The Navier-Stokes equations are the set of differential (vector) equations describing the motion of a fluid. F sity (p), constant-viscosity (u) Newtonian fluid, the time-dependent Navier-Stokes equation for the x-dire is dp J²u 8² u ² u + dx əx² მყ2 02² Ət ere ga is the x-direction gravitational acceleration and the x-direction pressure gradient. PGx +μ + ● Direction of flow 2L H | 9x = p ди pg₁+μ (24) = 0 = 2√9+(3 əy² وملح We want to use the differential equation to solve for the velocity profile of the fluid between the plates. differential equation only contains spatial derivatives, and so we must determine the appropriate bounda describing the flow of interest. Consider the situation where both plates are stationary. An import this particular flow is that it must be symmetric about the center line between the plates; that is, the velocity profile is effectively mirrored across the center line between the plates. To see this, r the physical problem looks identical to an observer standing on the center line looking toward eithe symmetry is generally leveraged by taking y = 0 to be the center line between the plates (as shown and defining one of the boundary conditions at the center line. Develop appropriate boundary condi 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 the plates.

Chemistry

10th Edition

ISBN:9781305957404

Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste

Chapter1: Chemical Foundations

Section: Chapter Questions

Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...

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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 dp + μl dx ² u 8² u 8² u + + 9x² dy² əz² ● where 9x is the x-direction gravitational acceleration and the x-direction pressure gradient. dx Direction of flow 2L x L y Į = 9x ди Ət tu pg₁+(354)= 0 - 2 093 +(24) ut 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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