Microscopic Momentum Balances

Transcription

1 013 Fluids ectue 6 7 Moison CM //013 CM3110 Tanspot I Pat I: Fluid Mechanics Micoscopic Momentum Balances Pofesso Faith Moison Depatment of Chemical Engineeing Michigan Technological Uniesity 1 Micoscopic Balances We hae been doing a micoscopic shell balance; these ae specific to whatee poblem we ae soling. We seek equations fo micoscopic mass, momentum (and enegy) balances that ae geneal. equations must not depend on the choice of the contol olume, dx d dy equations must captue the appopiate balance 1

2 013 Fluids ectue 6 7 Moison CM //013 Abitay Contol olume in a Flow b `` S ds nˆ Mass Balance On an abitay contol olume: ate of incease net flux of of mass in mass into t d ate of incease of mass Net conection in d (just as we did with the shell balance) Micoscopic mass balance fo any flow

3 013 Fluids ectue 6 7 Moison CM //013 Continuity Equation ds S nˆ Micoscopic mass balance witten on an abitaily shaped olume,, enclosed by a suface, S t x x y y x x y y Gibbs notation: t 5 Momentum Balance On an abitay contol olume: ate of incease net flux of of momentum in momentum sum of into foces on t d d g d ate of incease of momentum Net conection in Foce due to gaity d iscous foces (just as we did with the shell balance) Micoscopic momentum balance fo any flow 3

4 013 Fluids ectue 6 7 Moison CM //013 Equation of Motion ds S nˆ micoscopic momentum balance witten on an abitaily shaped olume,, enclosed by a suface, S Gibbs notation: Gibbs notation: P g t P t Naie Stokes Equation g geneal fluid Newtonian fluid 7 Continuity Equation (And Non Newtonian Equation) on the FONT t The one with is fo non Newtonian fluids 8 Faith A. Moison, Michigan 4

5 013 Fluids ectue 6 7 Moison CM //013 Naie Stokes (Newtonian Fluids Only) is on the back: P t g 9 Poblem Soling Pocedue soling fo elocity and stess fields amended: when using the micoscopic balances 1. sketch system. choose coodinate system 3. simplify the continuity equation (mass balance) 4. simplify the 3 components of the equation of motion (momentum balance) (note that fo a Newtonian fluid, the equation of motion is the Naie Stokes equation) 5. sole the diffeential equations fo elocity and pessue (if applicable) 6. apply bounday conditions 7. calculate any engineeing alues of inteest (flow ate, aeage elocity, foce on wall) T 10 5

6 013 Fluids ectue 6 7 Moison CM //013 EXAMPE I: Flow of a Newtonian fluid down an inclined plane eisited g g x g sin x g g cos ai x H fluid g g g g x y x g sin 0 g cos 11 A coss-section A: EXAMPE II: Pessue dien flow of a Newtonian fluid in a tube: Poiseuille flow steady state well deeloped long tube () fluid g 1 6

7 013 Fluids ectue 6 7 Moison CM // Naie Stokes:

8 013 Fluids ectue 6 7 Moison CM //013 Poiseuille flow of a Newtonian fluid: Po P P ( ) P g Po P () g P o P Engineeing Quantities of Inteest (tube flow) aeage elocity olumetic flow ate component of foce on the wall Q F d d d d d d d d 16 8

9 013 Fluids ectue 6 7 Moison CM //013 Engineeing Quantities of Inteest (any flow) aeage elocity nˆ A A da da component of foce on olumetic Q flow ate nˆ A da ˆ ˆ the wall F e n pi da A at suface 17 Poiseuille flow of a Newtonian fluid: Po P P ( ) P0 g Po P () g P o P g Po P Q 8 Hagen Poiseuille Equation** 18 9

10 013 Fluids ectue 6 7 Moison CM //013 Poiseuille flow of a Newtonian fluid: a,max dd g P 4 o g P 8 o P P 1 dd 19 Poiseuille flow of a Newtonian fluid: a 1.5 p p 0 p p

11 013 Fluids ectue 6 7 Moison CM //013 EXAMPE II: Pessue dien flow of a Newtonian fluid in a tube: Poiseuille flow /<> 1 11