Keywords:-
Keywords:
Williamson fluid, Cattaneo-Christov heat flux, MHD, thermal radiation, HAM
Article Content:-
Abstract
The two-dimensional steady incompressible MHD boundary layer flow of convective Williamson fluid flow of the Cattaneo-Christov heat flux type over an exponentially stretching surface in the presence of heat generation and thermal radiation is considered. The nonlinear governing partial differential equations are reduced to ordinary differential equations by using a similarity transformation. The Homotopy Analysis Method (HAM) is applied to solve the reduced equations and the effects of importance of fluid parameters are explained through graphs. Also, the HAM solutions were compared with the numerical solutions and good agreement was observed.
References:-
References
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Christov, C. I. ,On Frame Indifferent Formulation of the Maxwell-Cattaneo Model of Finite-Speed Heat Conduction, Mechanics Research Communications, Vol. 36, No. 4, 2009, 481–486.
Han, S., Zheng, L., Li, C. and Zhang, X., Coupled Flow and Heat Transfer in Viscoelastic Fluid With Cattaneo-Christov Heat Flux Model,Applied Mathematics Letters, Vol. 38, 2014, 87–93.
Khan, M. S., Rahman, M. M., Arifuzzaman, S. M., Biswas, P. and Karim, I., Williamson Fluid Flow Behaviour Of MHD Convectiveradiative Cattaneo–Christov Heat Flux type Over A Linearly Stretched-Surface With Heat Generation and Thermal-Diffusion, Frontiers in Heat and Mass Transfer, Vol. 9, No. 15, 2017.
Bashir, S., Ramzan M., Ghazwani, H. A. S., Nisar, K. S., Saleel, C. A., and Abdelrahman A., Magnetic Dipole and Thermophoretic Particle Deposition Impact on Bioconvective Oldroyd-B Fluid Flow over a Stretching Surface with Cattaneo–Christov Heat Flux, Nanomaterials, Vol. 12, 2022.
Bilal, S., Shah, M. I., Khan, N. Z., Akgul, A. A., and Nisar, K. S., Onset about non-isothermal flow of Williamson liquid over exponential surface by computing numerical simulation in perspective of Cattaneo Christov heat flux theory, Alexandria Engineering Journal, Vol. 61, 2022, 6139–6150.
Faizan, M., Ali, F., Loganathan, K., Zaib, A., Reddy, C. A., Abdelsalam, S. I., Entropy Analysis of Sutterby Nanofluid Flow over a Riga Sheet with Gyrotactic Microorganisms and Cattaneo–Christov Double Diffusion, Mathematics, Vol. 10, 2022, 3157.
Alharbi, K. A. M., Alshahrani, M. N., Ullah, N., Khan, N. M., Marek, K., Mousa, A. A. A. and Ali, S., Cattaneo–Christov heat flow model for copper–water nanofluid heat transfer under Marangoni convection and slip conditions, Scientific Reports, Vol. 12, 2022, 5260.
Mahabaleshwar, U. S., Sneha, K. N. and Hatami, M., Effect of Cattaneo_Christov approximation for viscoelastic fluid with carbon nanotubes on flow and heat transfer, Scientific Reports, Vol. 12, 2022, 9485.
Shahzad, A., Imran, M., Tahir, M., Khan, S. A., Akgul, A., Abdullaev, S., Park, C. and Zahran, H. Y., Brownian motion and thermophoretic diffusion impact on Darcy-Forchheimer flow of bioconvective micropolar nanofluid between double disks with Cattaneo-Christov heat flux, Alexandria Engineering Journal, 2022.
Reddy, K. V., Reddy, G. V. R. and Hari Krishna, Y., Effects of Cattaneo–Christov heat flux analysis on heat and mass transport of Casson nanoliquid past an accelerating penetrable plate with thermal radiationandSoret–Dufour mechanism, Heat Transfer, Vol. 50, 2021, 3458–3479.
Ahmad, M. W., McCash, L. B., Shah, Z. and Nawaz, R., Cattaneo-Christov Heat Flux Model for Second Grade Nanofluid Flow with Hall Eject through Entropy Generation over Stretchable Rotating Disk, Coatings, Vol. 10, 2020, 610.
Ahmed, A., Khan, M., Sarfraz, M., Ahmed, J., and Iqbal, Z., Forced convection in 3D Maxwell nanofluid flow via Cattaneo–Christov theory with Joule heating, J Process Mechanical Engineering, 2021, 1-11.
