ADM Solution for Cu/CuO –Water Viscoplastic Nanofluid ‎Transient Slip Flow from a Porous Stretching Sheet with Entropy ‎Generation, Convective Wall Temperature and Radiative Effects

Document Type : Research Paper


1 Department of Mathematics, B V Raju Institute of Technology, Narsapur, Medak, 502313, Telangana State, India‎

2 Department of Mathematics, Siksha ‘O’ Anusandhan Deemed to be University, Bhubaneswar, 751030, Odisha, India‎

3 Fluid Mechanics, Propulsion and Nanosystems, Aeronautical and Mechanical Engineering, Newton Bldg., University of Salford, Manchester, M54WT, UK‎


A mathematical model is presented for entropy generation in transient hydromagnetic flow of an electroconductive magnetic Casson (non-Newtonian) nanofluid over a porous stretching sheet in a porous medium. The model employed is Cattaneo-Christov heat flux to simulate non-Fourier (thermal relaxation) effects. A Rosseland flux model is implemented to model radiative heat transfer. The Darcy model is employed for the porous media bulk drag effect. Momentum slip is also included to simulate non-adherence of the nanofluid at the wall. The transformed, dimensionless governing equations and boundary conditions (featuring velocity slip and convective temperature) characterizing the flow are solved with the Adomian Decomposition Method (ADM). Bejan’s entropy minimization generation method is employed. Cu-water and CuO-water nanofluids are considered. Extensive visualization of velocity, temperature, and entropy generation number profiles is presented for variation in pertinent parameters. The calculation of skin friction and local Nusselt number are also studied. The ADM computations are validated with simpler models from the literature. The solutions show that with elevation in the volume fraction of nanoparticle and Brinkman number, the entropy generation magnitudes are increased. An increase in Darcy number also upsurges the friction factor and heat transfer at the wall. Increasing volume fraction, unsteadiness, thermal radiation, velocity slip, Casson parameters, Darcy, and Biot numbers are all observed to boost temperatures. However, temperatures are reduced with increasing non-Fourier (thermal relaxation) parameter. The simulations are relevant to the high temperature manufacturing fluid dynamics of magnetic nano liquids, smart coating systems.


