A film of liquid with kinematic viscosity and density p spreads over a flat horizontal surface due to gravity as shown. Assume that the spread is planar in x - y plane with unit width into page. The height of the film is h(x, t) which varies in a direction and time t. The flow is incompressible and the x-velocity is u(x, y, t) which is governed by the lubrication theory due to the small thickness of the film (h< L). The pressure outside the film is uniform and atmospheric. Inside the film the pressure variation is hydrostatic in y direction. At x = 0, h = h, and his symmetric in x (i.e., h(L,t) = h(-L,t)). The gravitational acceleration is g. x=-L h(x₂t) y u(x, y, t) = x Show that +(udy) Using lubrication theory show that the velocity profile y(2h - y). = x=L ↓g Assuming that h/t = constant, use (a) and (b) results to find h(r, t) in term of ho, x, ah/at, v, L and g.
A film of liquid with kinematic viscosity and density p spreads over a flat horizontal surface due to gravity as shown. Assume that the spread is planar in x - y plane with unit width into page. The height of the film is h(x, t) which varies in a direction and time t. The flow is incompressible and the x-velocity is u(x, y, t) which is governed by the lubrication theory due to the small thickness of the film (h< L). The pressure outside the film is uniform and atmospheric. Inside the film the pressure variation is hydrostatic in y direction. At x = 0, h = h, and his symmetric in x (i.e., h(L,t) = h(-L,t)). The gravitational acceleration is g. x=-L h(x₂t) y u(x, y, t) = x Show that +(udy) Using lubrication theory show that the velocity profile y(2h - y). = x=L ↓g Assuming that h/t = constant, use (a) and (b) results to find h(r, t) in term of ho, x, ah/at, v, L and g.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:A film of liquid with kinematic viscosity and density p spreads over a flat
horizontal surface due to gravity as shown. Assume that the spread is planar in x - y plane
with unit width into page. The height of the film is h(x, t) which varies in a direction and
time t. The flow is incompressible and the x-velocity is u(x, y, t) which is governed by the
lubrication theory due to the small thickness of the film (h < L). The pressure outside
the film is uniform and atmospheric. Inside the film the pressure variation is hydrostatic
in y direction. At x = 0, h = h, and h is symmetric in z (i.e., h(L,t) = h(-L, t)). The
gravitational acceleration is g.
x=-L
hexat) ấy
x
u(x, y, t) = -2y(2h - y).
x=L
Show that +(udy) = 0
Using lubrication theory show that the velocity profile
↓g
Assuming that Oh/dt = constant, use (a) and (b) results to find h(r, t) in term
of ho, x, ah/ot, v, L and g.
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