hw8 solution

hw8 solution - 7/1 71:. = [a a:af@i¢€ Mao...

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VI. fluid 8 Fpr [-miw>$\.w WWK QWVAL g) Q :5 g Vgrdrde 7”“) m ; f : PK a M (3313’ KR‘Ckfi)‘ gym“ 4 A Q gnfi HM“ r r ‘T‘ r. htfifir r a “f .. QM“ mm” mm ’ W“ «*K : “MK RM“ ’9 (ix-mu LKU—mufl 9 q 62‘: -C‘q'g) + ESE—m "' U'fl) Q -— 3—3,?( 14—19)“ + Elm—mt “fiat—193‘; A w 3 ‘ ’3?" 4?: RAG-W" %L\—m ~§wfl~0 o :2 Q“F)1'- iU'flYJc ‘; =. 0-390- 1;; «.3: gé“%~"‘5“ “3 9‘ “ «(2.2-0 LI. Solution (a) Given v, = 0, V9 = va(r, 6), and v¢ = 0, examine the continuity equation for an incompressible fluid in spherical coordinates 1 a ., ‘1 6 1 5 ~-;— r‘v + -— v sine + , --— vi :0 r‘6r{ r) rsinEaBC 9 ) 1‘5anan ‘9) Drop the terms involving v, =0 and vq, = 0, and the equation simplifies to 1 <3 . r Sin 6 (v3 3m 8) - 0 Integrate the equation to get 299 sin 6 = m?) where u(r) arises from the integration of a partial differential equation (as opposed to a constant from an ordinary differential equation). (b) Examine the r—component of the Navier—Stokes equation in spherical coordinates dropping the left side of the equation for creeping flow and removing all the terms containing v, and vq, (equal to zero) a? + [ 2 6129 2 t8] =—-—— 7 -—:—————:v co 6?" 2 g"; 68 r“ g _ .1 Substitute v9 = 2;; from continuity into the Navier-Stokes equation 2 avg 2 a (fl)__2u(r)i( 1 )__2u(r)(53;_a) _§¥=—;§55 sing r2 69 sins? 1‘2 shit? 2 2 Vur 2M?“ cota r“ 7“ sine 7’3 53215 6? Zufi‘) ’cot lab“) "cotfi a? =-_—+7l 2 " 3 =“—?lla} 3r 7‘ sm 6 1* sm 6 6r _ a? '— 6? Thus, IP is not a function of r. Examine the (p ~component of the Navier-Stokes equation in spherical coordinates dropping the left side of the equation for creeping flow and removing all the terms containing v, and vq, (equal to zero) 0— 1 “+331 ‘ Tsinaacp '5' 6? = - -an- Thus, 1? is not a function of go. Therefore, 3° can only be a function of 6. (c) Begin with the B—component of the Navier-Stokes equation in spherical coordinates dropping the left side of the equation for creeping flow and removing all the terms containing v, and v(,, (equal to zero) 0 1d?+ [1 (3(76199)+ 1 3(‘ 861%) V9 ] = " —— T 7—— 7” —— — n—_T_— «r d0 } r‘é‘r 61‘ r3 sin a as 5’“ 68 7'“ sin“ 9 Substitute 1.5g = 2:: from continuity into the Navier-Stokes equation 13(‘r26_'vE)__1__6_ r26(u(~r} _ 1 16 1A7,.«3*(())_ 1 152(20221) r26? 87‘ -7'261' 6r 'sinB usinflfiar 61‘ u r -5i116r2 ctr r dr 1 a(. 83::,;)_ 1 a qga um) _ Mr) 5 .aa( 1 _ risineae 5m 69 _rzsin868 8m 68 same “rfisinaaa fan 86‘ sine. " 11(5) 3 L A ’ cote" 116') 6 iii?) 1 uflr) ...= a, 7—- 5111 ~-.——) 2 ,,_ —(-cot6)= ,,, ,4, 2:71—5— r‘smfidfl 51:16 7"“ 5111558 r‘smBsm‘fi 3" sm 6 W) r ___E§__ ._ “ith .. _ifi‘)_ r2 sin2 8 *rzsing 6 — 1’2 sin3 8 1d? 1 ‘1 d ,du’ u? 11'?“ 0 = ““—'+?} "")+'a—{:)~—‘ "1‘19?" rdE smBWdr d7“) r“ Sin°5 1“ 3m 8 0_ 1w+ [1 1 d “an.” - rdé‘ 7? sinargdr r dr, (d) Separate the differentials and multiply by v0 sin 6 1d? [ '1 1 d ( 76511)] _ _ = ,3, __ .. __ r .118 mu? ~r3dr dr rsian?_ [rsinfi 1 d ( gain” 7' d6 "3 sine err r «if .aczzp' [1 d 2cm” 5m :18 "’ Mir r an Now, the left side of the equation is a function of 6 alone while the right side of the equation is a function of r alone. This means both sides must be equal to some constant. Solve for the constant by integrating the differential equation that contains pressure shed? B 1 —: d6 Integrate from the given boundaries where 5P 1 and 39 2 are the values of the modified pressure at 6 = a and (9 = 7r - a, respectively, 5’: 3—: B d? = J. . d8 3: sm 6 I}; ‘ _81 tancgg) _ mtg) .— 25 8 2"?1 — n —8in tang) —. Bin<cot =28in<cot(—2-D Solve for B 92 “ «731 B : 2111(cot(s/2)) Next, integrate the differential equation that contains velocity 1; d ( i, du) B To solve the constants of integration, use the no slip condition on the surface of the spheres u = 0 at r = KR u=0Mr=R Such that o _ 8 NR C1 + C _ 27; KR 2 o—BR Q+C — 21} R 2 Solve the two equations for the constants C q BR: 1 — 27; K BR C3 = -"‘— (K + m Thus, u(r) is Br BR2 K BR 8R r~ R BR 1* R =—+ -—— + =——1——-——m-— =-———— —— —— “(a 2?} 21} 'r 2?} {X 1) 2n [ ( R) K (1 'r 27; [(1 R) + K (1 Substituting for B leads to ‘“(r>=a,§i”{o—%[(l“%)”(i'§)i ...
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hw8 solution - 7/1 71:. = [a a:af@i¢€ Mao...

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