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Outer pipe wall Consider the steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius R, and outer radius Ro. Assume that the pressure is constant everywhere there is no forced pressure gradient driving the flow, P¡ = P.. However, let the inner cylinder be moving at steady velocity V to the right, essentially a piston. The outer cylinder is stationary. This makes an axisymmetric Couette flow. Use cylindrical coordinates and the equations of motion to generate an expression for the x-component of velocity u as a function of r. Ignore the effects of gravity. Fluid: p, µ R; R, aP_ P2-P X2 %3D

Outer pipe wall Consider the steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius R, and outer radius Ro. Assume that the pressure is constant everywhere there is no forced pressure gradient driving the flow, P¡ = P.. However, let the inner cylinder be moving at steady velocity V to the right, essentially a piston. The outer cylinder is stationary. This makes an axisymmetric Couette flow. Use cylindrical coordinates and the equations of motion to generate an expression for the x-component of velocity u as a function of r. Ignore the effects of gravity. Fluid: p, µ R; R, aP_ P2-P X2 %3D

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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### Text Transcription Consider the steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius \( R_i \) and outer radius \( R_o \). Assume that the pressure is constant everywhere—there is no forced pressure gradient driving the flow, \( P_1 = P_2 \). However, let the inner cylinder be moving at steady velocity \( V \) to the right, essentially a piston. The outer cylinder is stationary. This makes an axisymmetric Couette flow. Use cylindrical coordinates and the equations of motion to generate an expression for the \( x \)-component of velocity \( u \) as a function of \( r \). Ignore the effects of gravity. ### Diagram Explanation The diagram shows a cross-section of an annulus between two cylinders. The fluid with properties \( \rho \) (density) and \( \mu \) (viscosity) fills the space between the cylinders. - **Inner Cylinder**: It has a radius \( R_i \) and moves with a velocity \( V \) to the right. - **Outer Cylinder**: It is stationary with a radius \( R_o \). - **Pressure**: The pressure is constant across the flow, indicated as \( P_1 = P_2 \). - **Flow Direction**: Arrows inside the fluid illustrate the direction of motion from left to right. The diagram includes a mathematical expression for the pressure gradient, written as: \[ \frac{\partial P}{\partial x} = \frac{P_2 - P_1}{x_2 - x_1} \] This illustrates the general concept of pressure gradient in a flow, although it's noted that here \( P_1 = P_2 \). The diagram is labeled, showing the coordinates along the axis labeled \( x_1 \) and \( x_2 \), and the cylindrical cross-section between the radii \( R_i \) and \( R_o \). An outer pipe wall is marked, emphasizing that the outer cylinder is fixed.
Transcribed Image Text:### Text Transcription Consider the steady, incompressible, laminar flow of a Newtonian fluid in an infinitely long round pipe annulus of inner radius \( R_i \) and outer radius \( R_o \). Assume that the pressure is constant everywhere—there is no forced pressure gradient driving the flow, \( P_1 = P_2 \). However, let the inner cylinder be moving at steady velocity \( V \) to the right, essentially a piston. The outer cylinder is stationary. This makes an axisymmetric Couette flow. Use cylindrical coordinates and the equations of motion to generate an expression for the \( x \)-component of velocity \( u \) as a function of \( r \). Ignore the effects of gravity. ### Diagram Explanation The diagram shows a cross-section of an annulus between two cylinders. The fluid with properties \( \rho \) (density) and \( \mu \) (viscosity) fills the space between the cylinders. - **Inner Cylinder**: It has a radius \( R_i \) and moves with a velocity \( V \) to the right. - **Outer Cylinder**: It is stationary with a radius \( R_o \). - **Pressure**: The pressure is constant across the flow, indicated as \( P_1 = P_2 \). - **Flow Direction**: Arrows inside the fluid illustrate the direction of motion from left to right. The diagram includes a mathematical expression for the pressure gradient, written as: \[ \frac{\partial P}{\partial x} = \frac{P_2 - P_1}{x_2 - x_1} \] This illustrates the general concept of pressure gradient in a flow, although it's noted that here \( P_1 = P_2 \). The diagram is labeled, showing the coordinates along the axis labeled \( x_1 \) and \( x_2 \), and the cylindrical cross-section between the radii \( R_i \) and \( R_o \). An outer pipe wall is marked, emphasizing that the outer cylinder is fixed.
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