For a microfluidic device, two Newtonian fluids (A and B) are flowing in a unidirectional flow inside a thin horizontal rectangular slit of thickness d due to an applied pressure gradient along the length L of the channel. The fluid flow rates have to be adjusted such that the slit is half filled with A and half fillled with B and the two remain in laminar flow to minimize mixing. Assume that Density of B < Density of A and that R = viscosity of B/ viscosity of A is the ratio of the viscosities of the two liquids. The liquids can be assumed to be immiscible.

(a) Draw a schematic of the flow. Clearly define and label the axes and any variables to be used. Indicate on the schematic the
fluid on top and bottom of the slit.

(b) Find the expression for the volumetric flow rates of the two fluids as a function R by applying equation of continuity and the Navier-Stokes equations. Clearly state assumptions for simplification of the N-S equations, the boundary conditions applied, and the solution of the differential equations. Then plot the ratio of volumetric flow rates of the two fluids as a function of R ranging from 0.1 to 10.

(c) If one of the fluids is water and the other fluid is 1,2-Dichlorobenzene and the Reynold's number of 2100 indicates the transition from laminar flow, what is the maximum pressure gradient (SI units) that can be applied along the length of a slit with width 5 cm? Also find the average velocity of the two fluids at this maximum pressure drop.