Steady fully developed laminar incompressible flow of a

Steady, fully developed, laminar, incompressible flow of a Newtonian fluid down an inclined plane under gravity Exercise 1: Show that, for steady, fully developed laminar flow down the slope (shown in the figure), the Navier-Stokes equations reduces to y z x dir where u is the velocity in the x-direction, ρ is the density, μ is the dynamic viscosity, g is acceleration due to gravity, and θ is the angle of the plane to the horizontal. Solve the above equation to obtain the velocity profile u ect ion of θ flo w

N-S equation therefore reduces to (similar to parallel plate flow x - component: y - component: y z x dir ect ion of z - component: No applied pressure gradient to drive the flow. Flow is driven by gravity alone. Therefore, we get flo w θ (4) x - component: y - component: z - component: p is not a function of z What was asked to be derived in Exercise 1

Solve the equation to obtain the velocity profile u (4) y z Equation (4) is a second order equation in u with respect to y. Therefore, we require two boundary conditions (BC) of u with respect to y. BC 1: At y = 0, u = 0 BC 2: At y = h, x h (no-slip boundary condition) θ (free-surface boundary condition) (5) Integrating equation (4), we get Applying BC 2, we get Combining equations (5) and (6), we get (6) (7)

y z x h θ (8) Integrating equation (7), we get Applying BC 1, we get B=0 Combining equations (8) and (9), we get (9) (10)

(10) y Volumetric flow rate through one unit width fluid film along the z-direction is given by z x h θ (11)