Akinbo, B. J. and Olajuwon, B. I.,Cattaneo-Christov Heat Flux And Heat Generation/Absorption Effect On Viscous Walters’ B Fluid Through A Porous Medium With Chemical Reaction, Journal of the Nigerian Mathematical Society, Vol. 40, No. 3, 2021, 205-226.
Amjad, M., Ahmed, K., Akbar, T., Muhammad, T., Ahmed, T. and Alshomrani, A.S.,Numerical investigation of double diffusion heat flux model in Williamson nanofluid over an exponentially stretching surface with variable thermal conductivity, Case Studies in Thermal Engineering, Vol. 36, 2022, 102231
Prasad, K. V., Vaidya H., Vajravelu, K. and Ramanjini, V., Analytical Study of Cattaneo-Christov Heat Flux Model for Williamson-Nanofluid Flow Over a Slender Elastic Sheet with Variable Thickness, Journal of Nanofluids, Vol. 7, 2018, 583–594.
Prakash, J., Prasad, P. D., Kumar, G. V., Kumar, R. V. M. S. S. K. and Varma, S. V. K., Heat and Mass Transfer Hydromagnetic Radiative Casson Fluid Flow over an Exponentially Stretching Sheet with Heat Source or Sink, International Journal of Engineering Science Invention, Vol. 5, No. 7, 2016, 12-23.
Shehzad, S. A., Abbasi, F. M., Hayat, T. and Ahmad, B., Cattaneo-Christov heat flux model for third-grade fluid flow towards exponentially stretching sheet, Applied Mathematics and Mechanics, Vol. 37, No. 6, 2016, 761–768.
Shehzad, S. A., Abbasi, F. M., Hayat, T. and Alsaedi, A., Cattaneo-Christov heat flux model for Darcy-Forchheimer flow of an Oldroyd-B fluid with variable conductivity and non-linear convection, Journal of Molecular Liquids, 2016.
Ray A. K., Flow of Electrically Conducting Williamson Fluid with Cattaneo-Christov heat flux due to Permeable Sheet, International Research Journal on Advanced Science Hub, Vol. 2, No. 12, 2020, 17-22.
Yahya, A. U., Salamat, N., Habib, D., Ali, B., Hussain, S. and Abdal, S., Implication of Bio-convection and Cattaneo-Christov heat flux on Williamson Sutterby nanofluid transportation caused by a stretching surface with convective boundary, Chinese Journal of Physics, Vol. 73, 2021, 706–718.
Mangathai, P. and Sidhartha, B. R., Unsteady Mhd Williamson And Casson Nano Fluid Flow In The Presence Of Radiation And Viscous Dissipation, Turkish Journal of Computer and Mathematics Education, Vol.12, No.13, 2021, 1036-1051.
Jamshed, W., Nisar, K.S., Ibrahim, R. W., Mukhtar, K., Vijayakumar, V. and Ahmad, F., Computational frame work of Cattaneo-Christov heat flux effects on Engine Oil based Williamson hybrid nanofluids: A thermal case study, Case Studies in Thermal Engineering, Vol. 26, 2021, 101179.
Hayat, T., Qayyum, S., Alsaedi, A. and Ahmad, B., Nonlinear convective flow with variable thermal conductivity and Cattaneo-Christov heat flux, Neural Comput and Applic, 2017.
Mondal, S. K., and Pal, D., Computational analysis of bioconvective flow of nanofluid containing gyrotactic microorganisms over a nonlinear stretching sheet with variable viscosity using HAM, Journal of Computational Design and Engineering, Vol. 7, No. 2, 2020, 251–267.
Rubab, K. and Mustafa, M., Cattaneo-Christov Heat Flux Model for MHD Three-Dimensional Flow of Maxwell Fluid over a Stretching Sheet, PLOS ONE, Vol. 11, No. 4, 2016, 1-16.
Salahuddin, T., Malik, M. Y., Hussain, A., Bilal, S. and Awais, M., MHD Flow of Cattanneo-Christov Heat Flux Model for Williamson Fluid Over a Stretching Sheet With Variable Thickness: Using Numerical Approach, Journal of Magnetism and Magnetic Materials, 2015, 1–14.
Kumar, K. A., Reddy J. V. R., Sugunamma V., and Sandeep N., Magnetohydrodynamic Cattaneo-Christov flow past a cone and a wedge with variable heat source/sink, Alexandria Engineering Journal, 2016, 1-9.
Bhatti, M. M. and Rashidi, M. M., Effects of Thermo-Diffusion and Thermal Radiation on Williamson Nanofluid over a Porous Shrinking/Stretching Sheet, 2016.
Elbashbeshy, E. M. A., Heat transfer over an exponentially stretching continuous surface with suction, Arch. Mech., Vol. 53, No. 6, 2001, 643-651.
