consider a liquid film of thickness in steady state laminar flow down a vertical, plane surface (wetted wall).assuming incompressible newtonian fluid with constant viscosity.and using the shell balance method. find the following:
(a) an expression for the velocity profile. (20 pts.)
(b) an expression for the shear stress distribution (10 pts)
(c) average velocity (15 pts)
(d) maximum velocity (5 pts)
(e) maximum shear stress (5 pts)
(f) volumetric flow rate (5pts)
(g) sketch the velocity and shear stress distribution (10 pts)
(h) adjust your results to parts (a) through (c) if the plane surface has angle of inclination with the
horizontal.(15 pts)
(a) Velocity profile: The velocity profile in a laminar flow film can be described using the Hagen-Poiseuille equation:
V(r) = (ΔP / (2ηL)) * (R^2 – r^2)
where V(r) is the velocity at radial distance r from the centerline, ΔP is the pressure drop over the length L of the surface, η is the viscosity of the fluid, and R is the radius of the film.
(b) Shear stress distribution: The shear stress distribution in a laminar flow film can be described using the following equation:
τ(r) = η * dV(r) / dr
where τ(r) is the shear stress at radial distance r from the centerline.
(c) Average velocity: The average velocity in the film can be found by integrating the velocity profile over the entire cross-sectional area of the film:
Vavg = (1 / A) * ∫V(r) * 2πr * dr
where A is the cross-sectional area of the film.
(d) Maximum velocity: The maximum velocity in the film can be found by taking the derivative of the velocity profile and setting it equal to zero:
dV(r) / dr = 0, solving for r = R, Vmax = V(R) = (ΔP / (2ηL)) * R^2
(e) Maximum shear stress: The maximum shear stress in the film can be found by taking the derivative of the shear stress distribution and setting it equal to zero:
dτ(r) / dr = 0, solving for r = 0, τmax = τ(0) = η * dV(0) / dr
(f) Volumetric flow rate: The volumetric flow rate in the film can be found by multiplying the average velocity by the cross-sectional area:
Q = Vavg * A
(g) Sketch: A sketch of the velocity and shear stress distributions in the film would show a parabolic shape, with the maximum velocity and shear stress occurring at the centerline of the film and decreasing towards the edges.
(h) Adjustments: If the plane surface has an angle of inclination with the horizontal, the velocity profile would need to be modified to account for the gravitational force acting on the fluid. The pressure drop would also need to be adjusted to account for the change in fluid height over the length of the surface. The average velocity, maximum velocity, and volumetric flow rate would also need to be adjusted accordingly.