(2) Starting with the Navier-Stokes equation with constant density and ignoring the gravitational acceleration, show that two fluid flows require the same Reynolds number in order to be dynamically similar. [5]

(3) Suppose that the power to drive a propeller of an airplane depends on diameter of the propeller $d$, free-stream velocity $U$, angular velocity $\omega$, speed of sound $c$, density of fluid $\rho$, and fluid's dynamic viscosity $\mu$. Find the dimensionless groups. In your opinion, which of these are the most important and should be duplicated in model testing? [7]

(4) Sketch the evolution of an initially rectangular fluid element, $|x| \leq 1, |y| \leq 1 \ (t=0)$, in a deformation flow, $u = ax, \ v = -ay$, where $a$ is a constant, and derive a formula for the evolution of the aspect ratio of the fluid element. [5]

(5) A viscous fluid of uniform density and uniform kinematic viscosity flows steadily in the $x$-direction along a tube of circular cross-section with radius $a$. It is subject to a constant pressure gradient $G$ in the $x$-direction. The flow speed $u$ depends only on $r$, the distance from the axis of the tube. Given that the $x$-component of $\nabla^2 \mathbf{u}$ in these circumstances is \[ \frac{1}{r} \frac{d}{dr} \left( r \frac{du}{dr} \right) \] show that \[ u(r) = A \left( a^2 - r^2 \right) \] and find $A$ in terms of $G, \rho$ and $\mu$. Find also the mass of fluid crossing the surface $x=0$ in unit time. [5]