Main Subjects

[1] Choi, S.U.S., Enhancing thermal conductivity of fluids with nanoparticles. In: Siginer DA and Wang HP (Eds.), Developments and Applications of Non-Newtonian Flows, ASME, New York, 66, 1995, 99-105.
[2] Buongiorno, J., Convective transport in nanofluids, ASME J Heat Transfer, 128, 2006, 240-250.
[3] Khan, MD. I., Qayyum, S., Hayat, T., Waqas, MD., Imran Khan, MD., and Alsaedi, A., Entropy generation minimization and binary chemical reaction with Arrhenius activation energy in MHD radiative flow of nanomaterial, Journal of Molecular Liquids, 259, 2018, 274-283.
[4] Zohra, F.T., Uddin, M.J., Ismail, A.I., and Bég, O. Anwar., Bioconvective electromagnetic nanofluid transport from a wedge geometry: simulation of smart electro-conductive bio-nano-polymer processing, Heat Transfer-Asian Res., 47, 2018, 231-250.
[5] Khan, MD. I., Qayyum, S., Hayat, T., Imran Khan, MD., Alsaedi, A., and Khan, T.A., Entropy generation in radiative motion of tangent hyperbolic nanofluid in presence of activation energy and nonlinear mixed convection, Physics Letters A, 382(31), 2018, 2017-2026.
[6] Prakash, J., Siva, E.P., Tripathi, D., Kuharat, S., and Bég, O. Anwar., Peristaltic pumping of magnetic nanofluids with thermal radiation and temperature-dependent viscosity effects: modelling a solar magneto-biomimetic nano pump, Renewable Energy, 133, 2019, 1308–1326.
[7] Li, Z., Asadi, S., Karimipour, A., Abdollahi, A., and Tlili, I., Experimental study of temperature and mass fraction effects on thermal conductivity and dynamic viscosity of SiO2-oleic acid/liquid paraffin nanofluid, International Communications in Heat and Mass Transfer, 110, 2020, 104436.
[8] Kumar, M., Reddy, G. J., Kumar, N. N., and Bég, O. Anwar., Computational study of unsteady couple stress magnetic nanofluid flow from a stretching sheet with Ohmic dissipation, Proc. IMechE-Part N: J Nanoengineering, Nanomaterials and Nano-systems, 233(2-4), 2019, 49-63.
[9] Ray, A.K., Vasu, B., Bég, O.A., Gorla, R.S.R., and Murthy, P.V.S.N., Magneto-bioconvection flow of a Casson thin film with nanoparticles over an unsteady stretching sheet: HAM and GDQ computation, Int. J. Numerical Methods Heat Fluid Flow, 29(11), 2019, 4277-4309.
[10] Ngiangia, A.T., and Akaezue, N.N., Heat transfer of mixed convection electro conductivity flow of copper nanofluid with different shapes in a porous microchannel provoked by radiation and first order chemical reaction, Asian J. of Physical and Chemical Sciences, 7(1), 2019, 1-14.
[11] Li, Z., Amin S., Al-Rashed, A.A.A.A., and Talebizadehsardari, P., Effect of porous medium and nanoparticles presences in a counter-current triple-tube composite porous/nano-PCM system, Applied Thermal Engineering, 167, 2020, 114777
[12] Ma, X., Sheikholeslami, M., Jafaryar, M., Shafee, A., Nguyen-Thoi, T., and Li, Z., Solidification inside a clean energy storage unit utilizing phase change material with copper oxide nanoparticles, Journal of Cleaner Production, 245, 2020, 118888.
[13] Bhatti, M.M., Abbas, T., Rashidi, M.M., and Ali, M.E.S., Numerical Simulation of Entropy Generation with Thermal Radiation on MHD Carreau Nanofluid towards a Shrinking Sheet, Entropy, 18, 2016, 200.
[14] Fourier, J.B.J., Theorie analytique de la chaleur, Didot. Paris, 1822, 499-508.
[15] Cattaneo, C., and Sulla conduzione Del Calore., Atti Del Seminario Matematico e Fisico dell’ Universita di Modena, 3, 1948, 83–101.
[16] Christov, C. I., On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mechanics Research Communications, 36, 2009, 481-486.
[17] Nagendramma, V., Raju, C.S.K., Mallikarjuna, B., Shehzad, S.A., and Leelaratnam, A., 3D Casson nanofluid flow over slandering surface in a suspension of gyrotactic microorganism with Cattaneo-Christov heat flux, Applied Mathematics and Mechanics, 39(5), 2018, 623-638.
[18] Makinde, O.D., Nagendramma, V., Raju, C.S.K., and Leelaratnam, A., Effect of Cattaneo-Christov heat flux on Casson nanofluid flow past a stretching cylinder, Defect and Diffusion Forum, 378, 2017, 28-38.
[19] Malik, M.Y., Khan, M., Salahuddin, T., and Khan, I., Variable viscosity and MHD flow in Casson fluid with Cattaneo-Christov heat flux model: using Keller box method, Int. J Eng. Sci. Tech., 19(4), 2016, 1985–92.
