Stability Analysis of Mass Transfer on a Continuous Flat Plate Moving in Parallel or Reversely to a Free Stream in the Presence of Chemical Reaction by Haar Wavelets

Document Type : Research Paper

Authors
Department of Mathematics, Rani Channamma University, Belagavi, Karnataka-591156, India
Abstract
This study investigates two-dimensional viscous incompressible boundary layer flow involving mass transfer above an uninterrupted flat surface in the presence of chemical reaction. Applicable similarity transformations, transform the leading equations into a system of nonlinear ordinary differential equations. These equations are solved via collocation approach using Haar wavelets. The double solutions exist and are presented through graphs. The obtained solutions are confirmed by comparing them with earlier findings. The various physical quantities are enfolded and convinced carefully using numerical and theoretical approaches. Enhancement in Schmidt number increases the mass transfer rate for upper branch solution and reduces for lower branch solution. Mass immersion arises for constructive chemical reaction and mass transfer enhances for destructive chemical reaction. Finally, the stability analysis is performed.
Keywords
Subjects

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[1] Blasius, H., Grenzschichten in Flssigkeitenmitkleiner Reibung, Zeitscherift fur Mathematik und Physik, 56(1), 1908, 1-37.
[2] Howarth, L., On the solutions of laminar boundary layer equations, Proceedings of the Royal Society A, London, 164(919), 1938.
[3] Riely, N., Magnetohydrodynamics free convection, Journal of Fluid Mechanics, 18(4), 1964, 577-586.
[4] Abu-Sitta, A.M.M., A note on a certain boundary-layer equation, Applied Mathematics and Computation, 64(1), 1994, 73-77.
[5] Sakiadis, B.C., Boundary-layer behaviour on continuous solid surfaces: Boundary-layer equations for two dimensional and axisymmetric flow, AIChE Journal, 7(1), 1961, 26-28.
[6] Abdulhafez, T.A., Skin friction and heat transfer on a continuous flat surface moving in a parallel free stream, International Journal of Heat and Mass Transfer, 28(6), 1985, 1234–1237.
[7] Afzal, N., Badaruddin, A., Elgarvi, A., Momentum and heat transport on a continuous flat surface moving in a parallel stream, International Journal of Heat and Mass Transfer, 36(13), 1993, 3399–3403.
[8] Watanabe, T., Pop, I., Hall effect on magneto-hydrodynamic boundary layer flow over a continuous moving flat plate, Acta Mechanica, 108(1), 1995, 35-47.
[9] Bataller, R.C., Radiation effects in the Blasius flow, Applied Mathematics and Computation, 198(1), 2008, 333-338.
[10] Cortell, R., Numerical solution of classical Blasius flat-plate problem, Applied Mathematics and Computation, 170(1), 2005, 706-710.
[11] Hussaini, M.Y., Lakin, W.D., Nachman, A., On similarity solutions of a boundary layer problem with an upstream moving wall, SIAM Journal of Applied Mathematics, 47(4), 1987, 699–709.
[12] Lin, H.T., Wu, K.Y., Hoh, H.L., Mixed convection from an isothermal horizontal plate moving in parallel or reversely to a free stream, International Journal of Heat and Mass Transfer, 36(14), 1993, 3547–3554.
[13] Afzal, N., Hussain, T., Mixed convection over a horizontal plate, Journal of Heat Transfer, 106(1), 1984, 240-241.
[14] Yao, L.S., Two-dimensional mixed convection along a flat plate, Journal of Heat Transfer, 109(2), 1987, 440-445.
[15] Hsu, C.T., Cheng, T., The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium, International Journal of Heat and Mass Transfer, 28(3), 1985, 683-697.
[16] Mukhopadhyay, S., Layek, G.C., Radiation effects on force convective flow and heat transfer over a porous plate in a porous medium, Meccanica, 44(5), 2009, 587-597.
[17] Wang, L., A new algorithm for solving classical Blasius equation, Applied Mathematics and Computation, 157(1), 2004, 1-9.
[18] Chowdhury, M.M.K., Effect of free convection flow of a visco-elastic fluid past an infinite vertical flat plate in presence of transverse magnetic field, Chemical Engineering Research Bulletin, 10, 2007, 11-31.
[19] Ishak, A., Nazar, R., Pop, I., Boundary-layer flow of a micropolar fluid on a continuously moving or fixed permeable surface, International Journal of Heat and Mass Transfer, 50(23-24), 2007, 4743–4748.
