Question 1: A fluid flows axially between two concentric circular cylinders. The inner cylinder has a radius of "a" and the outer radius is "b". For fully developed flow determine:
a) The velocity profile between r = a and r = b.
b) The relation between the flow rate and the pressure gradient.
c) The drag on a length L of the inner cylinder.
Question 2: Ideal flow issues from a reservoir into a parallel plate channel,
a) Using Schwarz-Christoffel Transformation find the ideal flow solution.
b) The relation between the flow rate and the pressure gradient.
Question 3: An incompressible viscous flow of uniform velocity U enters a parallel-plate channel of length L and width has shown. The flow is driven by the pressure difference between the entrance and the exit of the channel.
a) Write down the governing equations (2D) which determine the velocity and pressure of the flow in the channel State the boundary conditions.
b) If the channel length is much greater than the entrance length of the flow, determine the velocity profile in the fully developed region.
c) In the entrance region, the flow is developing from the uniform flow at the channel entrance to the fully developed velocity profile found in (b). Consider that the fully developed flow starts at the location where the two boundary layers (one over the upper plate and one over the lower plate) merge. Show from the order of magnitude analysis that the entrance length Le is proportional to the product of and Re where Re = ρUh/ µ is the Reynolds number, in which ρ is the fluid density and µ the viscosity.
Question 4: A square-duct wind tunnel of length L = 1 m is being designed to operate at room temperature and atmospheric conditions. A uniform airflow at U = 1 m/s enters through an opening of D = 20 cm. Due to the viscosity of air, it is necessary to design a variable cross-sectional area if a constant velocity is to be maintained in the middle part of the cross-section throughout the wind tunnel.
a) Determine the duct size D(x) as a function of x.
b) Plot the momentum and displacement thickness.
c) How will the result be affected if U = 20 m/s? At a given value of x, will D(x) be larger, smaller, or the same than the value obtained in part a)? Explain.
d) How will the result be affected if the wind tunnel is to be operated at 10 atm (and U = 1 m/s)? At a given value of x, will D(x) be larger, smaller, or the same than the value obtained in part (a)? Explain.
e) Does the airflow apply a net force to the wind tunnel? If so, indicate the direction of the force.
Question 5: Determine the boundary layer on a circular cylinder starting at the leading stagnation point. Determine the pressure distribution with ideal flow theory. Plot the non-dimensional results for the wall shear stress and the boundary layer versus location on the surface. Find the point of separation.
