20005a86_179

20005a86_179 - ON SIMILARITY SOLUTION OF CHEMICAL BOUNDARY...

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Unformatted text preview: ON SIMILARITY SOLUTION OF CHEMICAL BOUNDARY LAYER THEORY by K. M. CHAKRABART], Department of Mathematics, Regional Engineering College, Durgapur 9 (Received 5 December 1974) This paper presents an investigation of similarity solutions in chemical boundary layer theory. It is seen that when the flow is accompanied with nth order homogeneous volume reaction, we may get similarity solution when n ¢ 1. INTRODUCTION There is a possibility of an interesting and important investigation in boun- dary layer theory when the flow is along a soluble body and accompanied by chemical reactions. This type of study is useful in chemical physics and engineering. Some research work in this direction has been done by Ghosal et a1. (1972) and others. ANALYSIS Let us consider the flow along a flat plate that contains a species A slightly soluble in the fluid B. The concentration at the plate surface is assumed to be 5“, which is the solubility of A in B and the concentration far from the plate is assumed to be 6‘“; z 0. We are considering here the case when A reacts with B by an nth order homo- geneous chemical reaction (volume). We further assume here that the concen- tration of dissolved A is small enough to consider the physical properties p, ,u DAB to be constants throughout the fluid, where p and [I are respectively the density and the viscosity of the fluid B and DAB is the diflusivity of A in B. The boundary layer equations of the above system are adequately described by the following equations (Bird et al. 1960): 612 -aa —aak 62a E+ufi+v$~v$é W“) a; 55 ‘ a} + 5);; I 0 .42) ac — at —a& 026 —,‘ 6i + 140i + 226).) m DAB (3)72 + KC ...(3) Vol. 8, No. 2 180 K. M. CHAKRABARTl Where (.5 is the concentration of the solute which is a function of f, )7, f; n is an integer which indicates the order of the reaction and K the volume rate constant. The boundary conditions of the Sy:t€‘m are given by 5:0; 0. 5* C102,?» _ _ _ _ ...(4) y —>oo ; u —> U0° (constant), C ——>0 (subscript w is the wall condition). The variables u, v, x, y, t are made dimensionless as follows: i 5 i 7 t 7 V1? I} x “ L 7 .} ’ L 5 V L2 ;, ~ _ _ .2 ...(5). uL 'UL C l a: ’ 1’"? 6:5» where L is a representative length and C0 denotes a typical value of 5, say at x = 0 on the plate at t = 0. Introducing (5) in eqns. (1), (2) and (3) we get ' a: ax 5y “ 5y: ~16) flu (3'0 8*; + 5—); ~ 0 .,.(7) 0C BC 6C W 1 52C C73“1 LZK n 57 + u é—x- + Va v 3;— B}? + -~—r;fl-C ...(8) where V Sc = —~» DAB is the Schmidt’s number and 03-1 UK “Muff is a dimensionless constant. The boundary conditions now become y=0; u=v:0, C=Cw(x,t) y —->oo; u —>- Um (constant), C —>0 “(9) The equation of continuity is satisfied if we choose a dimensionless stream function 1,0 such that 0N SIMILARITY SOLUTION OF CHEMICAL BOUNDARY LAYER THEORY 181 If we choose, a similarity variable in the form n =y¢1(x,r) ...(10) and the dependent variables as follows: f("):l/I(xayat)/¢2(x7t) 0(a) : AC (x,y,t)/Cw (XJ) ...(12) (where A is a constant which is to be determined), the present problem reduces to one of determining the functions 461 (x, t), «#2 (x, t) and Cw (x, t) such that eqns. (6) and (8) are reducible to ordinary differential equations for f(11) and 007) with proper boundary conditions. With the following identities from eqns. (10), (ll) and ('12), V _0‘l/I u w a: : ¢i¢2.fl(") ...(13) v : _ _ _ f+ Wm] ...(14) C.” 2 7 9 ...(15) substituted in eqns. (6) and (8) We obtain f’” —- aznf” # (a2 + agif’ + a5fl” — (04 + a5)f’2: o ...(16) fi 9“ — ((1217 — a5f) 6’ — (a6 + a7f’) a + a1 9" : 0 ...(17) where ai = Cid/4’12 ...(18) a2 : din/(1)13 ...(19) as : dw/ («bf/«#2) ...(20) 04 : («Mira/W ...(21) as : «fizz/«#1 ...(22) as = Cit/(C.- ¢fl ...(23) a7 : 452 Cit/(451G) ...(24) It is to be noted here that A is made equal to CO (L2 k/ u)“ ""1’. We now try to find out for what values of n similarity solutions for eqns. ([6) and (17) are possible. From eqn. (19) we get ¢1 2 [B (x) — 202’].é ...(25) where B (x) is yet to be determined. Combining eqns. (l9) and (20) we get m = D (x) ch" ...(26) 182 K. M. CHAKRABARTI where D(x) is an arbitrary function and w 2 a3/a2. Thus (M x D(x) [B (x) — 2azt]“'“2. “(27) A similar combination of eqns. (21) and (22) gives m «I: E (I) W ...(28) where 8205/04, 04¢0. The case of (14:0 will be treated later. Equa- tion (21) may be SOIVed giving it ~1) Wu «#1 : x + Fm] ...(29) and therefore 452 is given by ‘14 (6.. 1) ens—1) 4,2 2 E(t) [ ~13“) x + F(t)] ...(30) Comparing eqns. (25), (27), (29) and (30), We get 32w:~—1, B(x):—vrx D(x) : E(t) 2 am, F(t) = -— 2a2t. With the above results we get $1 : (~ Zfi x — 2a2t )-m ---(31) “12 2 1/2 4,2 I a,2(w x _ 242; ) ...(32) “12 From relations (18) and (31) it appears that 20 —1/(n~1) szalll‘"“’(—— —" x —— 2a2t) -1i(n-1) = a13(——- 2—a‘ x - 2a2t) "(33) Now it is seen that the right-hand sides of relations (18)—(24) become constant with the above results for (ill, (fig and Cw. We see thatn may be any value except 1. Thus similarity solutions of eqns. (16) and (17) are possible if n;& 1. The case of a4 = 0 will be considered now. If a,1 = 0 then eqn. (25) becomes (fill 2 [a14 —— 2a2t]‘lt ...(34) ON SIMILARITY SOLUTION OF CHEMICAL BOUNDARY LAYER THEORY 183 where a” is a constant of integration. From eqns. (20), (22) and (34) we ultimately get 452 : (ac-,x + (115)(a14 —— 20201 ...(35)' where £115 is a constant. Also from eqns. (18) and (34) we get Cw : b ((114 —— 2a2t)‘1""'“ where b = af""‘1’. It can be easily verified that the right-hand sides of relations (23) and (24)- are constants with the abOVe results for (1),, (1)2 and C... Hence similarity solu- tion is also possible in this case for any value of n except n z: ACKNOWLEDGEMENT The author is grateful to Prof. S. K. Ghosal, Department of Mathematics, Jadavpur University, Calcutta, for his helpful suggestions. REFERENCES Bird, R. B., Stewart, W. E.. and Lightfoot, E. N. (1960). Transport Phenomena. John Wiley and Sons, Inc., New York, pp. 559. Ghosal, S. K., Sikdar, S. C., and Roy, S.C. (1972). On concentration boundary layer. Presented at the Seminar on Continuous Mechanics held at Centre of Advance Study in Applied Mathematics, Calcutta University, Calcutta, in March 1972. ...
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20005a86_179 - ON SIMILARITY SOLUTION OF CHEMICAL BOUNDARY...

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