Question

Problem 3 (30 Points) Consider a specific application where a synthetic engine oil is used as a working fluid in a journal bearing with a gap between the top & bottom surfaces of L=0.6 mm. Here, the top surface moves to the left with a constant velocity (Vr) of 16 m/s, while at the same time the bottom surface moves to the right with a constant velocity (Vb) of 9 m/s. The top surface is isothermal as it is maintained at a constant temperature of T=50 °C, and the bottom surface is well insulated. It is determined that viscosity of this synthetic engine oil is u=0.7 N.s/m² and its thermal conductivity is k=0.2 W/m.K. Place the origin of the coordinate system on the centerline plane, which is the geometric symmetry plane halfway between the top and bottom surfaces. Use the Systematic Methodology studied in class to answer the following questions (show all work!): i) Derive the custom-tailored governing equation and boundary conditions necessary to obtain the velocity profile (do not solve!)? ii) Derive the custom-tailored governing equation and boundary conditions necessary to obtain the temperature profile (do not solve!)?

Answer 1

V = 16m/sec Voo it = 50°c iv=o Vy=am/sec K = 0.2W/mk reco.7 Nosson? Z دیر 0.6mmah Q=0 Vb G Velocity profiles Assumptions: neglected (9) Inertia and body force terms are (6) No, Venation of pressure across flural film and long flurd film. dy (C) No slip at fluid-solid boundary. (d) No entennal forces act on the flurd film. ap ap =o / Jz dz flow is viscous and laminan f) au 2 D du ay an (9) LLL bearin bring length. The Navien stokes equation can be reduced to + u ap cre Ju an ay ay B.Cs At Y=0, U = Vb qm/see y = 0.6mm, U = v= -16m sem ap 0:6- Solution at *9*60* -16) are Ju Y CY-0.6) + 0.6

Temperature distribution: The heat conducton quation is given an. DT DI ao?T Dt { a = 1 1 1 ат at + u a T tv at an oy DT OZ 227 222 t کہ) az? a( 22т Ә2-т ау? considering steadly state, and, nawao.. so and Jy2 wat ay2 so 22T seand Day >schule >> агт Ju2 , 22T azz a ƏZT 2 B-cs :- IT Aty o ay At 1,420.6, T= 50°C