A rotating viscometer consists of two concentric cylinders - an inner cylinder of radius $R_i$ rotating at angular velocity (rotation rate) $\omega_r$, and a stationary outer cylinder of inside radius $R_o$. In the tiny gap between the two cylinders is the fluid of viscosity $\mu$. The length of the cylinders is $L . L$ is large such that end effects are negligible (we can treat this as a two-dimensional problem). Torque (T) is required to rotate the inner cylinder at constant speed. (a) Showing all of your work and algebra, generate an approximate expression for $\mathrm{T}$ as a function of the other variables. (b) Explain why your solution is only an approximation. In particular, do you expect the velocity profile in the gap to remain linear as the gap becomes larger and larger (i.e., if the outer radius $R_o$ were to increase, all else staying the same)?

Discuss stream speed profiles and provide a justification for the two measurement methods described next. (Adapted from Herschy, 1995) The speed of water in a channel varies considerably with depth. Due to friction, the speed reaches zero at the bottom and along the sides of the channel. The speed is greatest near the surface of the stream. To find the average speed for the vertical speed profile, two methods are frequently employed in practice: 1. The 0.6 depth method: The speed is measured at 0.6 of the depth from the surface, and this value is taken as the average speed. 2. The 0.2 and 0.8 depth method: The speed is measured at 0.2 and 0.8 of the depth from the surface, and the average of the two readings is taken as the average speed. The theoretical speed distribution of water flowing in an open channel is given approximately by $v(d)=\left(\frac{D-d}{a}\right)^{1 / c} \quad (6.19)$ where v(d) is the speed at depth d below the water surface, c is a constant varying from 5 for coarse beds to 7 for smooth beds, D is the total depth of the channel, and a is a constant that is equal to the distance above the bottom of the channel at which the speed has unit value. (a) Show by integration that the average speed $\bar{v}$ in the vertical profile is given by $\bar{v}=\frac{c}{c+1}\left(\frac{D}{a}\right)^{1 / c} \quad (6.20)$ (b) What fraction of the maximum speed is the average speed $\bar{v} ?$ (c) If you knew that the maximum speed occurred at the surface of the river [as predicted in the approximate formula for v(d)], how could you find $\bar{v} ?$ (In practice, the maximum speed may occur quite a bit below the surface due to friction between the water on the surface and the atmosphere. Therefore, the speed at the surface would not be an accurate measure of the maximum speed.)

Consider three reactions with different values of $E_2$ and $\Delta E$ : Reaction 1. $E_{\mathrm{a}}=20 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=-60 \mathrm{~kJ} / \mathrm{mol}$ Reacrion 2. $E_{\mathrm{a}}=10 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=-20 \mathrm{~kJ} / \mathrm{mol}$ Reacrion 3. $\quad E_{\mathrm{a}}=40 \mathrm{~kJ} / \mathrm{mol} ; \Delta E=+15 \mathrm{~kJ} / / \mathrm{mol}$ (a) Sketch a potential energy profile for each reaction that shows the potential energy of reactants, products, and the transition state. Include labels that define $E_1$ and $\Delta E$. (b) Assuming that all three reactions are carried out at the same temperature and that all three have the same frequency factor $A$, which reaction is the fastest and which is the slowest? (c) Which reaction is the most endothermic, and which is the most exothermic?

$$\text{\blue{SCIENTIFIC INQUIRY}}$$

$\text{\blue{Draw IT}}$ A researcher has developed an assay to measure the activity of an important enzyme present in liver cells growing in culture. She adds the enzyme’s substrate to a dish of cells and then measures the appearance of reaction products. The results are graphed as the amount of product on the y-axis versus time on the x-axis. The researcher notes four sections of the graph. For a short period of time, no products appear (section A). Then (section B) the reaction rate is quite high (the slope of the line is steep). Next, the reaction gradually slows down (section C). Finally, the graph line becomes flat (section D). Draw and label the graph, and propose a model to explain the molecular events occurring at each stage of this reaction profile.

A laboratory wind tunnel has a flexible upper wall that can be adjusted to compensate for boundary-layer growth, giving zero pressure gradient along the test section. The wall boundary layers are well represented by the $\frac{1}{7}$-power-velocity profile. At the inlet the tunnel cross section is square, with height $H_{1}$ and width $W_{1}$, each equal to 1 ft. With freestream speed $U_{1}$=90 ft/s, measurements show that $\delta_{1}$=0.5 in. and downstream $\delta_{6}$=0.65 in. Calculate the height of the tunnel walls at 6. Determine the equivalent length of a flat plate that would produce the inlet boundary layer thickness. Estimate the streamwise distance between sections 1 and 6 in the tunnel. Assume standard air.

