Diffusion of chemically reactive species in a porous medium
Authors:
K. Vajravelu, J. R. Cannon and D. Rollins
Journal:
Quart. Appl. Math. 64 (2006), 17-28
MSC (2000):
Primary 34B15, 76D03, 76S05, 76V05
DOI:
https://doi.org/10.1090/S0033-569X-06-01003-8
Published electronically:
January 24, 2006
MathSciNet review:
2211375
Full-text PDF Free Access
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Abstract: Solutions for a class of nonlinear second-order differential equations, arising in diffusion of chemically reactive species of a Newtonian fluid immersed in a porous medium over an impervious stretching sheet, are obtained. Using the Schauder theory, existence and uniqueness results are established. Moreover, the exact analytical solutions (for some special cases) are obtained and are used to validate the numerical solutions. The results obtained for the diffusion characteristics reveal many interesting behaviors that warrant further study of the effects of reaction rate on the transfer of chemically reactive species.
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2 R. S. R. Gorla, Unsteady mass transfer in the boundary layer on a continuous moving sheet electrode, J. Electrochem. Soc. 125 (1978), 865.
3 R. M. Griffith, Velocity, temperature, and concentration distributions during fibre spinning, Ind. Engng. Chem. Fundam. 3 (1964), 245.
4 L. E. Erickson, L. T. Fan and V. G. Fox, Heat and mass transfer on a moving continuous flat plate with suction or injection, Ind. Engng. Chem. Fundam. 5 (1966), 19.
5 L. J. Crane, Flow past a stretching plate, Z. Agnew. Math. Phys. 21 (1970), 645.
6 B. Siddappa and B. S. Khapate, Rivlin-Ericksen fluid flow past a stretching plate, Rev. Roum. Sci. Techn. Mec. Appl. 21 (1976), 497.
7 T. C. Chiam, Micropolar fluid flow over a stretching sheet, Z. Angew. Math. Mech. 62 (1982), 565.
8 K. R. Rajagopal, T. Y. Na and A. S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta 23 (1984), 213.
9 B. Siddappa and S. Abel, Non-Newtonian flow past a stretching plate, Z. Angew. Math. Phys. 36 (1985), 890.
10 H. I. Andersson and B. S. Dandapat, Flow of a power-law fluid over a stretching sheet, Stability Appl. Anal. Continuous Media 1 (1992), 339.
11 J. Vleggaar, Laminar boundary-layer behaviour on continuous accelerating surfaces, Chem. Engng. Sci. 32 (1977), 1517.
12 P. S. Gupta and A. S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Engng. 55 (1977), 744.
13 P. Carragher and L. J. Crane, Heat transfer on a continuous stretching sheet, Z. Angew. Math. Mech. 26 (1982), 564.
14 L. J. Grubka and K. M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, ASMA J. Heat Transfer 107 (1985), 148.
15 D. R. Jeng, T. C. A. Chang and K. J. DeWitt, Momentum and heat transfer on a continuous moving surface, ASMA J. Heat Transfer 108 (1986), 532.
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18 P. L. Chambre and J. D. Young, On the diffusion of a chemically reactive species in a laminar boundary layer flow, Phys. Fluids 1 (1958), 48.
19 P. L. Chambre, On the ignition of a moving combustible gas stream, J. Chem. Phys. 25 (1956), 417.
20 N. Dural and A. L. Hines, A comparison of approximate and exact solutions for homogeneous irreversible chemical reaction in the laminar boundary layer, Chem. Engng. Commun. 96 (1990), 1.
21 H. I. Andersson, O. R. Hansen and B. Holmedal, Diffusion of a chemically reactive species from a stretching sheet, Int. J. Heat Mass Transfer 37 (1994), 659.
22 S. Abel and P. H. Veena, Visco-elastic fluid flow and heat transfer in a porous medium over a stretching sheet, Internat. J. Non-linear Mech. 33 (1998), 531.
23 R. K. Gupta and T. Sridhar, Visco-elastic effects in non-Newtonian flow through porous media, Rheol. Acta 24 (1985), 148.
24 K. V. Prasad, M. S. Abel and A. Joshi, Oscillatory motion of a visco-elastic fluid over a stretching sheet in porous media, J. Porous Media 3 (2000), 61.
25 K. V. Prasad, M. S. Abel and S. K. Khan, Momentum and heat transfer in visco-elastic fluid flow in porous medium over a non-isothermal stretching sheet, Int. J. Num. Methods for Heat and Fluid Flow 10 (2001), 784.
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27 G. Johnson, M. Massoudi and K. R. Rajagopal, Flow of a fluid-solid mixture between flat plates, Chemical Engng. Sci. 46 (1991), 1713.
28 M. S. Abel, S. K. Khan and K. V. Prasad, Study of visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity, Int. J. Nonlinear Mech. 37 (2002), 81.
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- J. B. McLeod and K. R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary, Arch. Rational Mech. Anal. 98 (1987), no. 4, 385–393. MR 872753, DOI https://doi.org/10.1007/BF00276915
- K. R. Rajagopal and L. Tao, Mechanics of mixtures, Series on Advances in Mathematics for Applied Sciences, vol. 35, World Scientific Publishing Co., Inc., River Edge, NJ, 1995. MR 1370661
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1966. MR 0208797
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490
1 D.-T. Chin, Mass transfer to a continuous moving sheet electrode, J. Electrochem. Soc. 122 (1975), 643.
2 R. S. R. Gorla, Unsteady mass transfer in the boundary layer on a continuous moving sheet electrode, J. Electrochem. Soc. 125 (1978), 865.
