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Prove the following identities using simplified index notation: а) V-(u v) = u •Vy + v (V• u )

Prove the following identities using simplified index notation: а) V-(u v) = u •Vy + v (V• u )

Introduction to Chemical Engineering Thermodynamics
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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Prove the following identities using simplified index notation: V-(и у) — и-Vу +y (V.u) V-(v•!)= v-(V•") + !:Vy (where" denotes transpose) V (ч-у) - и х(Vxу) - ух(Vxu)- (и-V)у+(у:V)и a) T b) c) Dv ον 1 V(y•v)-vx(V xv) ôt 2 d) Dt D where Dt +v•V is known as the "substantial derivative" operator. Hint: for part (d), it will be easier to work from the right side toward the left.
Transcribed Image Text:Prove the following identities using simplified index notation: V-(и у) — и-Vу +y (V.u) V-(v•!)= v-(V•") + !:Vy (where" denotes transpose) V (ч-у) - и х(Vxу) - ух(Vxu)- (и-V)у+(у:V)и a) T b) c) Dv ον 1 V(y•v)-vx(V xv) ôt 2 d) Dt D where Dt +v•V is known as the "substantial derivative" operator. Hint: for part (d), it will be easier to work from the right side toward the left.
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