advanced fluids

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3. a) The uniform stream of velocity U approaches a channel (of width b). After flowing a distance le (“entrance length”) flow inside the channel will become “fully developed”. Show that the entrance length le is proportional to the Reynolds number Re (based on the channel width b). Fluid’s dynamic viscosity is  and density is . (10 points) U Boundary layer Potential core b Boundary layer Fully developed flow le Entrance region b) The uniform stream of velocity U and temperature T approaches stationary flat plate whose temperature is TW (TW > T). Fluid’s kinematic viscosity is , thermal diffusivity is , and Prandtl number Pr = 10. Estimate distances xv and xT (measured from the leading edge) where velocity boundary layer reaches thickness v ~ L and thermal boundary layer reaches thickness T ~ L, respectively. What is the ratio xT / xv ? (15 points)  U U T T TW x Page 2 of 4 4. A fluid is flowing from A to B through the network of tubes due to the known pressure difference pA - pB (see schematic). Obtain an expression for the flow rate Q as a function of fluid viscosity (), pressure difference (pA-pB), tube radius (R) and tube length (L). Neglect elbow effects and disturbance in various tube junctions. Consider simple laminar flow through straight circular pipe throughout the system. Neglect body forces. To save you some integration time: mean velocity in a circular pipe is one half of the maximum velocity. Page 3 of 4 Navier-Stokes Equation in Cylindrical Coordinates  1    v  vz r r  z 2 1     1  2 2    r  r r  r  r 2  2 z 2 V   vr r  momentum vr v 1 1 p 2 v    V   vr  v 2    g r     2vr  r2  2   t r  r r r      momentum v vv 1 p 2 v v    V   v  r     g     2 v  2 r  2  t r  r  r  r   z  momentum vz 1 p  V   vz    g z   2 v z t  z Page 4 of 4
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