Recall that viscous incompressible unidirectional flow with velocity u = wk is governed by Iw t 1 (1) where z is the coordinate in the k direction, t is time, p is pressure, p is the density, v = p/p is the kinematic viscosity and is the dynamic viscosity. OP-K. As = Consider steady incompressible unidirectional flow of a viscous fluid through a circular tube of radius R subject to a constant streamwise pressure gradient the volume flow rate through the tube is Q = u u.ndS -R4, 8 (2) where S denotes a cross-section of the tube. Suppose this fluid is very viscous, with large viscosity 1. According to (2), increasing viscosity will decrease the volume flow rate, other things being equal. The aim of this question is to see if it is possible to increase the volume flow rate of the high viscosity fluid (fluid 1) by introducing a second fluid of identical density but lower viscosity #2 (fluid 2). Let r denote the radial coordinate in a cylindrical coordinate system aligned with the tube axis. Suppose fluid 1 occupies the central circular region 0 < r < r, while the low viscosity fluid (fluid 2) occupies the surrounding annular region r1 r < R. The pressure gradient throughout the tube is a = (a) Find a general solution w(r) of (1) for steady axisymmetric flow subject to a constant pressure gradient Oz -K. (b) Let wi(r) denote the velocity in fluid 1 for 0 rr and w(r) denote the velocity in fluid 2 for r1 r < R. Use the general solution from part (a) together with appropriate boundary conditions to determine w(r) and w(r). Assume that the velocities and shear stress match at rr, that is, dwi == w(T1)=w2 (11) and pdr dw2 == 21 dr 71 (c) Hence calculate the volume flow rate Q1 of fluid 1. Write down a nondimensional expression for Q1/Q as a function of r/R and p/2. Hint: Remember that fluid 1 only occupies the region 0