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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;ﬂC ...(8)
where
V
Sc = —~»
DAB
is the Schmidt’s number and
031 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(n1)
= a13(—— 2—a‘ x  2a2t) "(33) Now it is seen that the righthand 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 righthand 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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