Liao, S. J., Homotopy analysis method in nonlinear differential equations, Springer, 2011.
Liao, S. J., Advances in the homotopy analysis method, World Scientific Publishing Co. Pte. Ltd., 2013.
Liao, S. J.,Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2003.
Liao, S. J., On the homotopy analysis method for nonlinear problems, Journal of Applied Mathematics and Computation, Vol. 147, 2004, 499-513.
Liao, S. J., Comparision between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computations, Vol. 169, 2005, 1186-1194.
Achala, L. N. and Sathyanarayana, S. B., Homotopy Analysis Method For Nonlinear Boundary Value Problems, JP of Applied Mathematics, Vol. 5, No. 1 & 2, 2012, 27-46.
Sathyanarayana, S. B. and Achala, L. N., Approximate Analytical Solution of Magnetohydrodynamics Compressible Boundary Layer flow with Pressure Gradient and suction/injection, Journal of Advances in Physics, Vol. 6, No. 3, 2014.
Sathyanarayana, S. B. and Achala, L. N., Approximate analytic Solution of Compressible Boundary Layer flow with an adverse Pressure Gradient by Homotopy Analysis Method, Theoritical Mathematics & Applications, Vol. 5, No. 1, 2015, 15-31.
Achala, L. N., Madusudhan, R. and Sathyanarayana, S. B., Semi Analytic Approximate Solution Of Nonlinear Partial Differential Equations, International Journal of Mathematics and Computer Research, Vol. 4, No. 6, 2016, 1418-1428.
Sushma,V. J., Raju B. T., Achala, L. N. and Sathyanarayana, S. B., MHD boundary layer flow of nanofluid over a moving surface in presence of thermal radiation by homotopy analysis method, Global Journal of Engineering Science and Researches, 2019, 435-443.
Rekha, K., Asha, C. S. and Achala, L. N., Williamson Nanofluid Flow over a Moving Surface on Boundary Layer by Homotopy Analysis Method, International Journal of Mechanical Engineering, Vol. 7, No. 2, 2022, 1723–1736.
Rekha, K., Asha, C. S. and Achala, L. N., Magnetohydrodynamic Boundary Layer Flow of Williamson Nanofluid over a Moving Surface in the presence of Gyrotactic Microorganism, International Journal of Mechanics and Thermodynamics, Vol. 13, No. 1, 2022, 15-31
Christov, C. I. ,On Frame Indifferent Formulation of the Maxwell-Cattaneo Model of Finite-Speed Heat Conduction, Mechanics Research Communications, Vol. 36, No. 4, 2009, 481–486.
Han, S., Zheng, L., Li, C. and Zhang, X., Coupled Flow and Heat Transfer in Viscoelastic Fluid With Cattaneo-Christov Heat Flux Model,Applied Mathematics Letters, Vol. 38, 2014, 87–93.
Khan, M. S., Rahman, M. M., Arifuzzaman, S. M., Biswas, P. and Karim, I., Williamson Fluid Flow Behaviour Of MHD Convectiveradiative Cattaneo–Christov Heat Flux type Over A Linearly Stretched-Surface With Heat Generation and Thermal-Diffusion, Frontiers in Heat and Mass Transfer, Vol. 9, No. 15, 2017.
Bashir, S., Ramzan M., Ghazwani, H. A. S., Nisar, K. S., Saleel, C. A., and Abdelrahman A., Magnetic Dipole and Thermophoretic Particle Deposition Impact on Bioconvective Oldroyd-B Fluid Flow over a Stretching Surface with Cattaneo–Christov Heat Flux, Nanomaterials, Vol. 12, 2022.
Bilal, S., Shah, M. I., Khan, N. Z., Akgul, A. A., and Nisar, K. S., Onset about non-isothermal flow of Williamson liquid over exponential surface by computing numerical simulation in perspective of Cattaneo Christov heat flux theory, Alexandria Engineering Journal, Vol. 61, 2022, 6139–6150.
Faizan, M., Ali, F., Loganathan, K., Zaib, A., Reddy, C. A., Abdelsalam, S. I., Entropy Analysis of Sutterby Nanofluid Flow over a Riga Sheet with Gyrotactic Microorganisms and Cattaneo–Christov Double Diffusion, Mathematics, Vol. 10, 2022, 3157.
Alharbi, K. A. M., Alshahrani, M. N., Ullah, N., Khan, N. M., Marek, K., Mousa, A. A. A. and Ali, S., Cattaneo–Christov heat flow model for copper–water nanofluid heat transfer under Marangoni convection and slip conditions, Scientific Reports, Vol. 12, 2022, 5260.
