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Find the equation of motion (Navier Stokes) for a viscous fluid between two rotating concentric cylinders (axle and shaft). The inner cylinder has the radius ro and rotates at angular speed wo. The outer cylinder has the radius R and is stationary. Write down each vector component of the equation in a separate line and use reasonable assumptions to simplify the equation, especially the derivatives. Be sure to use cylindrical coordinates for the convective operator and the other derivatives.

Find the equation of motion (Navier Stokes) for a viscous fluid between two rotating concentric cylinders (axle and shaft). The inner cylinder has the radius ro and rotates at angular speed wo. The outer cylinder has the radius R and is stationary. Write down each vector component of the equation in a separate line and use reasonable assumptions to simplify the equation, especially the derivatives. Be sure to use cylindrical coordinates for the convective operator and the other derivatives.

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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**Title: Derivation of the Navier-Stokes Equation for a Viscous Fluid Between Rotating Concentric Cylinders** **Introduction:** This exercise focuses on deriving the Navier-Stokes equation applicable to the motion of a viscous fluid situated between two rotating concentric cylinders. The inner cylinder, resembling an axle or shaft, has a radius denoted by \( r_0 \) and rotates with an angular speed \( \omega_0 \). The outer cylinder, with radius \( R \), remains stationary. **Objective:** Our goal is to find each vector component of the equation, documenting them separately, and to apply reasonable assumptions to simplify the overall equation, particularly the derivatives. Cylindrical coordinates must be used when formulating the convective operator and other derivatives. **Procedure:** 1. **Identify Parameters:** - Inner Cylinder: Radius = \( r_0 \), Angular Speed = \( \omega_0 \) - Outer Cylinder: Radius = \( R \), Stationary 2. **Navier-Stokes Equation Components:** - We seek to write the radial, angular, and axial components of the Navier-Stokes equation for this setup. 3. **Assumptions:** - Consider a steady flow with no radial velocity component. - Assume axial symmetry (no variation in the angular direction). - The flow is incompressible. - Boundary conditions include no-slip conditions at the surfaces of the cylinders. 4. **Utilizing Cylindrical Coordinates:** - Focus on expressing the vector components and derivatives in terms of radial (\( r \)), angular (\( \theta \)), and axial (\( z \)) coordinates. This systematic derivation ensures clarity in understanding fluid dynamics within rotating systems, illustrating key concepts in vector calculus and fluid mechanics.
Transcribed Image Text:**Title: Derivation of the Navier-Stokes Equation for a Viscous Fluid Between Rotating Concentric Cylinders** **Introduction:** This exercise focuses on deriving the Navier-Stokes equation applicable to the motion of a viscous fluid situated between two rotating concentric cylinders. The inner cylinder, resembling an axle or shaft, has a radius denoted by \( r_0 \) and rotates with an angular speed \( \omega_0 \). The outer cylinder, with radius \( R \), remains stationary. **Objective:** Our goal is to find each vector component of the equation, documenting them separately, and to apply reasonable assumptions to simplify the overall equation, particularly the derivatives. Cylindrical coordinates must be used when formulating the convective operator and other derivatives. **Procedure:** 1. **Identify Parameters:** - Inner Cylinder: Radius = \( r_0 \), Angular Speed = \( \omega_0 \) - Outer Cylinder: Radius = \( R \), Stationary 2. **Navier-Stokes Equation Components:** - We seek to write the radial, angular, and axial components of the Navier-Stokes equation for this setup. 3. **Assumptions:** - Consider a steady flow with no radial velocity component. - Assume axial symmetry (no variation in the angular direction). - The flow is incompressible. - Boundary conditions include no-slip conditions at the surfaces of the cylinders. 4. **Utilizing Cylindrical Coordinates:** - Focus on expressing the vector components and derivatives in terms of radial (\( r \)), angular (\( \theta \)), and axial (\( z \)) coordinates. This systematic derivation ensures clarity in understanding fluid dynamics within rotating systems, illustrating key concepts in vector calculus and fluid mechanics.
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