3B.5 Parallel-disk viscometer (Fig. 3B.5). A fluid, whose viscosity is to be measured, is placed in the gap of thick- ness B between the two disks of radius R. One measures the torque T, required to turn the upper disk at an angular velocity . Develop the formula for deducing the viscosity from these measurements. Assume creeping flow. Fluid with viscosity μ and density p is held in place by surface tension Fig. 3B.5. Parallel-disk viscometer. Disk at z = B rotates with angular velocity Disk at z = 0 is fixed Both disks have radius R and R >> B (a) Postulate that for small values of the velocity pro- files have the form v, = 0, v₂ = 0, and v, = rf(z); why does this form for the tangential velocity seem reasonable? Pos- tulate further that P = P(r, z). Write down the resulting simplified equations of continuity and motion. (b) From the 6-component of the equation of motion, ob- tain a differential equation for f(z). Solve the equation for f(z) and evaluate the constants of integration. This leads ultimately to the result v, r(z/B). Could you have guessed this result? (c) Show that the desired working equation for deducing the viscosity is μ = 2BT/TR (d) Discuss the advantages and disadvantages of this in- strument.