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Use the Shell Balance approach for Momentum conservation, create a sketch of a control volume in Cartesian coordinates, and indicate all fluxes and forces in your sketch. Use this Shell Balance to derive the following Momentum conservation equations. x-motion: p(u y-motion: p(u ax +v) = - + μ (3+0) + Bx ax +v) = − ² +μ (0²2² +0²2) - - + By

Use the Shell Balance approach for Momentum conservation, create a sketch of a control volume in Cartesian coordinates, and indicate all fluxes and forces in your sketch. Use this Shell Balance to derive the following Momentum conservation equations. x-motion: p(u y-motion: p(u ax +v) = - + μ (3+0) + Bx ax +v) = − ² +μ (0²2² +0²2) - - + By

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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**Momentum Conservation using the Shell Balance Approach** To apply the shell balance approach for momentum conservation, sketch a control volume in Cartesian coordinates. Clearly indicate all fluxes and forces in your diagram. Using this shell balance, derive the following momentum conservation equations: **Equations:** - **x-motion:** \[ \rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) + B_x \] - **y-motion:** \[ \rho \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = -\frac{\partial p}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) + B_y \] **Explanation of Terms:** - \( \rho \): Density of the fluid - \( u, v \): Velocity components in the x and y directions, respectively - \( p \): Pressure - \( \mu \): Dynamic viscosity - \( B_x, B_y \): Body forces in the x and y directions These equations represent the conservation of momentum in a fluid, accounting for convection, diffusion, and external forces.
Transcribed Image Text:**Momentum Conservation using the Shell Balance Approach** To apply the shell balance approach for momentum conservation, sketch a control volume in Cartesian coordinates. Clearly indicate all fluxes and forces in your diagram. Using this shell balance, derive the following momentum conservation equations: **Equations:** - **x-motion:** \[ \rho \left( u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \right) = -\frac{\partial p}{\partial x} + \mu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) + B_x \] - **y-motion:** \[ \rho \left( u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \right) = -\frac{\partial p}{\partial y} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right) + B_y \] **Explanation of Terms:** - \( \rho \): Density of the fluid - \( u, v \): Velocity components in the x and y directions, respectively - \( p \): Pressure - \( \mu \): Dynamic viscosity - \( B_x, B_y \): Body forces in the x and y directions These equations represent the conservation of momentum in a fluid, accounting for convection, diffusion, and external forces.
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