Foundations of Fluid Mechanics I
Foundations of Fluid Mechanics I M E 521
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This 0 page Class Notes was uploaded by Chester Goldner III on Sunday November 1, 2015. The Class Notes belongs to M E 521 at Pennsylvania State University taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/233073/m-e-521-pennsylvania-state-university in Mechanical Engineering at Pennsylvania State University.
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Date Created: 11/01/15
Equation Sheet for Midterm Exam ME 521 prepared by Professor J M Cimbala Tensor transformation rules for rotation Cy 2 cos 06 Where 04 angle between the old 139 and new j axes Then for tensorA A CWAHAV W CWC Ay H4 qu CbJWL etc Epsilondelta relation swam 5 5 75 5 xm jquot m m Cross product a x 1k eykaxb Gauss Stokes and Leibniz theorems 0F t t G Ja xdequdAx s qudAg ud L A C i I F7ctdV jmdw c FQJWAd dl V at 2 WI AU DF 0F 0F Material derivative u Dr 0t x for any uid Principal strain rates eigenvalues found from m D 0F Evy de 5ng iFquA following a uid particle Where F is some variable 7 l Strain rate tensor 8V 7 2 MW M Reynolds transport theorem Where F can be any quantity per unit volume Conservation of mass 0p 0 adv quuJaMJ CV CS 0p 0 0 at 0xI pu Conservation of 1puxdV japuxujdAJ ngde larvall CV CS CV CS For any fluid a a 0r 5pugpuujpg Constitutive equation relation between stress and strain with 0 defined as the deviatan39c stress tensor m For Newtonian fluid TV 7pde 2 lemma and the famous N avierStokes equation results aux aux 0p 0 7 m Bu 0 Bu p u 7 pgx u A 01 J 0x 0x 0x 0x 6x1 6x1 0x F 39 3912 T p5 Zye au a 01 02quot Or lncam reSSZ e OW I 7 I I u 7 I p I J J 0 0t 1 0x 0x pg axjax Incompressible conservation equations of mass and in Cartesian coordinates xyz Bu 0v 0w 2 02 02 02 a a 0 0 0 2 2 2 uVu v w 0x By 02 0x By 02 0x By 02 au au au au 0 azu azu 0qu 0 u v w pgxu 7 0x2 yz g avuavv w iiap azvazvazv 0 2 6y pgy axz yz 022 0w 0w 0w 0w 017 02w02w02w u V w 7 p at 0x ay 02 az pgl 1 0x2 yz 022 1 0M 0V 1 0v 1 ex 039gt6 g y 2 0y 0x 2 y W 0y 2y 039 wi i W 2 0x 0y 70v Bu Incompressible conservation equations of mass and in cylindrical coordinates re z 1 0 1 0 0 1 0 0 1 02 02 a 0 0 0 m u u0 V2 r 2 Z 2 aVu i uz r0r r09 02 r0r 0r r 09 02 Or r 09 02 au39 aailu90i uz au39 flu 7la p g v Via fizu 732 01 Or r 00 02 r p 0r r 00 0 0 0 n rlu9ampuz ilu39u9g9V Vzu9i2u9 2 01 Or r 00 02 r pr00 r r 00 0iur lu90i 27la pgzvVzuz 01 Or r 00 02 p 02 0a 1 10 a 1 r 0 M5 1 0M 1 err Urr eoo 55 er 0y5 0r 2y r 06 r 2y 20r r Zr 09 2 a 713W 1 z 7 r 0r 5 r 09 BE 0 a 0M Mechanical energy equation uE puxgx Tyux p iy Rate ofv1scous d1551pat10n of kinet1c 01 0x 0x 0x 0 energy per unit volume 2 0110 Deviatoric stress tensor Kinetic energy per unit volume m J First Law Heat equation 1pe lxuI Jdl i CJSpe ailI ujdA ngxude CJSTyudiJ 7 J qdix CV CS CV CS CS D 0 0g De g 0a eiuu u ru 7 7 7 th 2 mg 0xy 0x Dt 0x1 paxx DT air DT D 0 0T Ifincompressible pCFEkm2ueyey Ifidealgas pCpEFIaxka Ifidealgas alvery low DT OZT Mach number pC k F Di fixxfixI The Tds equations of Thermodynamics Tds ale de Tds db 7 Udp Where T temperature p pressure e specific internal energy h specific enthalpy s specific entropy and U 1 0 specific volume Bernoulli equations For incompressible steady irrolalional flow can be viscous q2 E g2 constant 0 I7 For incompressible steady inviscid flow can be rotational q2 g2 constant along a streamline 0 For steady compressible inviscid irrolalional isenlropic flow 11 q2 g2 constant 11 E e g 0
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