[20] Asif, M., Waseem, J., and Asim, A., Entropy and heat transfer analysis using Cattaneo-Christov heat flux model for a boundary layer flow of Casson nanofluid, Results in Physics, 10, 2018, 640-649.
[21] Ibrahim, W., and Makinde, O.D., Magnetohydrodynamic stagnation point flow and heat transfer of Casson nanofluid past a stretching sheet with slip and convective boundary condition, Journal of Aerospace Engineering, 29(2), 2016, 04015037.
[22] Kamran, A., Hussain, S., Sagheer, M., and Akmal, N., A numerical study of magneto hydro-dynamics flow in Casson nanofluid combined with Joule heating and slip boundary conditions, Results in Physics, 7, 2017, 3037-3048.
[23] Nadeem, S., Rizwan, U.H., Akbar, N., and Khan, Z.H., MHD three dimensional Casson fluid flow past a porous linearly stretching sheet, Alexandria Eng. J., 52(4), 2013, 577-582.
[24] Hussain, T., Shehzad, S.A., Alsaedi, A., Hayat, T., and Ramzan, M., Flow of Casson nanofluid with viscous dissipation and convective conditions: A mathematical model, J. Central South University, 22, 2015, 1132-1140.
[25] Sulochana, C., Ashwin Kumar, G.P., and Sandeep, N., Similarity solutions of 3D Casson nanofluid flow over a stretching sheet with convective boundary conditions, J. Nigerian Math Soc., 35(1), 2016, 28-141.
[26] Abolbashari, M.H., Freidoonimehr, N., Nazari, F., and Rashidi, M.M., Analytical modeling of entropy generation for Casson nano-fluid flow induced by a stretching surface, Adv. Powder Tech., 26(2), 2015, 542–552.
[27] Wasim, J., and Asim, A., Cattaneo-Christov based study of TiO2-Cuo/EG Casson hybrid nanofluid flow over a stretching surface with entropy generation, Applied Nanoscience, 8(4), 2018, 685-698.
[28] Qing, J., Bhatti, M.M., Abbas, M.A., Rashidi, M.M., and Ali, M.E.S., Entropy Generation on MHD Casson Nanofluid Flow over a Porous Stretching/Shrinking Surface, Entropy, 18, 2016, 123.
[29] Li, Z., Shahrajabian, H., Bagherzadeh, S.A., Hamid J., Arash K., and Tlili, I., Effects of nano-clay content, foaming temperature and foaming time on density and cell size of PVC matrix foam by presented Least Absolute Shrinkage and Selection Operator statistical regression via suitable experiments as a function of MMT content, Physica A: Statistical Mechanics and its Applications, 537, 2020, 122637.
[30] Ullah, I., Khan, I., and Shafie, S., MHD natural convection flow of Casson nanofluid over nonlinearly stretching sheet through porous medium with chemical reaction and thermal radiation, Nanoscale Research Letters, 11, 2016, 527.
[31] Afify, A.A., The influence of slip boundary conditions on Casson nanofluid flow over a stretching sheet in the presence of viscous dissipation and chemical reaction, Mathematical Problems in Engineering, 2017, 3804751, 1-12.
[32]  Ghadikolaei, S.S., Hosseinzadeh, K., Ganji, D.D., and Jafari, B., Nonlinear thermal radiation effect on magneto Casson nanofluid flow with Joule heating effect over an inclined porous stretching sheet, Case Studies in Thermal Eng., 12, 2018, 176-187.
[33] Thumma, T., Mishra, S.R., and Shamshuddin, M.D., Effect of heat generation and viscous dissipation on MHD 3D Casson nanofluid flow past an impermeable stretching sheet. In: Srinivasacharya D., Reddy K. (eds.) Numerical Heat Transfer and Fluid Flow, Lecture Notes in Mechanical Engineering, Springer, Singapore, 2019, 575-585.
[34] Adomian, G., A review of the decomposition method in applied mathematics, J. Anal. Appl., 135, 1988, 501-544.
[35] Adomian, G., Solving frontier problems of physics: The decomposition method, Springer, Netherlands 1st Ed., 1994.
[36] Hossein, J., and Varsha, D., Revised Adomian decomposition method for solving a system of nonlinear equations, Applied Mathematics and Computation, 175, 2006, 1-7.
[37] Khan, U., Ahmed, N., Khan, S.I., Bano, S., and Mohyud-din, S.T., Unsteady squeezing flow of a Casson fluid between parallel plates, World J. Modelling and Simulations 10(4), 2014, 308-319.
[38] Srinivas, J., Adesanya, S.O., Falade, J.A., and Nagaraju, G., Entropy generation analysis for a radiative micropolar fluid flow through a vertical channel saturated with non-Darcian porous medium, Int. J. Appl. Comput. Math. 3(4), 2017, 3759-3782.
[39] Ebaid, A., Aljoufi, M.D., and Wazwaz, A.M., An advanced study on the solution of nanofluid flow problems via Adomian’s method, Applied Mathematics Letters, 46, 2015, 117-122.
