![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.](https://image.slidesharecdn.com/homeworktransportphenomena-150901002915-lva1-app6891/85/Homework-transport-phenomena-1-320.jpg)
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Homework transport phenomena
- 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.