Mahabaleshwar, U. S., Sneha, K. N. and Hatami, M., Effect of Cattaneo_Christov approximation for viscoelastic fluid with carbon nanotubes on flow and heat transfer, Scientific Reports, Vol. 12, 2022, 9485.
Shahzad, A., Imran, M., Tahir, M., Khan, S. A., Akgul, A., Abdullaev, S., Park, C. and Zahran, H. Y., Brownian motion and thermophoretic diffusion impact on Darcy-Forchheimer flow of bioconvective micropolar nanofluid between double disks with Cattaneo-Christov heat flux, Alexandria Engineering Journal, 2022.
Reddy, K. V., Reddy, G. V. R. and Hari Krishna, Y., Effects of Cattaneo–Christov heat flux analysis on heat and mass transport of Casson nanoliquid past an accelerating penetrable plate with thermal radiationandSoret–Dufour mechanism, Heat Transfer, Vol. 50, 2021, 3458–3479.
Ahmad, M. W., McCash, L. B., Shah, Z. and Nawaz, R., Cattaneo-Christov Heat Flux Model for Second Grade Nanofluid Flow with Hall Eject through Entropy Generation over Stretchable Rotating Disk, Coatings, Vol. 10, 2020, 610.
Ahmed, A., Khan, M., Sarfraz, M., Ahmed, J., and Iqbal, Z., Forced convection in 3D Maxwell nanofluid flow via Cattaneo–Christov theory with Joule heating, J Process Mechanical Engineering, 2021, 1-11.
Akinbo, B. J. and Olajuwon, B. I.,Cattaneo-Christov Heat Flux And Heat Generation/Absorption Effect On Viscous Walters’ B Fluid Through A Porous Medium With Chemical Reaction, Journal of the Nigerian Mathematical Society, Vol. 40, No. 3, 2021, 205-226.
Amjad, M., Ahmed, K., Akbar, T., Muhammad, T., Ahmed, T. and Alshomrani, A.S.,Numerical investigation of double diffusion heat flux model in Williamson nanofluid over an exponentially stretching surface with variable thermal conductivity, Case Studies in Thermal Engineering, Vol. 36, 2022, 102231
Prasad, K. V., Vaidya H., Vajravelu, K. and Ramanjini, V., Analytical Study of Cattaneo-Christov Heat Flux Model for Williamson-Nanofluid Flow Over a Slender Elastic Sheet with Variable Thickness, Journal of Nanofluids, Vol. 7, 2018, 583–594.
Prakash, J., Prasad, P. D., Kumar, G. V., Kumar, R. V. M. S. S. K. and Varma, S. V. K., Heat and Mass Transfer Hydromagnetic Radiative Casson Fluid Flow over an Exponentially Stretching Sheet with Heat Source or Sink, International Journal of Engineering Science Invention, Vol. 5, No. 7, 2016, 12-23.
Shehzad, S. A., Abbasi, F. M., Hayat, T. and Ahmad, B., Cattaneo-Christov heat flux model for third-grade fluid flow towards exponentially stretching sheet, Applied Mathematics and Mechanics, Vol. 37, No. 6, 2016, 761–768.
Shehzad, S. A., Abbasi, F. M., Hayat, T. and Alsaedi, A., Cattaneo-Christov heat flux model for Darcy-Forchheimer flow of an Oldroyd-B fluid with variable conductivity and non-linear convection, Journal of Molecular Liquids, 2016.
Ray A. K., Flow of Electrically Conducting Williamson Fluid with Cattaneo-Christov heat flux due to Permeable Sheet, International Research Journal on Advanced Science Hub, Vol. 2, No. 12, 2020, 17-22.
Yahya, A. U., Salamat, N., Habib, D., Ali, B., Hussain, S. and Abdal, S., Implication of Bio-convection and Cattaneo-Christov heat flux on Williamson Sutterby nanofluid transportation caused by a stretching surface with convective boundary, Chinese Journal of Physics, Vol. 73, 2021, 706–718.
Mangathai, P. and Sidhartha, B. R., Unsteady Mhd Williamson And Casson Nano Fluid Flow In The Presence Of Radiation And Viscous Dissipation, Turkish Journal of Computer and Mathematics Education, Vol.12, No.13, 2021, 1036-1051.
Jamshed, W., Nisar, K.S., Ibrahim, R. W., Mukhtar, K., Vijayakumar, V. and Ahmad, F., Computational frame work of Cattaneo-Christov heat flux effects on Engine Oil based Williamson hybrid nanofluids: A thermal case study, Case Studies in Thermal Engineering, Vol. 26, 2021, 101179.
