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According to the lubrication approximation, flows with almost parallel streamlines can be treated as locally fully-developed. For example pressure-driven flow between slightly non-parallel plates can be treated by the lubrication approximation which leads to lubrication equations for flow in a channel. In a manner of speaking the flow "adjusts" itself locally to conform to the requirements laid down by the change in the cross-sectional area. This illustrates a case where the lubrication approximation can be used to take into account a change in geometry. The lubrication approximation can also be used to take into account a change in the flowrate. Consider pressure- driven steady flow of an incompressible Newtonian fluid with density p and viscosity µ in a parallel plate channel of width 2d. The channel walls are slightly permeable and allow the fluid to flow very slowly into the channel through each porous wall at a uniform flowrate of V per unit length per unit depth. V is thus the flowrate per unit length (in the x-direction) per unit depth (in the z-direction) of fluid through one of the walls. The initial flowrate per unit depth at the inlet of the channel is qo. Starting from the lubrication equations for pressure-driven flow in a channel, derive an expression for the change in the pressure drop per unit length due to addition of the fluid. You can assume that the no slip condition holds for vx at the walls. You can assume the plates have a large depth, so that there is no dependence on z, and vz = 0. yt Z., X ↓ y↑ ↓ d

According to the lubrication approximation, flows with almost parallel streamlines can be treated as locally fully-developed. For example pressure-driven flow between slightly non-parallel plates can be treated by the lubrication approximation which leads to lubrication equations for flow in a channel. In a manner of speaking the flow "adjusts" itself locally to conform to the requirements laid down by the change in the cross-sectional area. This illustrates a case where the lubrication approximation can be used to take into account a change in geometry. The lubrication approximation can also be used to take into account a change in the flowrate. Consider pressure- driven steady flow of an incompressible Newtonian fluid with density p and viscosity µ in a parallel plate channel of width 2d. The channel walls are slightly permeable and allow the fluid to flow very slowly into the channel through each porous wall at a uniform flowrate of V per unit length per unit depth. V is thus the flowrate per unit length (in the x-direction) per unit depth (in the z-direction) of fluid through one of the walls. The initial flowrate per unit depth at the inlet of the channel is qo. Starting from the lubrication equations for pressure-driven flow in a channel, derive an expression for the change in the pressure drop per unit length due to addition of the fluid. You can assume that the no slip condition holds for vx at the walls. You can assume the plates have a large depth, so that there is no dependence on z, and vz = 0. yt Z., X ↓ y↑ ↓ d

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 4 According to the lubrication approximation, flows with almost parallel streamlines can be treated as locally fully-developed. For example pressure-driven flow between slightly non-parallel plates can be treated by the lubrication approximation which leads to lubrication equations for flow in a channel. In a manner of speaking the flow "adjusts" itself locally to conform to the requirements laid down by the change in the cross-sectional area. This illustrates a case where the lubrication approximation can be used to take into account a change in geometry. The lubrication approximation can also be used to take into account a change in the flowrate. Consider pressure- driven steady flow of an incompressible Newtonian fluid with density p and viscosity µ in a parallel plate channel of width 2d. The channel walls are slightly permeable and allow the fluid to flow very slowly into the channel through each porous wall at a uniform flowrate of V per unit length per unit depth. V is thus the flowrate per unit length (in the x-direction) per unit depth (in the z-direction) of fluid through one of the walls. The initial flowrate per unit depth at the inlet of the channel is qo. Starting from the lubrication equations for pressure-driven flow in a channel, derive an expression for the change in the pressure drop per unit length due to addition of the fluid. You can assume that the no slip condition holds for vx at the walls. You can assume the plates have a large depth, so that there is no dependence on z, and vz = 0. y 2. y ▬▬▬▬▬▬▬▬▬ X
Transcribed Image Text:Problem 4 According to the lubrication approximation, flows with almost parallel streamlines can be treated as locally fully-developed. For example pressure-driven flow between slightly non-parallel plates can be treated by the lubrication approximation which leads to lubrication equations for flow in a channel. In a manner of speaking the flow "adjusts" itself locally to conform to the requirements laid down by the change in the cross-sectional area. This illustrates a case where the lubrication approximation can be used to take into account a change in geometry. The lubrication approximation can also be used to take into account a change in the flowrate. Consider pressure- driven steady flow of an incompressible Newtonian fluid with density p and viscosity µ in a parallel plate channel of width 2d. The channel walls are slightly permeable and allow the fluid to flow very slowly into the channel through each porous wall at a uniform flowrate of V per unit length per unit depth. V is thus the flowrate per unit length (in the x-direction) per unit depth (in the z-direction) of fluid through one of the walls. The initial flowrate per unit depth at the inlet of the channel is qo. Starting from the lubrication equations for pressure-driven flow in a channel, derive an expression for the change in the pressure drop per unit length due to addition of the fluid. You can assume that the no slip condition holds for vx at the walls. You can assume the plates have a large depth, so that there is no dependence on z, and vz = 0. y 2. y ▬▬▬▬▬▬▬▬▬ X
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