y=h air, p = Poo fluid y=-0 Ꮖ a g 1
An incompressible fluid of density ρ and viscosity μ flows down a plane inclined at an angle α.
Assume constant gravitational acceleration downward, fully-developed flow, constant pressure in
the air outside the fluid, and zero stress exerted by the air on the fluid.
i) Starting from the incompressible Navier-Stokes equations, derive the differential equation and
boundary conditions that govern the velocity u(y).
ii) Solve the equation from the previous part for u(y).
iii) Using your solution, calculate the following quantities:
The mass flow rate (per unit depth) down the channel.
The vorticity vector, ~ξ, and rate-of-strain tensor, epsilon at a point (x, y) in the channel.
The shear stress exerted by the fluid on the bottom wall
The viscous force in the fluid
iv) Consider a control volume consisting of a section of length L of the channel. Demonstrate
that the conservation of x momentum holds for this control volume by integrating appropriate
quantities over its perimeter and interior.
Please show all work!!!
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