Student ID

Department of Mechanical Engineering

Michigan State University

East Lansing, Michigan

Ph.D. Qualifying Exam in

Fluid Mechanics

Closed book and Notes, Some basic equations are provided on an attached information sheet.

Answer all questions.

All questions have the same weighting.

Exam prepared by

Professor F. Jaberi

Professor A. Naguib

January 2017

Problem 1: As seen in the figure, a jet strikes a vane on a cart that moves to the right at constant velocity Vc on a cart. Ignoring friction, compute:

The force components Fx and Fy required to maintain the motion of the cart as described

The power delivered to the cart

The cart velocity for which the power P is maximum

Problem 2: A fixed cylinder of diameter D and length L, immersed in a stream flowing normal to its axis at velocity U, will experience zero average lift. However, if the cylinder is rotating at angular velocity , a lift force F will arise. The fluid density  is important, but viscosity is secondary and can be neglected. Do the following:

Find the relevant dimensionless groups for this problem. Express the interdependence of the dimensionless groups in a function form that allows computation of the non-dimensional lift from other non-dimensional parameters

Considering the case with L >> D, the following experimental data are obtained for a cylinder 30 cm in diameter and 2 m long, rotating in air with density of 1.2 kg/m2, with U = 25 m/s:

 (rev/min)	0	3000	6000	9000	12000	15000
F (N)	0	850	2260	2900	3120	3300

Using the above data, predict the lift of a cylinder with D = 5 cm, L = 80 cm, rotating at 3800 rev/min in water (density of 1000 kg/m3) at U = 4 m/s.

Problem 3: An infinitely long and infinitely wide (two dimensional) flat plat oscillates sinusoidally in its own plane beneath a viscous fluid, as shown below. The fluid is at rest far above the plate. Making as many simplifying assumptions as you can, set up the governing equations and boundary conditions for finding the fluid velocity field u through the following steps:

(a) Starting from full Navier-Stokes equations (attached), simplify the equations for this problem as much as possible.

(b) Write all boundary/initial conditions for this problem.

(c) Explain how you may solve the simplified equations to find the velocity over the plate (you do not have to fully solve the equations)

Problem 4. A fluid with constant density of , steady velocity of Uo and static pressure of Po is moving over a quarter-cylinder object. The velocity on the top surface, obtained from the inviscid potential flow theory, is U=2Uo Sin(). The pressure at point b is Pb .

Find the pressure distribution over top and bottom surfaces of the quarter-cylinder. You may assume the pressure on the back surface to be uniform and equal to Pb .

Compute the lift and drag forces on the object.

Draw the flow streamline and the velocity vector on the cylinder surface.

Possibly useful information and formulas:

Bernoulli’s Equation:

Integral{sin2() cos()} = sin3()/3 , Integral{sin3()} = cos3()/3 – cos()

Pressure: p has units N/m2 = Pascals. One atmosphere = 1.01325 E+5 Pa. Force: 1 N = 1 kg-m/s2. Newton: 1 N = 1 kg-m/s2; Joule (Work) = N-m; Watt (Power) = N-m/s. 1 ft = 0.3048 m; 1 in = 2.54 cm; 1 mile = 5280 ft. 1 m3= 103L.

Integral equations:

Mass :

Momentum:

Angular momentum:

Differential Equations - Continuity

Differential Equations – Momentums for Incompressible, Constant Viscosity (μ)