Hayat, T., Qayyum, S., Alsaedi, A. and Ahmad, B., Nonlinear convective flow with variable thermal conductivity and Cattaneo-Christov heat flux, Neural Comput and Applic, 2017.
Mondal, S. K., and Pal, D., Computational analysis of bioconvective flow of nanofluid containing gyrotactic microorganisms over a nonlinear stretching sheet with variable viscosity using HAM, Journal of Computational Design and Engineering, Vol. 7, No. 2, 2020, 251–267.
Rubab, K. and Mustafa, M., Cattaneo-Christov Heat Flux Model for MHD Three-Dimensional Flow of Maxwell Fluid over a Stretching Sheet, PLOS ONE, Vol. 11, No. 4, 2016, 1-16.
Salahuddin, T., Malik, M. Y., Hussain, A., Bilal, S. and Awais, M., MHD Flow of Cattanneo-Christov Heat Flux Model for Williamson Fluid Over a Stretching Sheet With Variable Thickness: Using Numerical Approach, Journal of Magnetism and Magnetic Materials, 2015, 1–14.
Kumar, K. A., Reddy J. V. R., Sugunamma V., and Sandeep N., Magnetohydrodynamic Cattaneo-Christov flow past a cone and a wedge with variable heat source/sink, Alexandria Engineering Journal, 2016, 1-9.
Bhatti, M. M. and Rashidi, M. M., Effects of Thermo-Diffusion and Thermal Radiation on Williamson Nanofluid over a Porous Shrinking/Stretching Sheet, 2016.
Elbashbeshy, E. M. A., Heat transfer over an exponentially stretching continuous surface with suction, Arch. Mech., Vol. 53, No. 6, 2001, 643-651.
Liao, S. J., Homotopy analysis method in nonlinear differential equations, Springer, 2011.
Liao, S. J., Advances in the homotopy analysis method, World Scientific Publishing Co. Pte. Ltd., 2013.
Liao, S. J.,Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, 2003.
Liao, S. J., On the homotopy analysis method for nonlinear problems, Journal of Applied Mathematics and Computation, Vol. 147, 2004, 499-513.
Liao, S. J., Comparision between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computations, Vol. 169, 2005, 1186-1194.
Achala, L. N. and Sathyanarayana, S. B., Homotopy Analysis Method For Nonlinear Boundary Value Problems, JP of Applied Mathematics, Vol. 5, No. 1 & 2, 2012, 27-46.
Sathyanarayana, S. B. and Achala, L. N., Approximate Analytical Solution of Magnetohydrodynamics Compressible Boundary Layer flow with Pressure Gradient and suction/injection, Journal of Advances in Physics, Vol. 6, No. 3, 2014.
Sathyanarayana, S. B. and Achala, L. N., Approximate analytic Solution of Compressible Boundary Layer flow with an adverse Pressure Gradient by Homotopy Analysis Method, Theoritical Mathematics & Applications, Vol. 5, No. 1, 2015, 15-31.
Achala, L. N., Madusudhan, R. and Sathyanarayana, S. B., Semi Analytic Approximate Solution Of Nonlinear Partial Differential Equations, International Journal of Mathematics and Computer Research, Vol. 4, No. 6, 2016, 1418-1428.
Sushma,V. J., Raju B. T., Achala, L. N. and Sathyanarayana, S. B., MHD boundary layer flow of nanofluid over a moving surface in presence of thermal radiation by homotopy analysis method, Global Journal of Engineering Science and Researches, 2019, 435-443.
Rekha, K., Asha, C. S. and Achala, L. N., Williamson Nanofluid Flow over a Moving Surface on Boundary Layer by Homotopy Analysis Method, International Journal of Mechanical Engineering, Vol. 7, No. 2, 2022, 1723–1736.
Rekha, K., Asha, C. S. and Achala, L. N., Magnetohydrodynamic Boundary Layer Flow of Williamson Nanofluid over a Moving Surface in the presence of Gyrotactic Microorganism, International Journal of Mechanics and Thermodynamics, Vol. 13, No. 1, 2022, 15-31
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K., R., Sathyanarayana, S. B., V. Jakati, S., C. S., A., & Nargund, A. L. (2023). Convective Cattaneo-Christov Heat Flux and Heat Generation Effect on Mhd Williamson Fluid Flow Over an Exponentially Stretching Surface with Thermal Radiation. International Journal Of Mathematics And Computer Research, 11(8), 3694-3703. https://doi.org/10.47191/ijmcr/v11i8.10