Question

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a) As shown in the Figure below, a fluid flows down an inclined surface. Show that the velocity distribution of that fluid is u=[pgsin0/(2p)]/(2hy - y2)) by using Navier-Stokes equations. (Show assumptions that you make and every each step) [13 points]

Answer 1

^{h - gSino W=o, Assumption • Laminas, steady flow • P= Patm v=o paralled flow :'g -0 (Continuity equation) NS equation in 'n direction act down words Du On D+ puan g²u ai tv ou su +0²4 on " I - Р .Dtry > jon y ay ogte ay 20 at Simplifics to razu + 0 +0+0+0= rg sino 3y2 du ४५ - pg sino a Du - rg Sino y + C, Р og siney & Gy+(2 u am (No slip ) Boundary conditions Put U=0 at y=0 0= Ca Put auto at goh (No sheas at free sus face ) ay Ca = g sino u ²7 rgsino hy Therefore - pgs an U= reysing (ahy-y-) proved pgsinoh U= to 9 ar}