[40] Manzoor, N., Maqbool, K., Bég, O.A., Shaheen, S., Adomian decomposition solution for propulsion of dissipative magnetic Jeffrey biofluid in a ciliated channel containing a porous medium with forced convection heat transfer, Heat Transfer- Asian Research, 48(2), 2019, 556-581.
[41] Opanuga, A.A., Gbadeyan, J.A., and Iyase, S.A., Second law analysis of hydromagnetic couple stress fluid embedded in a non-Darcian porous medium, IAENG International Journal of Applied Mathematics, 47(3), 2017,287-294.
[42] Opanuga, A.A., Okagbue, H.I., Agboola, O.O., and Bishop, S.A., Second law analysis of Ion slip effect on MHD couple stress fluid, Int. J. Mechanics, 12, 2018, 96-101.
[43] Thirupathi, T., Bég, O.A., and Kadir, Numerical study of heat source/sink effects on dissipative magnetic nanofluid flow from a non-linear inclined stretching/shrinking sheet, J. of Molecular Liquids, 232, 2017, 159-173.
[44] Thirupathi, T., Chamkha, A.J., and Sheri, S.R., MHD Natural Convective Flow of Nanofluids past Stationary and Moving Inclined Porous Plate considering Temperature and Concentration gradients with Suction, Int. J. of Numerical Methods for Heat and Fluid Flow, 27(8), 2017, 1765-1794.
[45] Sheri, S.R., and Thirupathi, T., Numerical study of heat transfer enhancement in MHD free convection flow over vertical plate utilizing nanofluids, Ain Shams Engineering Journal, 9, 2018, 1169-1180.
[46] Thirupathi, T., and Mishra, S.R., Effect of viscous dissipation and joule heating on MHD Jeffery nanofluid flow with and without multi slip boundary conditions, J. Nanofluids, 7(3), 2018, 516–526.
[47] Thirupathi, T., Bég, O.A., and Sheri, S.R., Finite element computation of magnetohydrodynamic nanofluid convection from an oscillating inclined plate with radiative flux, heat source and variable temperature effects, Proc. IMechE Part N: J. Nanomaterials, Nanoengineering and Nanosystems, 231(4), 2017,179-194.
[48] Aziz, A., Jamshed, W., and Aziz, T., Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity, Open Physics, 6, 2018, 123-136.
[49] Thirupathi T., and Mishra, S.R., Effect of nonuniform heat source/sink, and viscous and Joule dissipation on 3D Eyring–Powell nanofluid flow over a stretching sheet, J. of Computational Design and Eng., 2020,
[50] Wakif, A., Boulahia, Z., Ali, F., Eid, M.R., and Sehaqui, R., Numerical analysis of the unsteady natural convection MHD Couette nanofluid Flow in the presence of thermal radiation using single and two-phase nanofluid models for Cu–Water nanofluids, Int. J. Appl. Comput. Math., 4, 2018, 81.
[51] Wakif, A., Boulahia, Z., Mishra, S.R., Rashidi, M.M., and Sehaqui, R., Influence of a uniform transverse magnetic field on the Thermo - hydrodynamic stability in water-based nanofluids with metallic nanoparticles using the generalized Buongiorno’s mathematical model, European Physical Journal Plus, 133, 2018, 181.
[52] Dogonchi, A.S., Selimefendigil, F., and Ganji, D.D., Magneto-hydrodynamic natural convection of CuO-water nanofluid in complex shaped enclosure considering various nanoparticle shapes, Int. J. Numer. Methods Heat Fluid Flow, 29(5), 2019, 1663-1679.
[53] Das, S., Chakraborty, S., Jana, R.N., and Makinde, O.D., Entropy analysis of unsteady magneto-nanofluid flow past accelerating stretching sheet with convective boundary conditions, Adv. Appl. Math. Mech., 36(2), 2015, 1610.
[54] Asai, S., Magnetohydrodynamics in Materials Processing, Electromagnetic Processing of Materials, Springer, Germany, 2011, 49-86.
[55] Bég, O.A., Abdul Gaffar, S., Ramachandra Prasad, V., and Uddin, M.J., Computational solutions for non-isothermal, nonlinear magneto convection in porous media with Hall/ Ion slip currents and Ohmic dissipation, Engineering Science and Technology - An International Journal, 19, 2016, 377-394.
[56] Thirupathi, T., and Magagula, V.M., Transient electromagnetohydrodynamic radiative squeezing flow between two parallel Riga plates using a spectral local linearization approach, Heat Transfer–Asian Res., 49(1), 2019, 67-85.
[57] Wakif, A., Chamkha, A.J., Thirupathi, T., Animasaun, I. L., and Rachid S., Thermal radiation and surface roughness effects on the thermo-magneto-hydrodynamic stability of alumina–copper oxide hybrid nanofluids utilizing the generalized Buongiorno’s nanofluid model, Journal of Thermal Analysis Calorimetry, 2020,