This document contains three problems related to transport phenomena: 1. Determining velocity components for a 2D steady flow given one velocity, and conditions for irrotational and satisfying Navier-Stokes equations. 2. Calculating the steady-state interface height of a liquid in a rotating container, assuming rigid body rotation and neglecting surface tension. 3. Analyzing start-up flow between a suddenly moving flat plate and initially stationary fluid, reducing the Navier-Stokes equation to ordinary differential equation and solving for velocity profile over time.

1. TRANSPORT PHENOMENA
Homework (30% exam I)
1. Assume that for a certain steady, two-dimensional flow in a región defined by 𝑥 ≥ 0 and 𝑦 ≥ 0, one
velocity component is given by:
𝑣𝑥(𝑥, 𝑦) = 𝐶𝑥 𝑚
Where 𝐶 and 𝑚 are constants.
A. Assuming that the fluid is incompressibe and that 𝑣 𝑦(𝑥, 0) = 0, determine 𝑣 𝑦(𝑥, 𝑦).
B. For what values of 𝑚 will the flow be irrotational?
C. For what values of 𝑚 can the Navier-Stokes equation be satisfied? For those cases
determine the dynamic pressure, 𝓅(𝑥, 𝑦), assuming 𝓅(0,0) = 𝓅0.
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 𝑅 that is rotated at an angular velocity 𝜔. 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, ℎ(𝑟), assuming that the ambient air is at a
constant pressure, 𝑃0. 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?):
ℎ(𝑟) = ℎ𝑜 +
𝜔2
𝑅2
2𝑔
[(
𝑟
𝑅
)
2
−
1
2
]
Where ℎ𝑜 ≡ 𝑉/(𝜋𝑅2
) is the liquid height under static conditions. 𝑉 is the volume of fluid.

2. 3. A flat plate at 𝑦 = 0 is in contact with a Newtonian fluid, initially at rest, which occupies the space
𝑦 > 0. At 𝑡 = 0 the place is suddenly set in motion in the 𝑥 direction at a velocity 𝑈, 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):
𝑑2
𝜑
𝑑𝜂2
+ 2𝜂
𝑑𝜑
𝑑𝜂
= 0
Where 𝜑 = 𝑣𝑥/𝑈 and 𝜂 =
𝑦
√4𝑣𝑡
. Moreover 𝜑 is a function of 𝜂.
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 𝜑 and
𝜂.
C. Plot the velocity profile for different times. Assume that the kinematic viscosity (𝑣) is equal
to 1 and 𝑦 is very small.