Investigate the possibility of various voting methods violating the monotonicity criterion.

A $14$-member committee is selecting a site for its next meeting. The choices are

$$\text{Anaheim} \ a, \text{New Orleans} \ n, \text{Denver} \ d, \text{and} \text{Yakima} \ y.$$

$\textbf{(a)}$ The committee members decide to use the pairwise comparison method to select a site in a nonbinding decision. Prior to any discussion, the $14$ members rank the choices according to the following voter profile.

$$\begin{array}{c|c} \small \text{Number of Voters} & \small \text{Ranking} \\ \hline 5 & \mathrm{a}>\mathrm{n}>\mathrm{d}>\mathrm{y} \\ \hline 4 & \mathrm{y}>\mathrm{d}>\mathrm{n}>\mathrm{a} \\ \hline 3 & \mathrm{y}>\mathrm{d}>\mathrm{a}>\mathrm{n} \\ \hline 2 & \mathrm{n}>\mathrm{a}>\mathrm{d}>\mathrm{y} \\ \hline \end{array}$$

Show that Anaheim is selected by the pairwise comparison method in the preliminary nonbinding decision.

$\textbf{(b)}$ The $2$ committee members with the bottom ranking in the table rearrange their ranking after listening to the discussions. The other $12$ committee members stick with their original rankings of the cities. For the official vote, the $14$ members rank the choices according to the following voter profile.

$$\begin{array}{c|c} \small \text{Number of Voters} & \small \text{Ranking} \\ \hline 5 & \mathrm{a}>\mathrm{n}>\mathrm{d}>\mathrm{y} \\ \hline 4 & \mathrm{y}>\mathrm{d}>\mathrm{n}>\mathrm{a} \\ \hline 3 & \mathrm{y}>\mathrm{d}>\mathrm{a}>\mathrm{n} \\ \hline 2 & \mathrm{a}>\mathrm{y}>\mathrm{n}>\mathrm{d} \\ \hline \end{array}$$

Use the pairwise comparison method to determine the site selection of the committee.

$\textbf{(c)}$ Does the pairwise comparison method violate the monotonicity criterion in this selection process?

A laboratory wind tunnel has a test section that is square in cross section, with inlet width $W_{1}$ and height $H_{1}$, each equal to 1 ft. At freestream speed $U_{1}$=80 ft/s, measurements show the boundary-layer thickness is $\delta_{1}$=0.4 in. with a $\frac{1}{7}$-power turbulent velocity profile. The pressure gradient in this region is given approximately by dp/dx=-0.035 in. $\mathrm{H}_{2} \mathrm{O}$/in. Evaluate the reduction in effective flow area caused by the boundary layers on the tunnel bottom, top, and walls at section 1. Calculate the rate of change of boundary-layer momentum thickness, $d \theta / d x$, at section 1. Estimate the momentum thickness at the end of the test section, located at L=10 in downstream.

Aluminum heat sinks of rectangular profile are commonly used to cool electronic components. Consider a $7.62$-cm-long and $9.68$-cm-wide commercially available heat sink whose cross section and dimensions are as shown in the figure. The heat sink is oriented vertically and is used to cool a power transistor that can dissipate up to $125 \mathrm{~W}$ of power. The back surface of the heat sink is insulated. The surfaces of the heat sink are untreated, and thus they have a low emissivity (under $0.1$ ). Therefore, radiation heat transfer from the heat sink can be neglected. During an experiment conducted in room air at $22^{\circ} \mathrm{C}$, the base temperature of the heat sink was measured to be $120^{\circ} \mathrm{C}$ when the power dissipation of the transistor was $18 \mathrm{~W}$. Assuming the entire heat sink to be at the base temperature, find the average natural convection heat transfer coefficient for the case.

Read Karine's profile on her social networking site. Answer the question that follows in a complete sentence.

• Elle est s$\'e$rieuse (serious)?

Karine Duplessis:

$\^A$ge: J'ai seize ans (et demi).

Personnalit$\'e$: Je suis intelligente, $\'e$nergique, et dr$\^o$le. J'aide ma m$\`e$re et man p$\`e$re avec leur magasin $\'e$lectronique.

Musique: J'aime bien le hip-hop et la musique alternative. Je n'aime pas le rock et la musique classique!

Mati$\`e$res: J’aime bien les sciences. J'$\'e$tudie la biologie et la chimie parce que c'est int$\'e$ressant!

