1. Consider steady Couette flow of a constant-property fluid as illustrated in Fig. 3-1. The two plates are porous and f

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1 Consider Steady Couette Flow Of A Constant Property Fluid As Illustrated In Fig 3 1 The Two Plates Are Porous And F 1 (67 KiB) Viewed 41 times

1 Consider Steady Couette Flow Of A Constant Property Fluid As Illustrated In Fig 3 1 The Two Plates Are Porous And F 2 (74.46 KiB) Viewed 41 times

1. Consider steady Couette flow of a constant-property fluid as illustrated in Fig. 3-1. The two plates are porous and fluid enters the interplate space through the bottom plate and leaves at the same steady and uniform rate pu through the top plate. (a) Show that the x-motion equation is du d²u dy dy2 subject to the boundary conditions that u(y=0)=0 and u(y = L) = U from which it follows that the velocity distribution is U Homework u U D euy/v1 CUL/V-1 (b) Plot u/U against y/L for several values of the dimensionless Reynolds number uL/v, and comment on the effect the imposed "blowing" has on the velocity profile. Show that the viscous drag at the bottom plate Tis μU UL/v Tw= LeUL/v-1 When u is positive, the upward "blowing" is representative of that which occurs in evaporation or transpiration cooling of a turbine
blade. When u is negative, the downward "suction" is representa- tive of that which occurs in condensation or boundary-layer suc- tion to control flow over an airfoil. (c) Show that if the bottom and top plates are maintained at tempera- tures To and T₁, respectively, the energy equation is U dT dy with the boundary conditions of 9w =α T(y=0) = To and T(y=L) = T₁ from which it follows that the temperature distribution is ePruy/v1 ePrul/v-1 T-To T₁ - To d²T dy² (d) Plot (T-To)/(T₁ - To) against y/L and comment as in part b. Show also that the diffusive heat flux q, at the bottom plate is (ToT)k Pr vL/v erul/v1 L As "blowing" increases the wall heat flux decreases, an effect called transpiration cooling that can be used to protect surfaces from hot atmospheres. Comment on the ratio of diffusive to convective energy transport rates. To Figure 3-1 Geometry and coordinate system for Couette flow.