The Navier-Stokes equations are the set of differential (vector) equations describing the motion of a fluid. For a constant- ensity (p), constant-viscosity (u) Newtonian fluid, the time-dependent Navier-Stokes equation for the x-direction velocity ) is 8² u Ja P9x ou + Qy2 8₂² here g, is the x-direction gravitational acceleration and the x-direction pressure gradient. ди P Ət Direction of flow 2L H y 9x pg₁+μ (3²4)= 0 - (2 √pg₁+1 (34) Əy² We want to use the differential equation to solve for the velocity profile of the fluid between the plates. Our simplified differential equation only contains spatial derivatives, and so we must determine the appropriate boundary conditions describing the flow of interest. Consider the situation where both plates are stationary. An important aspect of this particular flow is that it must be symmetric about the center line between the plates; that is, the shape of the velocity profile is effectively mirrored across the center line between the plates. To see this, recognize that the physical problem looks identical to an observer standing on the center line looking toward either plate. This symmetry is generally leveraged by taking y = 0 to the center line between the plates (as shown in the figure) and defining one of the boundary conditions at the center line. Develop appropriate boundary conditions for this problem. To help you do this, consider that • the no-slip condition applies at either plate, and the symmetry condition tells us something about the derivative of the velocity profile at the center line between the plates.

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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Please help me understand this. The equation simplifies but then I think I have done the boundary correct but I am not sure.

Transcribed Image Text:The Navier-Stokes equations are the set of differential (vector) equations describing the motion of a fluid. For a constant- density (p), constant-viscosity (u) Newtonian fluid, the time-dependent Navier-Stokes equation for the x-direction velocity (u) is P9x J² u Ox² ● dx Direction of flow 2L x dr where 9x is the x-direction gravitational acceleration and the x-direction pressure gradient. 2² 242 y L U əz² Į ди 9x = p. Ət P9₁ +μ (24) = 0 - 2 Jpg.+ (241) x u əy² Əy² We want to use the differential equation to solve for the velocity profile of the fluid between the plates. Our simplified differential equation only contains spatial derivatives, and so we must determine the appropriate boundary conditions describing the flow of interest. Consider the situation where both plates are stationary. An important aspect of this particular flow is that it must be symmetric about the center line between the plates; that is, the shape of the velocity profile is effectively mirrored across the center line between the plates. To see this, recognize that the physical problem looks identical to an observer standing on the center line looking toward either plate. This symmetry is generally leveraged by taking y = 0 to be the center line between the plates (as shown in the figure) and defining one of the boundary conditions at the center line. Develop appropriate boundary conditions for this problem. To help you do this, consider that • the no-slip condition applies at either plate, and the symmetry condition tells us something about the derivative of the velocity profile at the center line between the plates.

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