Pr 1 (25 pt): Water flows down a stationary inclined plane as shown. Beginning with the full Navier-Stokes equations giv

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Pr 1 25 Pt Water Flows Down A Stationary Inclined Plane As Shown Beginning With The Full Navier Stokes Equations Giv 1 (37.39 KiB) Viewed 53 times

Pr 1 (25 pt): Water flows down a stationary inclined plane as shown. Beginning with the full Navier-Stokes equations given below, ∂x∂u​+∂y∂y​=0ρ(∂t∂u​+u∂x∂u​+v∂y∂u​)=−∂x∂P​+μ(∂x2∂2u​+∂y2∂2u​)+ρg,ρ(∂t∂v​+u∂x∂v​+v∂y∂v​)=−∂y∂P​+μ(∂x2∂2y​+∂y2∂2v​)+ρg,​ Solve the differential equations for the velocity profile and show that it is written as u(y)=2μρgsinθ​(2hy−y2) The assumptions are: 1. Fully developed, u=u(y) only. 2. Steady 3. Incompressible 4. No pressure gradient in the x-direction, ∂P/∂x=0. 5. 2-dimensional (a) (5 pt) With the above assumptions, reduce the Navier-Stokes equations to the form specific to this problem. CLEARLY STATE ASSUMPTIONS for dropping terms. (b) (5pt) State the boundary conditions for the equations. (c) (5 pt) Solve the simplified equations for the velocity profile. For parts (d) and (e), if you can't get the velocity profile, use the one above to solve. (d) (5 pt) Develop an expression for the shear stress at the wall. (e) (5 pt) Develop an expression for the volumetric flow rate per unit width