Assume that the flowrate. Q, of a gas from a smokestack is a function of the density of the ambient air,$\rho_{a}$, the density of the gas,$\rho_{g}$, within the stack, the acceleration of gravity, g, and the height and diameter of the stack, h and d, respectively. Use $\rho_{c}$, d, and g as repeating variables to develop a set of pi terms that could be used to describe this problem.

Solve the equation.

$$\ln (x-1)+\ln (x+1)=2 \ln \sqrt{12}$$

At a sudden contraction in a pipe the diameter changes from $D_{1}$ to $D_{2}$. The pressure drop, $\Delta p$, which develops across the contraction is a function of $D_{1}$ and $D_{2}$, as well as the velocity, V, in the larger pipe, and the fluid density, $\rho$, and viscosity, $\mu$ . Use $D_{1}$, V and $\mu$ as repeating variables to determine a suitable set of dimensionless parameters. Why would it be incorrect to include the velocity in the smaller pipe as an additional variable?

Solve the equation using square roots. $x^{2}-64=0$

The Mach number for a body moving through a fluid with velocity V is defined as VIc, where c is the speed of sound in the fluid. This dimensionless parameter is usually considered to be important in fluid dynamics problems when its value exceeds 0.3. What would be the velocity of a body at a Mach number of 0.3 if the fluid is $\textbf{(a)}$ air at standard atmospheric pressure and $20^{\circ} \mathrm{C}$, and $\textbf{(b)}$ water at the same temperature and pressure?

A mixing basin in a sewage filtration plant is stirred by a mechanical agitator with a power input $\dot{W} \doteq F \cdot L / T$ . Other parameters describing the performance of the mixing process are the fluid absolute viscosity $\mu \doteq F \cdot T / L^{2}$, the basin volume $V=L^{3}$ , and the velocity gradient $G \doteq 1 / T$ . Determine the form of the dimensionless relationship.

A vertical shaft passes through a bearing and is lubricated with an oil having a viscosity of $0.2 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^2$ as shown in Fig. P6.84. Assume that the flow characteristics in the gap between the shaft and bearing are the same as those for laminar flow between infinite parallel plates with zero pressure gradient in the direction of flow. Estimate the torque required to overcome viscous resistance when the shaft is turning at 80 rev/min.

The Reynolds number, $\rho V D / \mu$ , is a very important parameter in fluid mechanics. Verify that the Reynolds number is dimensionless using both the FLT system and the MLT system for basic dimensions, and determine its value for ethyl alcohol flowing at a velocity of 3 m/s through a 2 -in.-diameter pipe.

Flow characteristics for a 30-ft-diameter prototype parachute are to be determined by tests of a 1-ft-diameter model parachute in a water tunnel. Some data collected with the model parachute indicate a drag of 17 lb when the water velocity is 4 ft/s. Use the model data to predict the drag on the prototype parachute falling through air at 10 ft/s. Assume the drag to be a function of the velocity, V , the fluid density, $\rho$, and the parachute diameter, D.

Determine whether the function has an inverse (is one-to-one). If so, find the inverse and graph both the function and its inverse. $f(x)=x^{4}-2 x-1$

The Keystone Pipeline has D = 36 in. and an oil flow rate Q = 590,000 barrels per day (1 barrel = 42 U.S. gallons). Its pressure drop per unit length, Δp/L, depends on the fluid density ρ, viscosity μ, diameter D, and flow rate Q. A water-flow model test, at 20°C, uses a 5-cm-diameter pipe and yields Δp/L ≈ 4000 Pa/m. For dynamic similarity, estimate Δp/L of the pipeline. For the oil take ρ = 860 kg/m³ and μ = 0.005 kg/m · s.

What role did the policy of containment play in the involvement of the United States in wars in Korea and Vietnam?

A soda can in the shape of a cylinder is to hold 16 fluid ounces. Find the dimensions of the can that minimize the surface area of the can.

For the flow of a thin film of a liquid with a depth $h$ and a free surface, two important dimensionless parameters are the Froude number, $V / \sqrt{g h}$. and the Weber number, $\rho V^2 h / \sigma$. Determine the value of these two parameters for glycerin (at $20^{\circ} \mathrm{C}$ ) flowing with a velocity of $0.5 \mathrm{~m} / \mathrm{s}$ at a depth of $2 \mathrm{~mm}$.