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P2, 42 2h P1 Po P1, Hi L Figure 3: Two layer flow for (b) (b) Consider two immiscible (do not mix) fluid layers with different densities and viscosities which glide smoothly between two parallel horizontal walls, as shown in figure 3. Let pı and p2 be the densities and µ1 and µ2 be the viscosities of the two fluids, respectively. The flow is the result of an imposed constant pressure gradient, 2, similar to that of the Poiseuille flow. Assume a steady, incompressible, parallel viscous flow with p1 > P2 and µi > µ2- (i) Using the Navier-Stokes equations find the velocity profiles of both fluids. (ii) Ensure the fluid obeys the following conditions: • The shear stress is continuous at the interface between the two fluids. • The velocity profile is continuous at the interface between the two fluids. No-slip boundary at the walls. (iii) Sketch the velocity profile. (iv) Determine where the maximum velocity occurs and explain why. (Hint: think about the respective viscosities and densities).

P2, 42 2h P1 Po P1, Hi L Figure 3: Two layer flow for (b) (b) Consider two immiscible (do not mix) fluid layers with different densities and viscosities which glide smoothly between two parallel horizontal walls, as shown in figure 3. Let pı and p2 be the densities and µ1 and µ2 be the viscosities of the two fluids, respectively. The flow is the result of an imposed constant pressure gradient, 2, similar to that of the Poiseuille flow. Assume a steady, incompressible, parallel viscous flow with p1 > P2 and µi > µ2- (i) Using the Navier-Stokes equations find the velocity profiles of both fluids. (ii) Ensure the fluid obeys the following conditions: • The shear stress is continuous at the interface between the two fluids. • The velocity profile is continuous at the interface between the two fluids. No-slip boundary at the walls. (iii) Sketch the velocity profile. (iv) Determine where the maximum velocity occurs and explain why. (Hint: think about the respective viscosities and densities).

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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P2, µ2 2h P1 Po ρ1, μι L Figure 3: Two layer flow for (b) (b) Consider two immiscible (do not mix) fluid layers with different densities and viscosities which glide smoothly between two parallel horizontal walls, as shown in figure 3. Let pı and p2 be the densities and µ1 and µ2 be the viscosities of the two fluids, respectively. The flow is the result of an imposed constant pressure gradient, 2, similar to that of the Poiseuille flow. Assume a steady, incompressible, parallel viscous flow with p1 > P2 and µi > µ2- (i) Using the Navier-Stokes equations find the velocity profiles of both fluids. (ii) Ensure the fluid obeys the following conditions: The shear stress is continuous at the interface between the two fluids. • The velocity profile is continuous at the interface between the two fluids. No-slip boundary at the walls. (iii) Sketch the velocity profile. (iv) Determine where the maximum velocity occurs and explain why. (Hint: think about the respective viscosities and densities). to
Transcribed Image Text:P2, µ2 2h P1 Po ρ1, μι L Figure 3: Two layer flow for (b) (b) Consider two immiscible (do not mix) fluid layers with different densities and viscosities which glide smoothly between two parallel horizontal walls, as shown in figure 3. Let pı and p2 be the densities and µ1 and µ2 be the viscosities of the two fluids, respectively. The flow is the result of an imposed constant pressure gradient, 2, similar to that of the Poiseuille flow. Assume a steady, incompressible, parallel viscous flow with p1 > P2 and µi > µ2- (i) Using the Navier-Stokes equations find the velocity profiles of both fluids. (ii) Ensure the fluid obeys the following conditions: The shear stress is continuous at the interface between the two fluids. • The velocity profile is continuous at the interface between the two fluids. No-slip boundary at the walls. (iii) Sketch the velocity profile. (iv) Determine where the maximum velocity occurs and explain why. (Hint: think about the respective viscosities and densities). to
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