Viscous Flow.pptx

LAMINAR FLOW

1. VISCOUS FLOWS L AM I NAR FLOW D e f i ni t i on: T h e f lo w in w hich t he p a rtic l es of f lu id b e ha v e i n o r d erly m a n n er w i t h o ut i n t e r m i x i n g w i th e a c h o t h e r a n d t h e f low ta k es pla c e in n u m ber of sh e e t s, lay e rs or laminar , ea c h slidi n g o v er t h e o t h e r is c a ll e d as l a m i n ar f l o w . Ch a r a c t e r is t i cs of Lam i n a r Fl o w : P a r t ic l e of f lu id b e h a v e in discipl i n e d m a nn e r. N o i n t er - m ix i n g of p artic l e. F l o w tak e s p la c e i n l a y e rs wh i ch g l i de o v er o n e an ot h e r . 2. V e l o ci t y of f low at a po i nt is n e a r l y c o nst a nt in m a g n i t u d e a n d di r e c t i o n. 3. V i s c o us f or c e pl a y s a n i m po r t ant r o l e i n f l uid f l o w ( as c o m pa r e d to o t h e r f o r ces ) . 4. S h e ar s t r e ss is o b t a i n e d b y t h e Ne w t on’s L a w of Viscosi t y . 5. A ny d i s tu rb a nce c au s ed i s q u ick l y da mp ed by visco u s f orc e s 6. Du e t o N o -slip c o ndit i o n , v elo c i t y ac r oss t h e s e c t ion i s n o t u ni f o r m . V el o ci t y g rad i e n t and h e n c e , th e shear s t r e ss g r a d i e nt i s est a blis h ed at ri g ht a n gles t o the d i r ec t i o n of f l ow .

2. Hagen Poiseuille Equation

3. LAMINAR FLOW THROUGH A CIRCULAR PIPE Shear Stress Distribution: sh e ar s t r e s s is m axi m u m . t o = ( ¶ p / ¶ x)(R/ 2 ) VELOCITY DISTRIBUTIONS u = (1 / 4μ ) (- ¶ p / ¶ x) ( R2 -r 2 )

4. (R 2 / 4μ)(- ¶ p/ ¶ x ) = um ax u= u m ax [ 1 - ( r/R ) 2] DISCHARGE AND AVERAGE VELOCITY , u av = [ u m ax / 2 ] Pressure Drop over a Length of Pipe ( p 1 – p 2) = [ 3 2 μuL / D2] f = [16 / Re] POWER REQUIRED TO MAINTAIN THE FLOW ( P ) = r g q hf

5. LA M I N AR F L O W BET W EEN P ARA LLEL P L A T E S – B OTH P L A T ES F IX ED

6. ( ¶ p / ¶ x) = ( ¶ t / ¶ y ) Velocity Distribution: u = 1 / 2μ ( – ¶ p/ ¶ x) [ t y – y 2 ] U m ax = B 2/ 8 μ ( – ¶ p / ¶ x) DISCHARGE AND AVERAGE VELOCITY uav = (2/3) Umax PRESSURE DROP OVER A GIVEN LENGTH OF PLATES ( p 1 – p 2) = [ 1 2 μ u av /B2] L SHEAR STRESS ( t ) = (– ¶ p/ ¶ x ) (B/2)