Question

Figure shows two parallel plates (upper plate at \( x = b \) and lower one at \( x = -b \) of length L (aligned in \( z \)-direction and infinite width (in \( y \)-direction, normal to the plane of the figure)). Two immiscible, incompressible liquids are flowing steadily in the \( z \)-direction through the thin passage between the plates under the influence of a horizontal pressure gradient \( \left( \frac{P_0 - P_L}{L} \right) \). During the flow, the passage is always half-filled with denser fluid I (viscosity \( \mu_1 \)) at the bottom and rest is occupied by lighter fluid II (viscosity \( \mu_2 \)). Considering exactly in between the fluids and no instabilities in the flow, the shear stress \( \tau_{xz} \) is expressed as:
\[ \tau_{xz} = \frac{(P_0 - P_L) b}{L} \left( \frac{x}{b} - \frac{1}{2} \right) \left( \frac{\mu_1' - \mu_2'}{\mu_1' + \mu_2'} \right) \] Which one of the following options correctly identifies the location of the point having maximum velocity of the flow?

• A. Above the interface
• B. Below the interface
• C. At the interface
• D. At the top plate

Hint:

In flow between two parallel plates with different fluids, the maximum velocity is found above the interface between the two fluids, where the velocity gradient is maximum and shear stress is zero.

Answer 1

The Correct Option is A

In this problem, we are dealing with a laminar flow between two parallel plates, where two immiscible incompressible fluids are flowing through the gap. The passage between the plates is half-filled with denser fluid I (viscosity \( \mu_1 \)) at the bottom and the lighter fluid II (viscosity \( \mu_2 \)) at the top. We are asked to determine the location where the maximum velocity of the flow occurs. The relationship for shear stress \( \tau_{xz} \) is provided, and it depends on the position \( x \) and the viscosities of the two fluids. From the equation, we can observe that the shear stress varies linearly with the position \( x \) within the flow. The shear stress is zero at the interface between the two fluids, which means that the velocity gradient is steepest just above the interface of the two fluids. Since the flow is parabolic, the maximum velocity will occur where the velocity gradient is maximum, which is just above the interface between fluid I and fluid II.
This conclusion is drawn from the fact that in a laminar flow between parallel plates, the maximum velocity occurs at the point where the shear stress is zero (which is at the interface), and the flow accelerates as we move away from the interface toward the region of maximum velocity. Therefore, the maximum velocity will be located just above the interface.
Thus, the correct answer is (A) Above the interface.