### Sequential Implicit Numerical Scheme for Pollutant and Heat Transport in a Plane-Poiseuille Flow

Document Type: Research Paper

Author

Department of Mathematics, Rivers State University, Port Harcourt, Nigeria

Abstract

A sequential implicit numerical scheme is proposed for a system of partial differential equations defining the transport of heat and mass in the channel flow of a variable-viscosity fluid. By adopting the backward difference scheme for time derivative and the central difference scheme for the spatial derivatives, an implicit finite difference scheme is formulated. The variable-coefficient diffusive term in each equation is first expanded by differentiation. The next step of the sequential approach consists of providing a solution of the temperature and concentration, before providing a solution for the velocity. To verify the numerical scheme, the results are compared with those of a Matlab solver and a good agreement are found. We further conduct a numerical convergence analysis and found that the method is convergent. The numerical results are investigated against the model equations by studying the time evolution of the flow fields and found that the data, such as the boundary conditions, are perfectly verified. We then study the effects of the flow parameters on the flow fields. The results show that the Solutal and thermal Grashof numbers, as well as the pressure gradient parameter, increase the flow, while the Prandtl number and the pollutant injection parameter both decrease the flow. The conclusion of the study is that the sequential scheme has high numerical accuracy and convergent, while a change in the pollutant concentration leads to a small change in the flow velocity due to the opposing effects of viscosity and momentum source.

Keywords

Main Subjects

[1] G.V. Ramana Reddy, N. Bhaskar Reddy, R.S.R. Gorla, Radiation and chemical reaction effects on mhd flow along a moving vertical porous plate. International Journal of Applied Mechanics and Engineering, 21(1), 2016, 157–168.

[2] T. Chinyoka, O.D., Makinde, Analysis of nonlinear dispersion of a pollutant ejected by an external source into a channel flow. Mathematical Problems in Engineering, 2010, Article ID 827363, 17 p.

[3] O.D. Makinde, T. Chinyoka, Numerical investigation of transient heat transfers to hydromagnetic channel flow with radiative heat and convective cooling. Communications in Nonlinear Science and Numerical Simulation, 15(12), 2010, 3919–3930.

[4] J.C. Umavathi, M.A. Sheremet, S. Mohiuddin, Combined effect of variable viscosity and thermal conductivity on mixed convection flow of a viscous fluid in a vertical channel in the presence of first order chemical reaction. European Journal of Mechanics-B/Fluids, 58, 2016, 98–108.

[5] J.C. Umavathi, J.P. Kumar, M.A. Sheremet, Heat and mass transfer in a vertical double passage channel filled with electrically conducting fluid. Physica A: Statistical Mechanics and its Applications, 465, 2017, 195–216.

[6] R. Bhargava, R. Sharma, O.A. Beg, Oscillatory chemically-reacting mhd free convection heat and mass transfer in a porous medium with soret and dufour effects - finite element modelling. International Journal of Applied Mathematics and Mechanics, 5(6), 2009, 15–37.

[7] P. Mebine, Radiation effects on mhd couette flow with heat transfer between two parallel plates. Journal of Pure Applied Mathematics, 3(2), 2007, 191–202.

[8] C. Israel-Cookey, E. Amos, C. Nwaigwe, Mhd oscillatory couette flow of a radiating viscous fluid in a porous medium with periodic wall temperature. American Journal of Scientific and Industrial Research, 1(2), 2010, 326–331.

[9] C. Israel-Cookey, C. Nwaigwe, Unsteady mhd flow of a radiating fluid over a moving heated porous plate with time-dependent suction. American Journal of Scientific and Industrial Research, 1(1), 2010, 88–95.

[10] C. Nwaigwe, Mathematical modelling of ground temperature with suction velocity and radiation. American Journal of Scientific and Industrial Research, 1(2), 2010, 238–241.

