter is \(4 \mathrm{cm},\) find the maximum and average velocities in the pipe as well as the volume flow rate.

The vented storage tank shown in Fig. \(\mathrm{P} 8.90\) is used to refuel race cars at a race track. A total of \(40 \mathrm{ft}\) of steel pipe \((\mathrm{I.D},=0.957 \text { in. })\) two \(90^{\circ}\) regular elbows, and a globe valve make up the system. Calculate the time needed to put 20 gal of fuel in a car tank. The pressures, \(p_{2}\) and \(p_{1},\) are aqual, the connections are threaded, and the fuel has the properties of \(68^{\circ} \mathrm{F}\) normal octane \(\left(\nu=8.31 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}\right)\)
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