b) A continuous rigid belt, moving with uniform velocity U, passes vertically upwards through an incom- pressible viscous fluid, having density p and constant kinematic viscosity , contained within a large tank. In doing so the belt picks up a thin fluid film. A steady, fully-developed flow is maintained by the movement of the belt overcoming gravity's tendency to return the fluid to the tank, so that sufficiently far enough away from the bulk fluid in the tank the film can be assumed to have a uniform thickness h. (vi) Considering just one side of the belt, draw a sketch of the problem using a Cartesian coordinate system (x, y) with x lying along the belt and pointing in the direction of motion and y perpendicular to it pointing through the fluid film, with the origin at the belt surface. On your sketch indicate the expected form of the velocity profile through the fluid film. (vii) Given that the pressure throughout the the fully-developed film is constant, list the necessary assump- tions required to reduce the governing continuity and momentum (i.e. the Navier-Stokes) equations to a form that is solvable analytically and write down the accompanying boundary conditions for the given problem in order to do so. (viii) Show that the velocity distribution, u(y), within the fluid film is given by: g u(y) = ²/2 (2/2² - hy) + ₁ U₂ V and, if the belt has width W, the corresponding volumetric flow rate to be equal to: gh³ 3v For what value of U, written in terms of the other quantities, will the volumetric flow rate be zero? W (Uh

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:b) A continuous rigid belt, moving with uniform velocity U, passes vertically upwards through an incom- pressible viscous fluid, having density p and constant kinematic viscosity, contained within a large tank. In doing so the belt picks up a thin fluid film. A steady, fully-developed flow is maintained by the movement of the belt overcoming gravity's tendency to return the fluid to the tank, so that sufficiently far enough away from the bulk fluid in the tank the film can be assumed to have a uniform thickness h. (vi) Considering just one side of the belt, draw a sketch of the problem using a Cartesian coordinate system (x, y) with x lying along the belt and pointing in the direction of motion and y perpendicular to it pointing through the fluid film, with the origin at the belt surface. On your sketch indicate the expected form of the velocity profile through the fluid film. (vii) Given that the pressure throughout the the fully-developed film is constant, list the necessary assump- tions required to reduce the governing continuity and momentum (i.e. the Navier-Stokes) equations to a form that is solvable analytically and write down the accompanying boundary conditions for the given problem in order to do so. (viii) Show that the velocity distribution, u(y), within the fluid film is given by: g u(y) = 1/2 ( 12/²2 - hy) + U₁ V and, if the belt has width W, the corresponding volumetric flow rate to be equal to: 9h³ For what value of U, written in terms of the other quantities, will the volumetric flow rate be zero? 3v W (Uh

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