A layer of viscous fluid of constant thickness (no velocity perpenducilar to plate) flows steadily down an infinite, inclined plane. Determine, with the Navier Stokes equations, the flowrate per unit width as a function of flow height.

Answer:

q = (ρg/μ)(sin θ)(h³/3)

Explanation:

I've attached an image of a figure showing the coordinate system.

In this system: the velocity components v and w are equal to zero.

From continuity equation, we know that δu/δx = 0

Now,from the x-component of the navier stokes equation, we have;

-δp/δx + ρg(sin θ) + μ(δ²u/δy²) = 0 - - - - - (eq1)

Due to the fact that we have a free surface, it means we will not have a pressure gradient in the x-component and so δp/δx = 0

Then our eq 1 is now;

ρg(sin θ) + μ(δ²u/δy²) = 0

μ(δ²u/δy²) = -ρg(sin θ)

Divide both sides by μ to get;

(δ²u/δy²) = -(ρg/μ)(sin θ)

Integrating both sides gives;

δu/δy = -(ρg/μ)(sin θ)y + b1 - - - - (eq2)

Now, the shear stress is given by the formula;

τ_yx = μ[δu/δy + δv/δx]

From the diagram, at the free surface,τ_yx = 0 and y = h

This means that δu/δy = 0

Thus, putting 0 for δu/δy in eq 2, we have;

0 = -(ρg/μ)(sin θ)h + b1

b1 = h(ρg/μ)(sin θ)

So, eq 2 is now;

δu/δy = -(ρg/μ)(sin θ)y + h(ρg/μ)(sin θ)

Integrating both sides gives;

u = -[(y²/2) × (ρg/μ)(sin θ)] + h(ρg/μ)(sin θ)y + b2 - - - eq3

Because u = 0 when y = 0, it means that b2 = 0 also because when we plug 0 for u and y into eq3, we will get b2 = 0.

Thus, we now have:

u = -[(y²/2) × (ρg/μ)(sin θ)] + h(ρg/μ)(sin θ)y

Factorizing like terms, we have;

u = (ρg/μ)(sin θ)[hy - y²/2] - - - (eq 4)

The flow rate per unit width is gotten by Integrating eq 4 between the boundaries of h and 0 to give;

∫u = (h,0)∫(ρg/μ)(sin θ)[hy - y²/2]

q = (ρg/μ)(sin θ)[hy²/2 - y³/6] between h and 0

q = (ρg/μ)(sin θ)[h³/2 - h³/6]

q = (ρg/μ)(sin θ)(h³/3)