TRANSPORT PHENOMENA

Homework (30% exam I)

1. Assume that for a certain steady, two-dimensional flow in a regin defined by 0 and 0, one velocity component is given by:

(, ) =

Where and are constants.

A. Assuming that the fluid is incompressibe and that (, 0) = 0, determine (, ).

B. For what values of will the flow be irrotational?

C. For what values of can the Navier-Stokes equation be satisfied? For those cases determine the dynamic pressure, (, ), assuming (0,0) = 0.

2. Figure shows an example of a system with a gas-liquid interface of unknown shape, consisting of

a liquid in an open container of radius that is rotated at an angular velocity . If the container is rotated long enough, a steady state is reached in which the liquid is in rigid-body rotation. It is

desired to determine the steady-state interfase height, (), assuming that the ambient air is at a constant pressure, 0. The viscous stress vanishes for rigid-body rotation. The effects of surface tension will be neglected.

Show that interface height is (Newtonian or non-Newtonian fluid?):

() = +22

2[(

)2

1

2]

Where /(2) is the liquid height under static conditions. is the volume of fluid.

3. A flat plate at = 0 is in contact with a Newtonian fluid, initially at rest, which occupies the space > 0. At = 0 the place is suddenly set in motion in the direction at a velocity , and that plate velocity is maintained indefinitely. This problema may be viewed, for example, as representing the

early time (or penetration) phase in the start-up of a Couette viscometer.

A. Show that the non-dimensional formulation of the problem is (analyze the Navier-Stokes

equation):

2

2+ 2

= 0

Where = / and =

4. Moreover is a function of .

B. Solve the ordinary differential equation and determine the fluid velocity as a function of time

and position. For the mathematical solution, associate the boundary conditions with and .

C. Plot the velocity profile for different times. Assume that the kinematic viscosity () is equal to 1 and is very small.