where p is surface tension at temperature T .26 Thermocapil-
study, a constant-velocity, constant-form reaction wave tra-
verses the reaction domain, leaving behind a temporarily
unstable steady state of the previously well stirred reaction
solution. This is exactly what we have reported here.
0
0
lary forces are estimated by the Marangoni number, Ma:
Ma \ [(dp/dT )(*T l/gi)
(VI)
where l is the length of the free surface and i is the thermal
di†usivity.
We thank Dr. M. J. B. Hauser for numerous helpful dis-
cussions and for measuring the surface tension. We also thank
Professor S. K. Scott for many helpful suggestions during the
preparation of the manuscript. This work was initially sup-
ported by a professorial grant to R.H.S. from the University of
Zimbabwe Research Board and then by a grant from the
National Science Foundation (Grant No. CHE-9632592) to
R.H.S.
The thermocapillary e†ect is dominant at small d and the
buoyancy e†ects dominate at large d. Thermocapillary e†ects
are controlled by the amount of heat produced by the chemi-
cal reaction at the wave front. If a Ðnite rate of reaction is
assumed for the chemical reaction, then the amount of heat is
proportional to the extent of reaction. Initially, the autocat-
alysis increases the rate of the reaction and hence the amount
of heat evolved. The temperature di†erence generated at the
wave front leads to surface tension changes and thus the wave
accelerates.15 The wave deceleration is due to the chemical
kinetics being unable to supply enough heat to sustain the
same thermocapillary e†ect. On reaching the lateral wall the
wave front dies. However, the chemical reaction can still
proceed as there is unreacted solution below the surface. The
heat evolved leads to double di†usive convection and subse-
quently to the formation of a convective roll.
Further studies being undertaken in our laboratories
involve the evaluation of the numerical values of the Grashof
(or Rayleigh) and Marangoni numbers. In both cases, it is
possible to evaluate a Grashof or Marangoni factor in which
all the parameters are evaluated except for the temperature
jump, *T . Thus we should be able to estimate the strength of
each e†ect with *T . Previous work on non-reacting Ñuids had
deduced that the Marangoni e†ect is, by a temperature-
independent factor 2.5, larger than the Grashof e†ect.27 Thus
as the temperature increases, we expect a dominance of the
surface e†ects and surface velocities with very little activity in
the vertical plane. Our data so far are inconclusive.
We have evaluated, in the chloriteÈthiourea reaction, a
positive isothermal density change *d \ 2.7 kg m~3 (i.e. the
product solution of the reaction is heavier than the reactant
solution at the same temperature). The density change is so
small that a product solution at 1.5 ¡C higher than the reac-
tant solution becomes lighter. Thus the hot reacted solution
can overcome the isothermal density change [see eqn. (I)] and
become (temporarily) lighter than the reactant solution and
slide on top. Heat exchange at the wave front can bring
thermal equilibrium between the two solutions in which the
reacted solution will become heavier than the unreacted solu-
tion and produce the Ðngers shown in Fig. 1(e).
The di†erence in the velocities at the surface and in the bulk
of the Ñuid leaves inhomogeneously reacted pockets of solu-
tion which have the appearance of mosaic-like patterns when
viewed from the top. It appears, also, that the vessel geometry
strongly inÑuences the type of mosaic patterns observed. The
circular nature of the petri dishes used in this study gives what
appears to be cylindrical divergence of the patterns. A recent
theoretical study appears to support our results.28 In this
References
1
A. T. Winfree, in T he Geometry of Biological T ime, Springer
Verlag, New York, 1980.
2
G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium
Chemical Systems, Wiley, New York, 1977.
3
4
A. M. Turing, Philos. T rans. R. Soc. L ondon B, 1952, 37, 237.
Oscillations and T ravelling W aves in Chemical Systems, ed. R. J.
Field and M. Burger, Wiley, New York, 1985.
5
6
Hydrodynamic Instabilities and the T ransition to T urbulence, ed.
H. L. Swinney and J. P. Gollub, Springer, New York, 1981.
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,
Oxford University Press, Oxford, 1961.
7
8
9
J. A. Pojman and I. R. Epstein, J. Phys. Chem., 1990, 94, 4966.
Q. Ouyang and H. L. Swinney, Nature (L ondon), 1991, 352, 610.
J. W. Wilder, B. F. Edwards and D. A. Vasquez, Phys. Rev. A,
1992, 45, 2320.
10 I. R. Epstein, K. Kustin and R. H. Simoyi, J. Phys. Chem., 1992,
96, 5852.
11 H. Taube and H. Dodgen, J. Am. Chem. Soc., 1949, 71, 3330.
12 C. R. Chinake, E. Mambo and R. H. Simoyi, J. Phys. Chem.,
1994, 98, 2908.
13 C. R. Chinake and R. H. Simoyi, J. Phys. Chem., 1994, 98, 4012.
14 M. J. B. Hauser and R. H. Simoyi, Chem. Phys. L ett., 1994, 227,
593.
15 M. J. B. Hauser and R. H. Simoyi, Phys. L ett. A, 1994, 191, 31.
16 R. H. Simoyi, J. Phys. Chem., 1985, 89, 3570.
17 D. M. Kern and C-H. Kim, J. Am. Chem. Soc., 1965, 87, 5309.
18 R. H. Simoyi, J. Masere, C. Muzimbaranda, M. Manyonda and
S. Dube, Int. J. Chem. Kinet., 1991, 23, 419.
19 M. J. B. Hauser and R. H. Simoyi, unpublished results.
20 J. Ross, S. C. Mueller and C. Vidal, Science, 1988, 240, 460.
21 D. A. Vasquez, J. W. Wilder and B. F. Edwards, Phys. Fluids A,
1992, 4, 2410.
22 J. Metzger and D. Schwabe, Physico. Chem. Hyrodyn., 1988, 10,
263.
23 H. E. Huppert and J. S. Turner, J. Fluid Mech., 1981, 106, 299.
24 J. S. Turner, Annu. Rev. Fluid Mech., 1985, 17, 11.
25 C. Perez-Garcia and G. Carneiro, Phys. Fluids A, 1991, 3, 292.
26 Ming-I Char and Ko-Ta Chiang, J. Phys. D: Appl Phys., 1994,
27, 748.
27 K. Nitschke and A. Thess, Phys. Rev. E, 1995, 52, 5772.
28 J. H. Merkin, V. Petrov, S. K. Scott and K. Showalter, J. Chem.
Soc., Faraday T rans., 1996, 92, 2911.
Paper 6/07541J; Received 5th November, 1996
1350
J. Chem. Soc., Faraday T rans., 1997, V ol. 93