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Oct 30, 2023

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*P4.95 Two immiscible liquids of [ L 4 V : : h equal thickness / are being sheared . 1ty v e between a fixed and a moving plate, y h Pis as in Fig. P4.95. Gravity is neglected, v X - > | Fixed and there is no variation with x. - Fig. P4 .95 Find an expression for (a) the velocity at the interface; and (b) the shear stress in each fluid. Assume steady laminar flow. Solution: Treat this as a Ch. 4 problem (not Ch. 1), use continuity and Navier-Stokes: _— du dv dv . Continuity : = - o 0+ —= 0 : thus v =const = 0 forno -=slip at the walls X )y This tells us that there 1s no velocity v, hence we need only consider u(y) in Navier-Stokes: du du ap a’u @ d*u U— + V—) = =— + + or: 0+0=0+ 0+ p|.2( ox ay) ox I"I.2( axz ('9)’2 ) ‘u].z( ).-‘2 ) Thus w =a + by 4-62 Solutions Manual * Fluid Mechanics, Eighth Edition The velocity profiles are linear in y but have a different slope in each layer. Let u; be the velocity at the interface. (a) The shear stress 1s the same in each layer: Uy V=u, Solve for u; = By Ans.(a) h h My + o (b) In terms of the upper plate velocity, IV, the shear stress is r o= Ans.(b) uy+uy h
In this problem, water enters the turbine nozzles at 800 kPa absolute with £ s dow velocity. And exits at a pressure of 100 kPa, the maximum velocity %6 which water can be accelerated by the nozzles before striking the turbine blades can be determined as following: Assume that point 1 is at the inlet of the nozzle and point 2 is at the exit of the nozzle(as shown in the fig.) By applying Bernoullis equation between the points (1) and (2) as following: Where Vi = 0, and 2; = 29, by substituting in the Bernoullis equation as following: By substituting with given values as following: | [800%10% 100+ 10 Vo =/ 2 L \ 1000 1000 =37.4166 m/s The maximum velocity to which water can be accelerated by the nozzles before striking the turbine blades, V5 = 37.4166 m/s
Possible drag reduction in pipes = Shear Thinning = Newtonian «== Shear Thickening yH VIV Figure 3. Velocity profiles for different fluids 2 10. Bernoulli’s Equation is as follows: f + VT + gZ = Constant a) (4 pts) Bemoulli’s 1738 treatise Hydrodynamica contains many excellent sketches of flow patterns related to his frictionless relation. One, however, redrawn below, seems physically misleading. Identify what is wrong with the figure and explain why. Assume flow to be | P_[—L\,e_ l/\/U%A/\A' e +o (Y /.[\ < Wvo be esquod to Har (10« Wy ™, ele vok on (1) | [ com we Bermouds Hdrom (x) - (2) lo show Hrek Dne b\,e.\a\/u\' must be fhe saxme fvom (A) () v
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