To increase the flow-rate of a very viscous liquid through a circular tube (of radius R) for a given pressure drop, a second less viscous immiscible liquid (with the same density) is introduced to "lubricate" the tube wall. Consider an idealization of this problem in which the steady flow of both fluids is laminar (with constant properties) and the interface between the two fluids is perfectly concentric with the tube walls (no waves due to instabilities) at a radius r. Thus, the core (central region for 0 rr) of the tube consists of the very viscous fluid with viscosity H., and the annular region (r, r R) is occupied by the low-viscosity Hlubricating fluid. (a) (b) (c) Calculate the velocity distribution in both fluids for a constant pressure gradient =-G along the tube (z z direction). Calculate the volumetric flow-rate of the "core fluid" (i.e., the central very viscous fluid). Calculate the value of the radius for the fluid interface r that maximizes the volumetric flow of core fluid for the case in which the ratio of viscosities / is very large. L low-viscosity fluid He very viscous fluid Circular Tube