Liquid velocity profile. This problem involves a viscous liquid (oil) flowing do

Question

Liquid velocity profile. This problem involves a viscous liquid (oil) flowing down an inclined surface under the effect of gravity. The velocity of the liquid will vary with position perpendicular to the surface (y dimension). A function is provided for the velocity as a function of y. Use this function to determine the volumetric and mass flowrate of the oil by integrating the velocity across the y dimension from 0 to the top of the layer. Note that this integration is relatively simple, as it involves integrating a second order polynomial.

Explanation / Answer

velocity profile of a viscous liquid down an incline is given by

vx(y) = rho*g*sin(alpha)*y(2h - y)/2*mu

where alpha is incline angle, mu is viscocity of the liquid, and y coordinate is perpendicular to the incline

so volume flow rate , Q = integrate vx(y) dy from y = 0 to y = h 9 where h is heigjt of the liquid layer over the surface)

Q = integrate [rho*g*sin(alpha)*y(2h - y)/2*mu] dy

Q = g*sin(alpha)*rho*h^3/3*mu

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