401-2020.20

29.10.20 Lect ① Last time : - For stead , - state , ^ ← 9=4 × 2 pco , 1ham ' e ay PC " " " ' if 1h ' 1=111141 , ÷i÷÷ :# ox have nearly unidirectional [ y= - hcx , flow , with Ivylnlh 'lv × Kr × c- L > > h For x component of N - S : [ dominates ftp.K , for heal 10 q¥rxtoxExrx)= - Et tu Eyck - - steady , 2 Showed last time that term I a term 2 9 by facto - of quite 1h ' l - Re 㱺 # P = µ ¥pr × ( x. yl as for purely unidirectional I flow , except here , Vx and ¥ P may depend on X

② What about y - dependence of P ? Consider y - component of N - S s Ivy In 1h44 , P¥ery = - Typ + µ Try a Vx note that every term in this equation is is smaller by of order ¥ - Ih 't than corresponding terms for x component . 㱺 Typ n Ih 't EP < < QP , so neglect y deep approximate P as a function of x alone , leading to f × py, a function of × . • ° . can write the x - component of N - S as ftp.vxcx.y ) = Ly P' Cx ) ← no y - dependence

const . ④ 㱺 p ' = - sua f Cx ) - = - 31 of , where 9- I ¥ 2Wh ? 2h }x ) flow rate inlet exit ( per width ) w b L 㱺 Ap = Pco , - PA = ' II fo " " " " note strong ~ 3M . this dependence on Z - h

Consider a " slider bearing " ⑤ yq fixed solid lot ho=hCo ) Po n Po and he -_ ha ) ⊥ ( !hCx , 3h , where ho > he - → u ° - → × 1h 't Kl h ' -=F × hCx7 Similar arguments as above 㱺 PEX ) =ud¥r × c × ,y , = p × p where GP can be neglected & Wy ) = Pz y ' + achy t bcx ) Now , Bf . 㱺 v × Cx.y=o ) = U ( ie , bad - U ) and ⊥ ( x ,y=hCx ) ) = O 㱺 aah - - PI h ' - u [ had 㱺 Easy ) = ( y ' - hy ) + ( i - E) U 4 ) ( x ) = E- = g- = Ieds = ( I - E) + Uch - ÷ )

hohe ⑦ 㱺 of = U - ( heh . ) [ lift T → Po ⊥ s/ Po → u x recall , Phil = 61¥ , - ' 27¥ , = EP ✓ hcx ) From this , we can get Ff Guhl Cho 4) ( h - he ) PCX ) Po = [ P' Cx ) dx = . . o = - - 2 O h' Cx ) hi - hi for KIKI ^ AP =P - po Arma , y ' Pm , = I 9.774.7¥ ⊥ × → ÷m÷n ' O L R h 's 0 KIKI

⑧ Fliff ~ L ' W g p ~ L W MU 442 Feist Farag ~ L wore IT T ~ L Wu Uch Po Po ~ → u T smaller than lift by ~ hk