3 R. M. Griffith, Velocity, temperature, and concentration distributions during fibre spinning, Ind. Engng. Chem. Fundam. 3 (1964), 245.
4 L. E. Erickson, L. T. Fan and V. G. Fox, Heat and mass transfer on a moving continuous flat plate with suction or injection, Ind. Engng. Chem. Fundam. 5 (1966), 19.
5 L. J. Crane, Flow past a stretching plate, Z. Agnew. Math. Phys. 21 (1970), 645.
6 B. Siddappa and B. S. Khapate, Rivlin-Ericksen fluid flow past a stretching plate, Rev. Roum. Sci. Techn. Mec. Appl. 21 (1976), 497.
7 T. C. Chiam, Micropolar fluid flow over a stretching sheet, Z. Angew. Math. Mech. 62 (1982), 565.
8 K. R. Rajagopal, T. Y. Na and A. S. Gupta, Flow of a viscoelastic fluid over a stretching sheet, Rheol. Acta 23 (1984), 213.
9 B. Siddappa and S. Abel, Non-Newtonian flow past a stretching plate, Z. Angew. Math. Phys. 36 (1985), 890.
10 H. I. Andersson and B. S. Dandapat, Flow of a power-law fluid over a stretching sheet, Stability Appl. Anal. Continuous Media 1 (1992), 339.
11 J. Vleggaar, Laminar boundary-layer behaviour on continuous accelerating surfaces, Chem. Engng. Sci. 32 (1977), 1517.
12 P. S. Gupta and A. S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Engng. 55 (1977), 744.
13 P. Carragher and L. J. Crane, Heat transfer on a continuous stretching sheet, Z. Angew. Math. Mech. 26 (1982), 564.
14 L. J. Grubka and K. M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, ASMA J. Heat Transfer 107 (1985), 148.
15 D. R. Jeng, T. C. A. Chang and K. J. DeWitt, Momentum and heat transfer on a continuous moving surface, ASMA J. Heat Transfer 108 (1986), 532.
16 C.-K. Chen and M.-I. Char, Heat transfer of a continuous stretching surface with suction or blowing, J. Math. Anal. Appl. 135 (1988), 568.
17 K. Vajravelu and D. Rollins, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl. 158 (1991), 241.
18 P. L. Chambre and J. D. Young, On the diffusion of a chemically reactive species in a laminar boundary layer flow, Phys. Fluids 1 (1958), 48.
19 P. L. Chambre, On the ignition of a moving combustible gas stream, J. Chem. Phys. 25 (1956), 417.
20 N. Dural and A. L. Hines, A comparison of approximate and exact solutions for homogeneous irreversible chemical reaction in the laminar boundary layer, Chem. Engng. Commun. 96 (1990), 1.
21 H. I. Andersson, O. R. Hansen and B. Holmedal, Diffusion of a chemically reactive species from a stretching sheet, Int. J. Heat Mass Transfer 37 (1994), 659.
22 S. Abel and P. H. Veena, Visco-elastic fluid flow and heat transfer in a porous medium over a stretching sheet, Internat. J. Non-linear Mech. 33 (1998), 531.
23 R. K. Gupta and T. Sridhar, Visco-elastic effects in non-Newtonian flow through porous media, Rheol. Acta 24 (1985), 148.
24 K. V. Prasad, M. S. Abel and A. Joshi, Oscillatory motion of a visco-elastic fluid over a stretching sheet in porous media, J. Porous Media 3 (2000), 61.
25 K. V. Prasad, M. S. Abel and S. K. Khan, Momentum and heat transfer in visco-elastic fluid flow in porous medium over a non-isothermal stretching sheet, Int. J. Num. Methods for Heat and Fluid Flow 10 (2001), 784.
26 G. Johnson, M. Massoudi and K. R. Rajagopal, Flow of a fluid infused with solid particles through a pipe, Int. J. Engng. Sci. 29 (1991), 649.
27 G. Johnson, M. Massoudi and K. R. Rajagopal, Flow of a fluid-solid mixture between flat plates, Chemical Engng. Sci. 46 (1991), 1713.
28 M. S. Abel, S. K. Khan and K. V. Prasad, Study of visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity, Int. J. Nonlinear Mech. 37 (2002), 81.
29 W. C. Troy, E. A. Overman II, G. B. Ermentrout and J. P. Keener, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math. 44 (1987), 753.
30 J. B. McLeod and K. R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary, Arch. Rational Mech. Anal. 98 (1987), 385.
31 K. R. Rajagopal and L. Tao, Mechanics of Mixtures, World Scientific, River Edge, NJ, 1995.
32 M. Abramowitz and I. Stegun (Eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover, New York, 1965.
33 M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1967.
34 P. Hartman, Ordinary Differential Equations, Second Edition, Birkhäuser, Boston-Basel-Stuttgart, 1982.
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Additional Information
K. Vajravelu
Affiliation:
Dept. of Mathematics, University of Central Florida, Orlando, Florida 32816
J. R. Cannon
Affiliation:
Dept. of Mathematics, University of Central Florida, Orlando, Florida 32816
D. Rollins
Affiliation:
Dept. of Mathematics, University of Central Florida, Orlando, Florida 32816
Received by editor(s):
May 16, 2004
Published electronically:
January 24, 2006
Article copyright:
© Copyright 2006
Brown University
