MHD, Nonlinear thermal radiation and non-uniform heat source/sink effect on Williamson nanofluid over an inclined stretching sheet
Downloads
DOI:
https://doi.org/10.26637/MJM0803/0106Abstract
In this article, we have deliberate the impact of non-linear thermal radiation and non-uniform heat source/sink on MHD boundary layer flow of Williamson nanofluid above an inclined stretching surface. Using the similarity transformations, the governing PDE's equations of this problem, are converted into ODE'S. The changed ODE'S with boundary conditions are solved numerically by using RK scheme with shooting technique. The effect of governing parameters on dimensionless flow distributions are plotted and examined. The Numerical outcomes for skin friction, Nusselt number, and Sherwood number are tabulated for various values of physical parameters. It is obtained that temperature increases with enhance in non-uniform heat source and velocity profile diminishes with increase in an angle of inclination.
Keywords:
Williamson nanofluid, MHD, Inclined stretching sheet, Non-uniform heat source/sink, RK SchemeMathematics Subject Classification:
Mathematics- Pages: 1337-1345
- Date Published: 01-07-2020
- Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bhatti ,Tehseen Abbas ,Mohammad Mehdi Rashidi and Mohammad El-Sayed Ali. Numerical Simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy. 18(2016).
Sandeep Naramgiri and Sulochana,c., MHD flow over a permeable stretchin/shrinking sheet of a nanofluid with suction/injection, Alexandia Engineering Journal, $55(2016), 819-827$.
Bakr,A.A, Effect of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference, Commun Nonlinear Sci NumerSimulat., 16(2011), 698-710.
KrishnenduBhatachaarya and Layak,G.C, Magnetohydrodynamic boundary layer flow of nanofluid over an exponentially stretching permeable sheet, Physicks Research International, 2014.
$[delta]$ Shankar Bandari and YohannesYirga,Unsteady heat anf mass transfer in MHD flow of nanofluids over a stretching sheet with a non-uniform heat source/sink, International Journal of Mathematical, Physical, Nuclear Science and Engineering, vol.7(12)(2013).
Waqas ,M.,Hayat,T.,Shehzad,S.A, and Alsaedi,A. Transfort of magnetohydrodynamic nanomaterial in as tratified medium considering gyrotactic microorganisms, Physica. B Physics of Condensed Matter, 5292018, 33-40.
C.Y.Wang, Free convection on a vertical stretching surface, J.Appl. Math Mech.(ZAMM), 69(1989), 418-420.
S.Nadeem, S.T.Hussain and Changhoon Lee(2013): Flowof a Williamson fluid over a stretching sheet, Brazilian Journal Chemical Engineering, 30(3)(2013), pp.619-625
Hayat,T., Gulnaz Bashir, Waqas ,M.andAlsaedi , MHD 2D flow of Williamson nanofluid over an nonlinear variable thicked surface with heattransfer, Journal of Molecular Liquids, 223(2016), 836-844.
Krishnamurthy,M.R,Prasannakumara,B.C.,Gireesha,B.J and Ramasubba Reddy Gorla, Effect of chemical reaction on MHD boundary flow and melting heat transfer of Williamson nanofluid in porous medium, Engineering Science and Technology, 192016, 53-61.
Nadeem,S.,Rizwan UI Haq and Khan,Z.H, Numerical solution of non-Newtonian flow over a stretching sheet, ApplNanosci., 4(2014), 625-631.
Hayat,T. and Hina,S. Effect of heat and mass transfer of peristaltic of Williamson fluid in a non-unioform channel, Int.J.Numer.Mech.Fluids, 67(2011), 1590-1604.
Nadeem, S. and S.T.hussain, Flow and heat transfer analysis of Williamson nanofluid, ApplNanosci., 4(2014), 1005-1012. DOI 10.1007/s13204-013-0282-1.
Ramesh,G.K and Gireesha,B.J. Non-Linear radiative flow of nanofluid past a moving/stationary riga plate, Frontiers in Heat and Mass Transfer(FHMT), 9(3)(2017).
Archana M. and Girisha,B.J, ?Numerical exploration of the combined effects of non-linear thermal radiation and variable thermo-physical properties on the flow of casson nanofluid over a wedge, MultidisplinaryModeling and Material Structure, 2017.
MahanteshM.Nandeppanavar, Effect of space and temperature dependent heat source on heat transfer of MHD nanofluid due to non-linear stretching sheet with chemical reaction, Journal of Nanofluids, 6(2017), pp.1-7.
Ramesh,G.K,Gireesha,B.J and Bagewadi,C.S, Heat transfer in MHD dusty boundary layer flow over an inclined stretching sheet with non-uniform heat source/sink, $A d$ vance in Mathematical Physics, 2012, ID, 657805.
Pragya and Vasanthakumari,R.V, Mixed convective nanofluid flow over an inclined stretching plate with MHD and effects of suction and internal heat generation, Annalas of Pure and Applied Mathematics, 12(2016), $69-83$.
Mohammad Ali,AbdulAlim,Md. and Mohammad Shah Alam,Heat transfer boundary layer flow past an inclined stretching sheet in the presence of magnetic field, International Journal of Advancements in Research and Technology, 3(2014), 1-09.
BesthapuPrabhakar ,Shankar Bandari and Rizwan UI Haq, ?Impact of inclined Lorentz forces on tangent hyperbolic nanofluid flow with zero normal flux of nanoparticles at the stretching sheet, Neural Computer and Applications, 2016, DOI 10.1007/s00521-016-2601-4.
Dr.Ahamad M., AbdalHadi , Ndhal.M., Abdal Ameer, Effect of magnetic field over an inclined stretching sheet of three dimensional Maxwell fluid in the absence of mixed convection, Mathematical Theory and Modeling,
Choi ,S.U.S and Eastman, J.A, Enhancing thermal conductivity of fluids with nanoparticles in Proceeding of ASME, International journal of Mechanical Engineering congress and Exposition, san Franscisco, USA, 1995, 12-17.
Mohammad Amon Makarem ,AliBakhtyari and Mohammad RezaRahimpour, A numerical investigation on the heat and fluid flow of various nanfluids on a stretching sheet, Heat transfer Asia, 2017, 1-19.
Hayat,T.,Waqas,M.,Shehazad,S.A and Alsaedi,A., A model of solar radiation and joule heating in magnetohydrodynamic convevtive flow thixotropic nanofluid, Journal of Molecular Liquids, vol.215(2016), 704-710
Mohammad Mubashir Bha
- NA
Similar Articles
- Uma Dixit, On bounds of perfect number , Malaya Journal of Matematik: Vol. 8 No. 03 (2020): Malaya Journal of Matematik (MJM)
You may also start an advanced similarity search for this article.
Metrics
Published
How to Cite
Issue
Section
License
Copyright (c) 2020 MJM
This work is licensed under a Creative Commons Attribution 4.0 International License.
