1. Writing the Navier-Stokes Equation in Cylindrical Coordinates Consider a point (x, y, z) in three-dimensional space. We project the point onto the ry plane, and connect the projected point in the plane, say A, to the origin. The angle that the ray connecting A to the origin makes with the r axis is 0, and its length is r. Then, x = r cos 0, y =r sin 0, and z remains unchanged, z = z. These constitute the cylindrical coordinates. (a) Use chain rule of differentiation to show that sin 0 (cos 0) Or cos e + ar (sin 0) and, of course, 0/dz does not change. (b) Write down the Navier-Stokes equation in Cartesian coordinates, and using (a) rewrite it in cylindrical coordinates.
1. Writing the Navier-Stokes Equation in Cylindrical Coordinates Consider a point (x, y, z) in three-dimensional space. We project the point onto the ry plane, and connect the projected point in the plane, say A, to the origin. The angle that the ray connecting A to the origin makes with the r axis is 0, and its length is r. Then, x = r cos 0, y =r sin 0, and z remains unchanged, z = z. These constitute the cylindrical coordinates. (a) Use chain rule of differentiation to show that sin 0 (cos 0) Or cos e + ar (sin 0) and, of course, 0/dz does not change. (b) Write down the Navier-Stokes equation in Cartesian coordinates, and using (a) rewrite it in cylindrical coordinates.
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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Transcribed Image Text:1. Writing the Navier-Stokes Equation in Cylindrical Coordinates
Consider a point (x, y, z) in three-dimensional space. We project the point onto the ry
plane, and connect the projected point in the plane, say A, to the origin. The angle that
the ray connecting A to the origin makes with the r axis is 0, and its length is r. Then,
x = r cos 0, y =r sin 0, and z remains unchanged, z = z. These constitute the cylindrical
coordinates.
(a) Use chain rule of differentiation to show that
sin 0
(cos 0)
Or
cos e
+
ar
(sin 0)
and, of course, 0/dz does not change.
(b) Write down the Navier-Stokes equation in Cartesian coordinates, and using (a) rewrite
it in cylindrical coordinates.
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