[20] Ishak, A., Nazar, R., Pop, I., Flow and heat transfer characteristics on a moving flat plate in a parallel stream with constant surface heat flux, Heat and Mass Transfer, 45(5), 2009, 563–567.
[21] Weidman, P.D., Kubitschek, D.G., Davis, A.M.J., The effect of transpiration on self-similar boundary layer flow over moving surfaces, International Journal of Engineering Science, 44(11-12), 2006, 730–737.
[22] Wang, C.Y., Stagnation flow towards a shrinking sheet, International Journal of Non-linear Mechanics, 43(5), 2008, 377–382.
[23] Ishak, A., Nazar, R., Pop, I., Dual solutions in mixed convection flow near a stagnation point on a vertical surface in a porous medium, International Journal of Heat and Mass Transfer, 51(5-6), 2008, 1150–1155.
[24] Bachok, N., Ishak, A., Pop, I., Unsteady boundary-layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet, International Journal of Heat and Mass Transfer, 55(7-8), 2012, 2102–2109.
[25] Chambre, P.L., Young, J.D., On diffusion of a chemically reactive species in a laminar boundary layer flow, The Physics of Fluids, 1(1), 1958, 48–54.
[26] Soundalgekar, V.M., Effects of mass transfer and free convective currents on the flow past an impulsively started vertical plate, ASME Journal of Applied Mechanics, 46(4), 1979, 757–760.
[27] Soundalgekar, V.M., Birajdar, N.S., Darvekar, V.K., Mass transfer Effects on the flow past an impulsively started infinite vertical plate with variable temperature or constant heat flux, Astrophysics and Space Science, 100(1), 1984, 159–164.
[28] Das, U.N., Deka, R., Soundalgekar, V.M., Effect of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction, Forschung im Ingenieurwesen, 60(10), 1994, 284–287.
[29] Muthucumaraswamy, R., Ganesan, P., First order chemical reaction on the flow past an impulsively started vertical plate with uniform heat and mass flux, Acta Mechanica, 147(1), 2001, 45–57.
[30] Anjalidavi, S.P., Kandasamy, R., Effected of chemical reaction, heat and mass transfer on laminar flow along a semi-infinite horizontal plate, Heat and Mass Transfer, 35(6), 1999, 465–467.
[31] Anjalidavi, S.P., Kandasamy, R., Effects of chemical reaction, heat and mass transfer on MHD flow past a semi infinite plate, Zeitschrift für Angewandte Mathematik und Mechanik, 80(10), 2000, 697–700.
[32] Postelnicu, A., Influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Heat and Mass Transfer, 43(6), 2007, 595–602.
[33] Bhattacharyya, K., Layek, G.C., Similarity solution of MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing, Meccanica, 47(4), 2012, 1043-1048.
[34] Anderson, H.I., Hansen, O.R., Holmedal, B., Diffusion of a chemically reactive species from a stretching sheet, International Journal of Heat and Mass Transfer, 37(4), 1994, 659–664.
[35] Chamkha, A.J., Aly, A.M., Mansour, M.A., Similarity solution for unsteady heat and mass transfer from a stretching surface embedded in a porous medium with suction/injection and chemical reaction effects, Chemical Engineering Communications, 197(6), 2010, 846–858.
[36] Kandasamy, R., Periasamy, K., Sivagnana Prabhu, K.K., Chemical reaction, heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects, International Journal of Heat and Mass Transfer, 48(21-22), 2005, 4557–4561.
[37] Bhattacharyya, K., Layek, G.C., Chemically reactive solute distribution in MHD boundary layer flow over a permeable stretching sheet with suction or blowing, Chemical Engineering Communications, 197(12), 2010, 1527–1540.
[38] Bhattacharyya, K., Layek, G.C., Slip effect on diffusion of chemically reactive species in boundary layer flow over a vertical stretching sheet with suction or blowing, Chemical Engineering Communications, 198(11), 2011, 1354–1365.
[39] Bhattacharyya, K., Mass transfer on a continuous flat plate moving in parallel or reversely to a free stream in the presence of a chemical reaction, International Journal of Heat and Mass Transfer, 55(13-14), 2012, 3482–3487.
[40] Sachdev, P.L., Bujurke, N.M., Awati, V.B., Boundary Value Problems for Third‐Order Nonlinear Ordinary Differential Equations, Studies in Applied Mathematics, 115(3), 2005, 303-318.
