Mechanical Engineering: Fundamentals of Fluid Flow exam

(Fall 2018) ME721 Fundamentals of Fluid Flow

Homework Assignment #3 (Due date: 11/08/2018)

(*) Please include all the intermediate steps.

1. (20 points) Under the influence of gravity a viscous liquid (density ρ and viscosity µ) flows down a

stationary vertical wall, forming a thin film of constant thickness h. An upflow of air next to the film

exerts an upward constant shear stress τ on the surface of the liquid layer, as shown below. The

pressure in the flow is uniform. Derive expressions for (a) the film velocity u(y) as a function of y and

the flow parameters ρ, µ, h and τ, and (b) the shear stress τ that would result in a zero net volume flow

rate in the film.

2. (30 points) Two immiscible liquids of viscosities µ1 and µ2 are flowing one above the other in a

horizontal channel of fixed height 2a and of infinite extent in the flow direction under the influence of a

constant negative pressure gradient dp*/dx. (The channel dimension W in the direction normal to the

flow plane is much larger than 2a.) Each of the fluids occupies half of the channel, as shown below.

Considering the flow in both the upper and lower fluids to be a combined Poiseuille and Couette flow

with the pressure gradient dp*/dx and an interfacial velocity Vi, derive expressions for (a) Vi, (b) the

volume flow rates Q1/W and Q2/W for the two fluids and (c) the velocity profile u1(y) in fluid 1 in terms

of the parameters a, µ1, µ2 and dp*/dx.