The flow of water around the bridge pier is to be studied using a model built to a scale of $1 / 15$. If the river flows at $0.8 \mathrm{~m} / \mathrm{s}$, find the corresponding velocity of the water in the model at the same temperature.

The boy has a mass of $50 \mathrm{~kg}$ and attempts to slide down the inclined plane. If a 0.3-mm-thick oil surface develops between his shoes and the surface, find his terminal velocity down the incline. Both of his shoes have a total contact area of $0.0165 \mathrm{~m}^2$. Take $\rho_\theta=900 \mathrm{~kg} / \mathrm{m}^3$ and $\mu_a=0.0638 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^2$.

The effect of drag on a model airplane is to be tested in a wind tunnel with a wind speed of $200\ \mathrm{mi} / \mathrm{h}$. If a similar test is performed on the same model underwater in a channel, what should the speed of the water be to achieve the same result when the temperature is $60^{\circ} \mathrm{F}$ ?

The power $\dot{W}$ supplied by a pump is thought to be a function of the discharge $Q$, the change in pressure $\Delta p$ between the inlet and outlet, and the density $\rho$ of the fluid. Derive a relation between $\dot{W}$ and these parameters.

The half body is defined by a combined uniform flow having a velocity of $U$ and a point source of strength $q$. Find the pressure distribution along the top boundary of the half body as a function of $\theta$, if the pressure within the uniform flow is $p_0$. Neglect the effect of gravity. The density of the fluid is $\rho$.

As the liquid drains from the pipette, the exit velocity $V$ is a function of the liquid's specific weight $\gamma$, viscosity $\mu$, the nozzle's diameter $d$, and the liquid's level $h$. Find the relation between $V$ and these parameters. Take $q=h$ and $\mu$.

A ship has a length of $180 \mathrm{~m}$ and travels in the sea where $\rho_s=1030 \mathrm{~kg} / \mathrm{m}^3$. A model of the ship is built to a $1 / 60$ scale, and it displaces $0.06 \mathrm{~m}^3$ of water such that its hull has a wetted surface area of $3.6 \mathrm{~m}^2$. When tested in a towing tank at a speed of $0.5 \mathrm{~m} / \mathrm{s}$, the total drag on the model was $2.25 \mathrm{~N}$. Calculate the drag on the ship and its corresponding speed. Find the power needed to overcome this drag. The drag due to viscous (frictional) forces can be determined using $\left(F_D\right)_f=\left(\frac{1}{2} \rho V^2 A\right) C_D$, where $C_D$ is the drag coefficient determined from $C_D=1.328 / \sqrt{\operatorname{Re}}$ for $\operatorname{Re}<10^6$ and $C_D=0.455 /\left(\log _{10} \mathrm{Re}\right)^{2.58}$ for $10^6<\operatorname{Re}<10^9$. Take $\rho=1000 \mathrm{~kg} / \mathrm{m}^3$ and $\nu=1.00\left(10^{-6}\right) \mathrm{m}^2 / \mathrm{s}$.

The drag coefficient on an airplane is defined by $C_D=2 F_D / \rho V^2 L^2$. If the drag acting on the model of a plane tested at sea level is $0.3 \mathrm{~N}$, find the drag on the prototype, which is 15 times larger and is flying at $20$ times the speed of the model at an altitude of $3 \mathrm{~km}$.

Show that the hydrostatic pressure $p$ of an incompressible fluid can be determined using dimensional analysis by realizing that it depends upon the depth $h$ in the fluid and the fluid's specific weight $\gamma$.

The optimum performance of mixing blades $0.5 \mathrm{~m}$ in diameter is to be tested using a model one-fourth the size of the prototype. If the test of the model in water reveals the optimum speed to be $8\ \mathrm{rad} / \mathrm{s}$, find the optimum angular speed of the prototype when it is used to mix ethyl alcohol. Take $T=20^{\circ} \mathrm{C}$.

The length $\lambda$ of small water waves is thought to be a function of the period of time $\tau$ of each wave cycle, the water depth $h$, gravitational acceleration $g$, and the surface tension $\sigma$ of the water. Find the relation between $\lambda$ and these parameters. Take $q=\lambda$ and $\tau$.

The Rankine oval is defined by the source and sink of equal strength. Find this strength $q$ if the length of the Rankine oval is to be $1.054 \mathrm{~m}$. Also, find its corresponding width. The velocity of the uniform flow is $4 \mathrm{~m} / \mathrm{s}$.

The 100 -mm-diameter shaft is supported by an oillubricated bearing. If the gap within the bearing is $2 \mathrm{~mm}$, find the torque $\mathbf{T}$ that must be applied to the shaft, so that it rotates at a constant rate of $180\ \mathrm{rev} / \mathrm{min}$. Assume no oil leaks out due to sealing, and the flow behavior is similar to that which occurs between parallel plates, since the gap size is very much smaller than the radius of the shaft. Take $\rho_o=840 \mathrm{~kg} / \mathrm{m}^3$ and $\mu_o=0.22 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^2$.

The torque $T$ developed by a turbine depends upon the depth $h$ of water at the entrance, the density of the water $\rho$, the discharge $Q$, and the angular velocity of the turbine $\omega$. Find the relation between $T$ and these parameters. Take $q=T$ and $h$.