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Question 4 a) Two incompressible immiscible liquids are sandwiched between two plates. The upper plate moves at a velocity V1= 3 m/s and the bottom plate is stationary. The upper plate drags the fluid along creating the steady velocity profile shown in figure 4.a. The liquid at the boundary moves at a velocity V2= 2 m/s. The upper and lower fluids have viscosities of 0.4 Pa.s and 0.2 Pa.s and heights h1= 0.02 m and h2= 0.02 m respectively. Moving plate V1 V2 h1 h2 Stationary plate Figure 4.a. The velocity profile of two liquid being dragged by the motion of the upper plate. The dashed line is used to show a cross section of a fixed control volume. (not to scale) (i) The viscosity of the fluids and the no-slip effect creates a shear stress by the walls of a channel on the fluid. Calculate the ratio of the shear stresses on the bottom plate and the top plate due to the different fluids. [8 marks] (ii) Shown by the dashed line in figure 4.a is the cross section of a box-shaped control volume. The control volume coincides with the system at time t = 0 s. In your answer booklet make a sketch to indicate a. The system at time t = 0.1 s and b. the fluid that has entered and exited the control volume in that time period. Clearly label your sketch. [2x2 marks] c. Calculate the volume of fluid which has entered the control volume in that 0.1 s time period given the control volume is 1 m wide. [12 marks]

          Question 4

a) Two incompressible immiscible liquids are sandwiched between two plates. The upper plate moves at a velocity V1= 3 m/s and the bottom plate is stationary. The upper plate drags the fluid along creating the steady velocity profile shown in figure 4.a. The liquid at the boundary moves at a velocity V2= 2 m/s. The upper and lower fluids have viscosities of 0.4 Pa.s and 0.2 Pa.s and heights h1= 0.02 m and h2= 0.02 m respectively.

Moving plate V1
V2
h1
h2
Stationary plate
Figure 4.a. The velocity profile of two liquid being dragged by the motion of the upper plate. The dashed line is used to show a cross section of a fixed control volume. (not to scale)

(i) The viscosity of the fluids and the no-slip effect creates a shear stress by the walls of a channel on the fluid. Calculate the ratio of the shear stresses on the bottom plate and the top plate due to the different fluids.
[8 marks]

(ii) Shown by the dashed line in figure 4.a is the cross section of a box-shaped control volume. The control volume coincides with the system at time t = 0 s. In your answer booklet make a sketch to indicate
a. The system at time t = 0.1 s and
b. the fluid that has entered and exited the control volume in that time period.
Clearly label your sketch.
[2x2 marks]
c. Calculate the volume of fluid which has entered the control volume in that 0.1 s time period given the control volume is 1 m wide.
[12 marks]

Added by Keith K.

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