Pr1 (25 pt): Water flows down a stationary inclined plane as shown. Beginning with the full Navier-Stokes equations give

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Pr1 25 Pt Water Flows Down A Stationary Inclined Plane As Shown Beginning With The Full Navier Stokes Equations Give 1 (52.09 KiB) Viewed 34 times

Pr1 (25 pt): Water flows down a stationary inclined plane as shown. Beginning with the full Navier-Stokes equations given below, ix∂u​+∂v∂v​=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∂2v​+∂y2∂2v​)+ρg,​ Solve the differential equations for the velocity profile and show that it is written as u(y)=2uρgsinθ​(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) ( 5 pt) 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 wal. (e) (5 pt) Develop an expression for the volumetric flow rate per unit widen