- Question 2 A Cell Culture Consists Of A Viscous Incompressible Newtonian Fluid Con Tained Between Two Fixed Parallel P 1 (124.26 KiB) Viewed 8 times
- Question 2 A Cell Culture Consists Of A Viscous Incompressible Newtonian Fluid Con Tained Between Two Fixed Parallel P 2 (87.84 KiB) Viewed 8 times
- Question 2 A Cell Culture Consists Of A Viscous Incompressible Newtonian Fluid Con Tained Between Two Fixed Parallel P 3 (81.37 KiB) Viewed 8 times
Question 2. A cell culture consists of a viscous incompressible Newtonian fluid con- tained between two fixed parallel plates inclined at an angle , as shown in Figure 2. Cells are grown on the bottom plate. The flow is driven by a constant pressure gradient (@p/ax = constant) and gravity. The distance between the two plates is 2H and the chosen system of coordinates is shown in the figure. YA /211 0 Figure 2: Flow between parallel plates. 0), answer the Assuming steady, purely in-plane, and parallel flow (v = w = following questions. [5 marks] (a) Show that the flow is fully developed. (b) Find the velocity profile u(y) in terms of 0, H,, the viscosity, , and density, p, of the fluid, and the acceleration of gravity, g. [15 marks] (c) Find the shear stress at the bottom plate where cells are positioned. [4 marks] (d) Provide two parameters that could be changed to increase the shear stress sensed by the cells, Twall. Indicate in which direction each pa- rameter should be increased to achieve a greater Twall in the positive x-direction. Assume that the pressure gradient Op/ax produces a flow in the positive x-direction. [6 marks]
ac Ət ac Ət ac Ət q" = €o(T - Tur) q" = hr(Ts - Tsur) hr = €o(Ts + Tsur)(T2 + Tur) q' =h(T, - To) dT "1 q = -k dx ac J=-D Әх Ә Ә 2 (3) 8 (3)+2 (2) + ду ду Ә 18 (%)-+8 (8)-* (*) + kr k + rər Ər Ә 10 r2 Jr kp2 k Ər r2 sin²0 дф 1 Ә r2 sin 0 80 (18)+0(137) + (kains) - o-w-w- (+zv19:) v Vov + (* + {v* oz) v - Jos Ә Q - W, – W, V2 + 9% û = at 2 h+- dû + p(V · V) = V . (kVT) + Ф dt 02C 82c 82c + + + R дх2 дуг дz2 18 ac 1 82C 82C + + + R r2 аф2 022 1 Ә r2 sin 0 80 = D = D = D rər 10 r2 ar Ər 0Ꮯ ? Ər + + ӘТ Ət +9=PCp ат k sin 0- +4 = PCp at + 92 p(V.n)dA sin 0 ac де +q= PCp + ат Ət 1 02с r2 sin² 0 дф2 (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) + R (24)
P Р Р дег Ət дve Ət дv. at d d at (Bayw) = an (L_Body ) + J _Sp(V-n) dA (Bsyst) BpdV dt dt CS Q= S (V . n) dA m = (V . n) dA 2 1 1) a = do A Js Р +-V²+gz = constant и P 2 др + V . (pV) = 0 at av + (V.V) V = Vp+V Tij +pg Ət aV + (V.V)V) =-Vp+μV2V+pg Ət др 10(prvr) 10(pvo) (pvz) + = 0 + + Ət r ar r 20 дz aur vедит avr 1) Ər где дz + н 7²4-17/14--2000 дv + pgr = Ur Ər p2 r2 дve vе vе дv Urve +uz az Ər r 20 r 1 др -11 ( 7² 00 + 72²2 000-1/2) + где r2 ро ди. vоди, ди. г де Az + μV2vz + P9= DA 1 02A 82 A + Ər r2 202 дz2 P Р +Ur + Ur + + +Ur + ar V²A = + Uz Әр + uz др дz 10 rər + + Pge (1) (2) (3) (4) (5) (6) (8) (10) (11) (12)