"Hudson Corporation is considering three options for managing its data warehouse: continuing with its own staff, hiring an outside vendor to do the managing, or using a combination of its own staff and an outside vendor. The cost of the operation depends on future demand. The annual cost of each option (in thousands of dollars) depends on demand as follows:

$$\begin{array}{lccc} & {\text { Demand }} \\ \text { Staffing Options } & \text { High } & \text { Medium } & \text { Low } \\ \text { Own Staff } & 650 & 650 & 600 \\ \text { Outside Vendor } & 900 & 600 & 300 \\ \text { Combination } & 800 & 650 & 500 \end{array}$$

Construct a risk profile for the optimal decision in the previous part. What is the probability of the cost exceeding $700,000?"

Show that the kinetic energy coefficient, $\alpha$, for the “power law” turbulent velocity profile of Eq. $\frac{\overline{u}}{U}=\left(\frac{y}{R}\right)^{1 / n}=\left(1-\frac{r}{R}\right)^{1 / n}$ is given by Eq. $\alpha=\left(\frac{U}{\overline{V}}\right)^{3} \frac{2 n^{2}}{(3+n)(3+2 n)}$. Plot $\alpha$ as a function of $R e_{\overline{V}}$, for $R e_{\overline{V}}=1 \times 10^{4}$ to $1 \times 10^{7}$. When analyzing pipe flow problems it is common practice to assume $\alpha \approx 1$. Plot the error associated with this assumption as a function of $R e_{\overline{V}}$, for $R e_{\overline{V}}=1 \times 10^{4}$ to $1 \times 10^{7}$.

Read Karine's profile on her social networking site. Answer the question that follows in a complete sentence.

• Elle est intelligente?

Karine Duplessis:

$\^A$ge: J'ai seize ans (et demi).

Personnalit$\'e$: Je suis intelligente, $\'e$nergique, et dr$\^o$le. J'aide ma m$\`e$re et man p$\`e$re avec leur magasin $\'e$lectronique.

Musique: J'aime bien le hip-hop et la musique alternative. Je n'aime pas le rock et la musique classique!

Mati$\`e$res: J’aime bien les sciences. J'$\'e$tudie la biologie et la chimie parce que c'est int$\'e$ressant!

Read Karine's profile on her social networking site. Answer the question that follows in a complete sentence.

• Elle aime $\'e$tudier les langues?

Karine Duplessis:

$\^A$ge: J'ai seize ans (et demi).

Personnalit$\'e$: Je suis intelligente, $\'e$nergique, et dr$\^o$le. J'aide ma m$\`e$re et man p$\`e$re avec leur magasin $\'e$lectronique.

Musique: J'aime bien le hip-hop et la musique alternative. Je n'aime pas le rock et la musique classique!

Mati$\`e$res: J’aime bien les sciences. J'$\'e$tudie la biologie et la chimie parce que c'est int$\'e$ressant!

The data in the table were gathered by a ship equipped with an echo sounder as it traveled the North Atlantic Ocean eastward from Cape Cod, Massachusetts, to a point somewhat beyond the center of the Atlantic Ocean. Use the data in the table to construct a generalized profile of the ocean floor in the North Atlantic. Begin by plotting the distance of each point from Cape Cod, at the indicated depth. Complete the profile by connecting the points.

$$\begin{matrix} \text{Point} & \text{Distance (KM)} & \text{DEPTH (M)}\\ \text{1} & \text{0} & \text{0}\\ \text{2} & \text{180} & \text{150}\\ \text{3} & \text{270} & \text{2700}\\ \text{4} & \text{420} & \text{3300}\\ \text{5} & \text{600} & \text{4000}\\ \text{6} & \text{830} & \text{4800}\\ \text{7} & \text{1100} & \text{4750}\\ \text{8} & \text{1150} & \text{2500}\\ \text{9} & \text{1200} & \text{4800}\\ \text{10} & \text{1490} & \text{4750}\\ \text{11} & \text{1750} & \text{4800}\\ \text{12} & \text{1800} & \text{3100}\\ \text{13} & \text{1860} & \text{4850}\\ \text{14} & \text{2120} & \text{4800}\\ \text{15} & \text{2320} & \text{4000}\\ \text{16} & \text{2650} & \text{3000}\\ \text{17} & \text{2900} & \text{1500}\\ \text{18} & \text{2950} & \text{1000}\\ \text{19} & \text{2960} & \text{2700}\\ \text{20} & \text{3000} & \text{2700}\\ \text{21} & \text{3050} & \text{1000}\\ \text{22} & \text{3130} & \text{1900}\\ \end{matrix}$$