[11] R.K. Selvi, R. Muthuraj, Mhd oscillatory flow of a jeffrey fluid in a vertical porous channel with viscous dissipation. Ain Shams Engineering Journal, 9(4), 2018, 2503-2516.

[12] T. Hayat, M. Mustafa, S. Asghar, Unsteady flow with heat and mass transfer of third grade fluid over a stretching surface in the presence of chemical reaction. Nonlinear Analysis, Real World Application, 11, 2010, 3186– 3197.

[13] K. Kavita, P.K., Ramakrishna, K.B., Aruna, Influence of heat transfer on mhd oscillatory flow of jeffrey fluid in a channel. Advanced Applied Scientific Research, 3, 2012, 2312–2325.

[14] R. Muthuraj, S. Srinivas, A.K. Shukla, T.R. Ramamohan, Effects of thermal-diffusion, diffusion-thermo and space porosity on mhd mixed convection flow of micropolar fluid in a vertical channel with viscous dissipation. Asian Research, 43, 2014, 561–578.

[15] J.K. Singh, N. Joshi, S.G. Begum, Unsteady hydromagnetic heat and mass transfer natural convection flow past an exponentially accelerated vertical plate with hall current and rotation in the presence of thermal and mass diffusions. Frontiers in Heat and Mass Transfer, 7(24), 2016, 1-12.

[16] J.C. Umavathi, M.A. Sheremet, Mixed convection flow of an electrically conducting fluid in a vertical channel using robin boundary conditions with heat source/sink. European Journal of Mechanics-B/Fluids, 55, 2016, 132–145.

[17] S.M. Ibrahim, G. Lorenzini, P.V. Kumar, C.S.K. Raju, Influence of chemical reaction and heat source on dissipative mhd mixed convection flow of a casson nanofluid over a nonlinear permeable stretching sheet. International Journal of Heat and Mass Transfer, 111, 2017, 346–355.

[18] L.N. Moresi, V.S. Solomatov, Numerical investigation of 2d convection with extremely large viscosity variations. Physics of Fluids, 7(9), 1995, 2154–2162.

[19] T. Chinyoka, O.D. Makinde, Computational dynamics of unsteady flow of a variable viscosity reactive fluid in a porous pipe. Mechanics Research Communications, 37(3), 2010, 347–353.

[20] J.C. Umavathi, M.A. Sheremet, Influence of temperature dependent conductivity of a nanofluid in a vertical rectangular duct. International Journal of Non-Linear Mechanics, 78, 2016, 17–28.

[21] O.D. Makinde, T. Chinyoka, Transient analysis of pollutant dispersion in a cylindrical pipe with a nonlinear waste discharge concentration. Computers and Mathematics with Applications, 60, 2010, 642–652.

[22] R.A Van Gorder, K. Makowski, K. Mallory, K. Vajravelu, Self-similar solutions for the nonlinear dispersion of a chemical pollutant into a river flow. Journal of Mathematical Chemistry, 53(7), 2015, 1523–1536.

[23] O.D. Makinde, R.J. Moitsheki, B.A. Tau, Similarity reductions of equations for river pollution. Applied Mathematics and Computation, 188(2), 2007, 1267–1273.

[24] R.J. Moitsheki, O.D. Makinde, Symmetry reductions and solutions for pollutant diffusion in a cylindrical system. Nonlinear Analysis: Real World Applications, 10(6), 2009, 3420–3427.

[25] O.D. Makinde, P. Olanrewaju, W.M. Charles, Unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture. Journal of African Mathematical Union, 22, 2011, 65–78.

[26] C. Nwaigwe, Coupling Methods for 2D/1D Shallow Water Flow Models for Flood Simulations. PhD thesis, University of Warwick, United Kingdom, 2016.

[27] T. Chinyoka, O.D. Makinde, Analysis of transient generalized couette flow of a reactive variable viscosity third-grade liquid with asymmetric convective cooling. Mathematical and Computer Modelling, 54(1-2), 2011, 160–174.