Consider a viscous, Newtonian, and incompressible oil draining uniformly down the side of a vertical rod (the rod has a radius r= a). The oil is in contact with air at a pressure P = Pa everywhere along its outside edge. You may assume that the viscosity of the air is virtually zero (la - 0). At some distance down the rod ... the film will approach a fully developed flow with a constant outer radius r = b. Note that the positive z-direction is down.
Consider a viscous, Newtonian, and incompressible oil draining uniformly down the side of a vertical rod (the rod has a radius r= a). The oil is in contact with air at a pressure P = Pa everywhere along its outside edge. You may assume that the viscosity of the air is virtually zero (la - 0). At some distance down the rod ... the film will approach a fully developed flow with a constant outer radius r = b. Note that the positive z-direction is down.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![Consider a viscous, Newtonian, and incompressible oil draining uniformly down
the side of a vertical rod (the rod has a radius r = a). The oil is in contact with air
at a pressure P = Pa everywhere along its outside edge. You may assume that the
viscosity of the air is virtually zero (la z 0). At some distance down the rod ... the
film will approach a fully developed flow with a constant outer radius r = b. Note
that the positive z-direction is down.
It is NOT your concern about how long it takes to reach the fully developed flow.
Pa
Fully
developed
region
Film
a
= 0.03 m
Radius of vertical rod
b
= 0.05 m
Outside edge of the fully developed fluid film
Absolute pressure of air phase
Viscosity of the oil
Density of the oil
Pa
= 101325 Pa
= 2.90 Pa-s
= 1250 kg/m³
(3.a) Write the simplified forms of the three momentum balances.
--- Clearly state ALL of your assumptions
(3.b) Derive the expression for the velocity profile vz =f (r).
--- Clearly state your boundary condition(s) --
--- The expression may contain a, b, p, g, µ, r, Pa, and/or numbers ---
(3.c) What is the maximum fluid velocity [ m/s ]?
- Think about where the maximum fluid velocity is
---
(3.d)
If this problem were performed not in air, but with water surrounding
the oil film, would the shape of velocity profile change? If we imagined the
water to have zero viscosity, how would the oil velocity change from (3.c)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5d528f6-f704-4253-8470-44f2fbc0de86%2F111e9539-b3bf-4d6b-99b9-4bee4f2f3570%2Fm670zh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a viscous, Newtonian, and incompressible oil draining uniformly down
the side of a vertical rod (the rod has a radius r = a). The oil is in contact with air
at a pressure P = Pa everywhere along its outside edge. You may assume that the
viscosity of the air is virtually zero (la z 0). At some distance down the rod ... the
film will approach a fully developed flow with a constant outer radius r = b. Note
that the positive z-direction is down.
It is NOT your concern about how long it takes to reach the fully developed flow.
Pa
Fully
developed
region
Film
a
= 0.03 m
Radius of vertical rod
b
= 0.05 m
Outside edge of the fully developed fluid film
Absolute pressure of air phase
Viscosity of the oil
Density of the oil
Pa
= 101325 Pa
= 2.90 Pa-s
= 1250 kg/m³
(3.a) Write the simplified forms of the three momentum balances.
--- Clearly state ALL of your assumptions
(3.b) Derive the expression for the velocity profile vz =f (r).
--- Clearly state your boundary condition(s) --
--- The expression may contain a, b, p, g, µ, r, Pa, and/or numbers ---
(3.c) What is the maximum fluid velocity [ m/s ]?
- Think about where the maximum fluid velocity is
---
(3.d)
If this problem were performed not in air, but with water surrounding
the oil film, would the shape of velocity profile change? If we imagined the
water to have zero viscosity, how would the oil velocity change from (3.c)?
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