An incompressible viscous liquid (with density p and dynamic viscosity u) slides down an inclined plane as seen in the figure. The film thickness is h, and the solid substrate moves with a speed U up the incline. Assume steady, fully developed, laminar flow in the x direction with no pressure gradient (in the x direction). Furthermore, assume that the atmosphere at y = h imposes no shear stress on the liquid. (a) tions in terms of the parameters given. Derive an expression for the velocity profile using the Navier-Stokes equa- pgh? sin 0 (b) If the magnitude of the substrate velocity is given as U = 4µ determine the y-coordinate, normalized by the thickness, where the velocity is zero, i.e. find y/h for u(y) = 0. (c) reference line, assuming the substrate velocity given in part (b). Draw a qualitative sketch of the velocity profile, including a zero-velocity U Patm 19
An incompressible viscous liquid (with density p and dynamic viscosity u) slides down an inclined plane as seen in the figure. The film thickness is h, and the solid substrate moves with a speed U up the incline. Assume steady, fully developed, laminar flow in the x direction with no pressure gradient (in the x direction). Furthermore, assume that the atmosphere at y = h imposes no shear stress on the liquid. (a) tions in terms of the parameters given. Derive an expression for the velocity profile using the Navier-Stokes equa- pgh? sin 0 (b) If the magnitude of the substrate velocity is given as U = 4µ determine the y-coordinate, normalized by the thickness, where the velocity is zero, i.e. find y/h for u(y) = 0. (c) reference line, assuming the substrate velocity given in part (b). Draw a qualitative sketch of the velocity profile, including a zero-velocity U Patm 19
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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![An incompressible viscous liquid (with density p and dynamic viscosity u) slides down
an inclined plane as seen in the figure. The film thickness is h, and the solid substrate
moves with a speed U up the incline. Assume steady, fully developed, laminar flow in
the x direction with no pressure gradient (in the x direction). Furthermore, assume
that the atmosphere at y = h imposes no shear stress on the liquid.
(a)
tions in terms of the parameters given.
Derive an expression for the velocity profile using the Navier-Stokes equa-
pgh? sin 0
4µ
(b)
If the magnitude of the substrate velocity is given as U
determine the y-coordinate, normalized by the thickness, where the velocity is zero,
i.e. find y/h for u(y) = 0.
(c)
reference line, assuming the substrate velocity given in part (b).
Draw a qualitative sketch of the velocity profile, including a zero-velocity
U
Patm
19](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F704e4f48-aa8b-4c89-846b-e3a31036e601%2F318380e8-7f52-42b0-b3e9-d1c506fb3508%2Ffwzq8hj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:An incompressible viscous liquid (with density p and dynamic viscosity u) slides down
an inclined plane as seen in the figure. The film thickness is h, and the solid substrate
moves with a speed U up the incline. Assume steady, fully developed, laminar flow in
the x direction with no pressure gradient (in the x direction). Furthermore, assume
that the atmosphere at y = h imposes no shear stress on the liquid.
(a)
tions in terms of the parameters given.
Derive an expression for the velocity profile using the Navier-Stokes equa-
pgh? sin 0
4µ
(b)
If the magnitude of the substrate velocity is given as U
determine the y-coordinate, normalized by the thickness, where the velocity is zero,
i.e. find y/h for u(y) = 0.
(c)
reference line, assuming the substrate velocity given in part (b).
Draw a qualitative sketch of the velocity profile, including a zero-velocity
U
Patm
19
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