Using Buckingham Pi theorem to find the coefficients of equation: Pi=[density, running speed, diameter, viscosity]=[ρa1, Nb1, Dc1, μ]

Which of the following dimensionless parameters is the correct arrangement of the given parameters? (select all that O a. Pi = (Dp)/(Rho*V) Dp: Pressure change Rho: Density V: Velocity O b. Pi = (V*L*(Rho))/(nu) V: Velocity L: length Rho: Density nu: Dynamic viscosity O. Pi = Q/((A^1.5)*w) Q: Volumetric discharge rate A: Area w: Angular velocity O d. Pi = (a)(t^2)/(L) L: length a: acceleration t time

(Dimensional Analysis) If the velocity V in a fluid flow depends on a dimension 1, the fluid density p, and the viscosity µ, show that this implies that the Reynolds number Vlp/µ is constant.

P2, µ2 2h P1 Po ρ1, μι L Figure 3: Two layer flow for (b) (b) Consider two immiscible (do not mix) fluid layers with different densities and viscosities which glide smoothly between two parallel horizontal walls, as shown in figure 3. Let pı and p2 be the densities and µ1 and µ2 be the viscosities of the two fluids, respectively. The flow is the result of an imposed constant pressure gradient, 2, similar to that of the Poiseuille flow. Assume a steady, incompressible, parallel viscous flow with p1 > P2 and µi > µ2- (i) Using the Navier-Stokes equations find the velocity profiles of both fluids. (ii) Ensure the fluid obeys the following conditions: The shear stress is continuous at the interface between the two fluids. • The velocity profile is continuous at the interface between the two fluids. No-slip boundary at the walls. (iii) Sketch the velocity profile. (iv) Determine where the maximum velocity occurs and explain why. (Hint: think about the respective viscosities and densities).…