Lecture_27_notes - ChE 37 4—Lecture 27—Navier Stokes...

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Unformatted text preview: ChE 37 4—Lecture 27—Navier Stokes Equations 0 Equations . . . a .. — Continuity Equation: 3—: + V - p2) = 0 e Or: g? + 96% + 9%? + 854;” = 0 (Cartesian) — Or, for constant ,0: V - r7: 0. — Momentum: 5393? + V - 2 —VP — V - ’T + pg'. — Using the product rule and applying the continuity equation: pg—iupt-W: —VP—V-T+p§. — For constant density: pg—‘g -— pa- vv = —VP + uv227+ pg: — This can be rewritten with the material derivative as: p%’ 2 —VP + Mv217+ pg: — THIS IS THE NAVIER STOKES EQUATION governing fluid flow, which is paired with continuity. >u< Four equations (xyz momentum and continuity) in four unkowns: u,v,w,P). # SEE PAGE 450 of your book for this equation in expanded form. 0 Boundary Conditions — Given velocity at inlet or outlet or far field: 112-”, now, 1100. 811 — m = 0 at symmetry points like a pipe centerline. —— v = 0 at walls. 0 Initial Conditions — For unsteady flows, also specify the initial conditions 77(Cc,y, z), P($, y, z). o Analytic solutions available only for laminar flows in simple geometries. o Solve the equations by reducing dimensions and cancelling terms. 0 Example: Barometric Equation: no velocity field a VP = epg‘i. 0 Example: Couette Flow: Unforced flow between parallel plates, one stationary, the other moving at a fixed speed. 0 Example: Flow down an inclined plane. 0 Example: Given a 2—D velocity field7 compute the pressure field: Book Example 9—13. Ml‘tk) M Codi” l/ 4 f. /\7 to 2’; 4 Marat? Q’av 3’ — «N; ’ ’3” “3% "37' “3‘2; PU p :3’0 "Du *9 “a? J n . “ __ “.33 4: “"13: .. COM / /U f V j / ‘73" "37‘ “9,2 ‘2) , r3 'J ,g t ._ I“ d... www- L" 4 vamw : Pi * W w v 3:” ’2‘ Coal /uw‘!‘ f” ’2) v .- w ,. ! M ___.—.. 1L V' '1 —. J F 4‘ m \7‘ VJ ‘4. (5,, «’6 75 v E) a: BOWQV (w 43‘ {1:33 Mummmmwm W“ I a ) 16L ‘4” j: I f,.,..rg,- “fl . ,, égom WWIJV 63 MM, “J “ é; ’ é "’ a . \);n was} I 3'", q} , 44 eynmmlr/ {70 134*» {/er 53??? [gmh’zfl *4 fix éy.o\~.~( }{‘. .—v-’>- yam/(o m wx 1‘“. u: o a" “’“k [w SW” 3 E79 “’9 it iw W ‘ ram/zaml'J’W'é CA5, “‘- 1 “be 3! ' 0,. 7C , rwu,?~'«6‘}“‘“ ‘ / D 7: «-..L F 4 "D (a? 1 9 * film§ + l (WW/k - I'D . 0 m- ‘ 91M , '1 . »/7, 4) P2» 3.: :0 A» no, 4M.wa m» Imam ’7‘/ “‘7. w/ . ,. Vz; ,ru rm” 1'2”; u=97530 , a! w 4% \/: H [A 7" U Riv}. a r H i a: 7‘ ’ TECDflL— "’3';- F L“) :‘xéi’wy ' ‘ “3‘ x ’ tip-«pm 07 i 3:? 7‘“ u 7 4 fix AL 7.7.1 1/ (“D (4:35 :9 6’7 d7 "‘9 “j, ¢o$9{‘/+ g7 6&19 (7’ $6) Eggnmpb W $70!: Ext—«yd 51.]; -/-=> (M P x - g S ( 1 r7 / {*‘(DH‘KP‘ i/IJG’KJ (/4: 470 la .2 s? 1 "\ fl.) 7) I 3795p I r :‘ /(flf£ a Vi; A V 17 + :07? 1M 7” 7“ M :m . \f -. 0 O 4; 4/M+b) O 0 ’9? f - 2R1, {HUM/{4 *3” T)[>‘/‘/’)” 'flbéy flffp/ + 6’64) " 0' 0/ Z \ 2:“; iii/(7“) ":1 “#éqx “PAL: r”? 7 ( 047 AM ’VP {1 7M‘ (7“ ...
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This note was uploaded on 03/11/2012 for the course CHE 374 taught by Professor Davidlignell during the Fall '12 term at Brigham Young University, Hawaii.

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Lecture_27_notes - ChE 37 4—Lecture 27—Navier Stokes...

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