3B.10 Radial flow between parallel disks (Fig. 3B.10). A part of a lubrication system consists of two circular disks between which a lubricant flows radially. The flow takes place because of a modified pressure difference P₁ P2 between the inner and outer radii r₁ and ½₂, respectively. 1 (a) Write the equations of continuity and motion for this flow system, assuming steady-state, laminar, incompress- ible Newtonian flow. Consider only the region r₁ ≤r≤12 and a flow that is radially directed. Fluid in Radial flow outward between disks r = 12 =11 z = +b z=-b Fig. 3B.10. Outward radial flow in the space between two parallel, circular disks. (b) Show how the equation of continuity enables one to simplify the equation of motion to give dP 1 d² ρ μ dr dz² (3B.10-1) in which = rv, is a function of z only. Why is o indepen- dent of r? (c) It can be shown that no solution exists for Eq. 3B.10-1 unless the nonlinear term containing is omitted. Omis- sion of this term corresponds to the "creeping flow as- sumption." Show that for creeping flow, Eq. 3B.10-1 can be integrated with respect to r to give 12 d² 0 = (P₁ - P₂) + In (3B.10-2) dz² (d) Show that further integration with respect to z gives (P₁ - P2)b² v,(r, z) 2μr In (r₂/r₁) [1-(0)] (e) Show that the mass flow rate is (3B.10-3) 4π(P₁ - P₂)b³p ω (3B.10-4) 3μ In (r₂/r₁) (f) Sketch the curves P(r) and v,(r, z).