[41] Kudenatti, R.B., Awati, V.B., Solution of pressure gradient stretching plate with suction, Applied Mathematics and Computation, 210(1), 2009, 151-157.
[42] Rasheed, A., Anwar, M.S., Numerical computations of fractional nonlinear Hartmann flow with revised heat flux model, Computers and Mathematics with Applications, 76(10), 2018, 2421-2433.
[43] Awati, V.B., Dirichlet series and approximate analytical method for the solution of MHD boundary layer flow of Casson fluid over a stretching/shrinking sheet, TWMS Journal of Applied and Engineering Mathematics, 7(2), 2017, 343-353.
[44] Makinde, O.D., Awati, V.B., Bujurke, N.M., Dirichlet series and closed-form exact solutions of MHD Casson fluid flow over a permeable stretching/shrinking sheet, Palestine Journal of Mathematics, 10(1), 2021, 109-119.
[45] Awati, V.B., Mahesh Kumar, N., Chavaraddi, K.B., Dirichlet series and approximate analytical solutions of MHD flow over a linearly stretching sheet, International Journal of Industrial Mathematics, 7(4), 2015, 343-350.
[46] Puneeth, V., Ali, F., Khan, M.R., Anwar, M.S., Ahammad, N.A., Theoretical analysis of the thermal characteristics of Ree–Eyring nanofluid flowing past a stretching sheet due to bioconvection, Biomass Conversion and Biorefinery, 14(7), 2024, 8649-8660.
[47] Awati, V.B., Dirichlet series and analytical solutions of MHD viscous flow with suction/blowing, Applied Mathematics and Nonlinear Sciences, 2(2), 2017, 341-350.
[48] Alzahrani, J., Vaidya, H., Prasad, K.V., Rajashekhar, C., Mahendra, D.L., Tlili, I., Micro-polar fluid flow over a unique form of vertical stretching sheet: Special emphasis to temperature-dependent properties, Case Studies in Thermal Engineering, 34, 2022, 102037.
[49] Hussain, Z., Muhammad, S., Anwar, M.S., Effects of first-order chemical reaction and melting heat on hybrid nanoliquid flow over a nonlinear stretched curved surface with shape factors, Advances in Mechanical Engineering, 13(4), 2021, 1-12.
[50] Anwar, M.S., Alam, M.M., Khan, M.A., Abouzied, A.S., Hussain, Z., Puneeth, V., Generalized viscoelastic flow with thermal radiations and chemical reactions, Geoenergy Science and Engineering, 232, 2024, 212442.
[51] Lepik, Ü., Solving PDEs with the aid of two dimensional Haar wavelets, Computers & Mathematics with Applications, 61(7), 2011, 1873-1879.
[52] Chen, C.F., Hsiao, C.H., Haar wavelet method for solving lumped and distributed parameter systems, IEE Proceedings-Control Theory & Application, 144, 1997, 87-94.
[53] Hsiao, C.H., State analysis of the linear time delayed systems via Haar wavelets, Mathematics and Computers in Simulation, 44(5), 1997, 457-470.
[54] Lepik, Ü., Numerical solution of evolution equations by the Haar wavelet method, Applied Mathematics and Computation, 185(1), 2007, 695–704.
[55] Lepik, Ü., Solving fractional integral equations by the Haar wavelet method, Applied Mathematics and Computation, 214(2), 2009, 468–478.
[56] Majak, J., Pohlak, M., Eerme, M., Application of the Haar wavelet-based discretization technique to problems of orthotropic plates and shells, Mechanics of Composite Materials, 45(6), 2009, 631–642.
[57] Awati, V.B., Kumar, M., Wakif, A., Haar wavelet scrutinization of heat and mass transfer features during the convective boundary layer flow of a nanofluid moving over a nonlinearly stretching sheet, Partial Differential Equations in Applied Mathematics, 4, 2021, 100192.
[58] Awati, V.B., Mahesh Kumar, N., Analysis of forced convection boundary layer flow and heat transfer past a semi-infinite static and moving flat plate using nanofluids-by Haar Wavelets, Journal of Nanofluids, 10(1), 2021, 106-117.
[59] Merkin, J.H., On dual solutions occurring in mixed convection in a porous medium, Journal of Engineering Mathematics, 20(2), 1986, 171-179.
[60] Harris, S.D., Ingham, D.B., Pop, I., Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip, Transport in Porous Media, 77(2), 2009, 267-285.