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7-83. Liquid is confined between a top plate having an area A and a fixed surface. A force F is applied to the plate and gives the plate a velocity U. If this causes laminar flow, and the pressure does not vary, show that the Navier-Stokes and continuity equations indi- cate that the velocity distribution for this flow is defined by u = U(y/h), and that the shear stress within the liquid is Txy = FIA. U F n.

7-83. Liquid is confined between a top plate having an area A and a fixed surface. A force F is applied to the plate and gives the plate a velocity U. If this causes laminar flow, and the pressure does not vary, show that the Navier-Stokes and continuity equations indi- cate that the velocity distribution for this flow is defined by u = U(y/h), and that the shear stress within the liquid is Txy = FIA. U F n.

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
Problem 1.1MA
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**Problem 7-83: Fluid Dynamics Analysis** In this problem, a liquid is confined between a top plate with an area \( A \) and a fixed surface below. A force \( F \) is exerted on the top plate, providing it with a velocity \( U \). Assuming the flow is laminar and that there is no pressure variation, you are required to demonstrate, using the Navier-Stokes and continuity equations, that the velocity distribution for this flow is represented by \( u = U(y/h) \) and that the shear stress within the liquid is \( \tau_{xy} = F/A \). **Diagram Explanation:** The diagram illustrates a side view of the setup: - A horizontal surface represents the fixed boundary at the bottom. - Above this lies the liquid, depicted in blue, limited at the top by a moveable plate. - The vertical axis \( y \) is indicated alongside the thickness \( h \) of the liquid layer. - Arrows within the liquid show the velocity distribution, starting from zero at the fixed boundary and increasing linearly (denoted by \( u \)). - The top plate moves to the right with velocity \( U \), driven by the applied force \( F \), also shown as an arrow pointing in the same direction. This setup models a classic fluid mechanics scenario of shear flow between two plates, also known as Couette flow, underlining key principles of fluid dynamics such as velocity gradient and shear stress.
Transcribed Image Text:**Problem 7-83: Fluid Dynamics Analysis** In this problem, a liquid is confined between a top plate with an area \( A \) and a fixed surface below. A force \( F \) is exerted on the top plate, providing it with a velocity \( U \). Assuming the flow is laminar and that there is no pressure variation, you are required to demonstrate, using the Navier-Stokes and continuity equations, that the velocity distribution for this flow is represented by \( u = U(y/h) \) and that the shear stress within the liquid is \( \tau_{xy} = F/A \). **Diagram Explanation:** The diagram illustrates a side view of the setup: - A horizontal surface represents the fixed boundary at the bottom. - Above this lies the liquid, depicted in blue, limited at the top by a moveable plate. - The vertical axis \( y \) is indicated alongside the thickness \( h \) of the liquid layer. - Arrows within the liquid show the velocity distribution, starting from zero at the fixed boundary and increasing linearly (denoted by \( u \)). - The top plate moves to the right with velocity \( U \), driven by the applied force \( F \), also shown as an arrow pointing in the same direction. This setup models a classic fluid mechanics scenario of shear flow between two plates, also known as Couette flow, underlining key principles of fluid dynamics such as velocity gradient and shear stress.
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