(Solved): Problem involving the Navier Stokes equations Problem involving the Navier Stokes equations A viscou ...

Problem involving the Navier Stokes equations A viscous incompressible fluid film of uniform constant thickness \( H \) everywhere and known fluid properties \( \mu, \rho \) is flowing under gravity down an infinite plane inclined at \( \alpha \) to the horizontal. (Take the \( x \) direction as pointing down the plane and the \( y \) axis pointing normally upwards from it). Assume the flow is steady, that there is no-slip on the plane \( y=0 \), and zero tangential and normal stress on the free surface (i.e., \( p=0 \) and \( \mu(d u / d y)= \) 0 on \( y=H) \). (a) Assuming the velocity field has the components \( \mathbf{u}=(u, v, 0) \), write down and explain the relevant boundary conditions on \( y=0 \). (b) Explain the physical significance of the boundary conditions on \( y= \) \( H \), given above. (c) Justify a further assumption that the velocity vector is of the form \( \mathbf{u}= \) \( (u(y), v(y), 0) \) (d) Verify from the continuity (incompressibility) condition that \( v=0 \) everywhere and write down the appropriate form of the Navier Stokes equations. (e) Non-dimensionalise the problem defining dimensionless variables \( \left(u^{*}, y^{*}, p^{*}\right) \) (including the boundary conditions), choosing a velocity scale by balancing the terms in the Navier Stokes equations. (f) Show that the nondimensionalised Navier Stokes equations reduce to the form \( \frac{d^{2} u^{*}}{d y^{2}}=-1 \). Use the non-dimensionalised boundary conditions to find and sketch the velocity profile.