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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 (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
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
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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
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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