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A convergence criterion for tangent hyperbolic fluid along a stretching wall subjected to inclined electromagnetic field

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Abstract

The homotopy-based approach is a useful tool for solving nonlinear partial differential equations (PDEs) in physics and engineering. Our aim here is to optimize this approach by generating a convergence criterion for tangent hyperbolic fluid along a stretching wall with magnetic force. To this end, the governing partial differential equations (PDEs) get transformed to the dimensionless form via similarity variables. A comparison of the homotopy-based approach for the skin friction coefficient with different solution methodologies shows that the 9th-order approximate solution together with \( \hbar = - \,0. 5 2 3 \) will certainly achieve a very minor error for the present system.

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Abbreviations

\( {\text{T}} \) :

Cauchy stress tensor [Pa]

\( p \) :

Hydrostatic pressure [Pa]

\( {\text{I}} \) :

Identity tensor

\( n \) :

Power-law index

\( u \), \( v \) :

Velocity components along \( x \)- and \( y \)-directions, respectively [m s−1]

\( B_{0} \) :

Magnetic field strength [kg s−2 A−1]

\( U_{w} \) :

Velocity at the wall [m s−1]

\( b \) :

Stretching rate [s−1]

\( f \) :

Similarity function

\( We \) :

Weissenberg number

\( M \) :

Magnetic field parameter

\( C_{f} \) :

Skin friction coefficient

\( Re_{x} \) :

Reynolds number

\( \varvec{\tau} \) :

Viscous stress tensor [Pa]

\( \mu_{0} \) :

Initial shear rate viscosity [kg m−1 s−1]

\( \mu_{\infty } \) :

Infinite shear rate viscosity [kg m−1 s−1]

\( \varGamma \) :

Time constant [s]

\( \dot{\gamma } \) :

Shear rate [s−1]

\( \varPi \) :

Second invariant of the viscous stress tensor

\( \upsilon \) :

Kinematic viscosity [m2 s−1]

\( \sigma \) :

Electrical conductivity [S m−1]

\( \rho \) :

Fluid density [kg m−3]

\( \alpha \) :

Inclination angle of the magnetic field

\( \eta \) :

Similarity variable

\( \tau_{w} \) :

Wall shear stress [Pa]

\( \infty \) :

Condition at the infinite medium

\( i \), \( j \) :

Tensor index

References

  1. Abbasbandy, S.: Homotopy analysis method for heat radiation equations. Int. Commun. Heat Mass Transfer 34, 380–387 (2007)

    Article  Google Scholar 

  2. Abbasbandy, S., Mustafa, M.: Analytical and numerical approaches for Falkner-Skan flow of MHD Maxwell fluid using a non-Fourier heat flux model. Int. J. Numer. Methods Heat Fluid Flow 28, 1539-1555 (2018)

  3. Abbasi, F.M., Mustafa, M., Shehzad, S.A., Alhuthali, M.S., Hayat, T.: Analytical study of Cattaneo-Christov heat flux model for a boundary layer flow of Oldroyd-B fluid. Chin. Phys. B 25, 1–7 (2016)

    Google Scholar 

  4. Ahmad Soltani, L., Shivanian, E., Ezzati, R.: Convection-radiation heat transfer in solar heat exchangers filled with a porous medium: exact and shooting homotopy analysis solution. Appl. Therm. Eng. 103, 537–542 (2016)

    Article  Google Scholar 

  5. Akbar, N.S., Nadeem, S., Haq, R.U., Khan, Z.H.: Numerical solutions of magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid towards a stretching sheet. Indian J. Phys. 87, 1121–1124 (2013)

    Article  Google Scholar 

  6. Fang, T., Zhang, J., Yao, S.: Slip MHD viscous flow over a stretching sheet-An exact solution. Commun. Nonlinear Sci. Numer. Simulat. 14, 3731–3737 (2009)

    Article  Google Scholar 

  7. Farooq, U., Zhao, Y.L., Hayat, T., Alsaedi, A., Liao, S.J.: Application of the HAM-based Mathematica package BVPh 2.0 on MHD Falkner-Skan flow of nano-fluid. Comput. Fluids 111, 59-75 (2015)

  8. Fitzpatrick, P.M.: Advanced calculus: A course in mathematical analysis. PWS Publishing Company, Boston (1996)

    Google Scholar 

  9. Hashmi, M.S., Khan, N., Mahmood, T., Shehzad, S.A.: Effect of magnetic field on mixed convection flow of Oldroyd-B nanofluid induced by two infinite isothermal stretching disks. Int. J. Therm. Sci. 111, 463–474 (2017)

    Article  Google Scholar 

  10. Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007)

    Article  MATH  Google Scholar 

  11. Hayat, T., Shafiq, A., Alsaedi, A.: Characteristics of magnetic field and melting heat transfer in stagnation point flow of Tangent-hyperbolic fluid. J. Magn. Magn. Mater. 405, 97–106 (2016)

    Article  Google Scholar 

  12. Hayat, T., Qayyum, S., Alsaedi, A., Shehzad, S.A.: Nonlinear thermal radiation aspects in stagnation point flow of tangent hyperbolic nanofluid with double diffusive convection. J. Mol. Liq. 223, 969–978 (2016)

    Article  Google Scholar 

  13. Hayat, T., Khan, M.I., Waqas, M., Alsaedi, A.: Stagnation point flow of hyperbolic tangent fluid with Soret-Dufour effects. Results Phys. 7, 2711–2717 (2017)

    Article  Google Scholar 

  14. Hayat, T., Khan, M.I., Waqas, M., Alsaedi, A.: Radiative flow of hyperbolic tangent liquid subject to Joule heating. Results Phys. 7, 2197–2203 (2017)

    Article  Google Scholar 

  15. Hayat, T., Mumtaz, M., Shafiq, A., Alsaedi, A.: Stratified magnetohydrodynamic flow of tangent hyperbolic nanofluid induced by inclined sheet. Appl. Math. Mech.-Engl. Ed. 38, 271–288 (2017)

    Article  MathSciNet  Google Scholar 

  16. Hayat, T., Muhammad, K., Alsaedi, A., Asghar, S.: Thermodynamics by melting in flow of an Oldroyd-B material. J. Braz. Soc. Mech. Sci. Eng. 40, 1–11 (2018)

    Article  Google Scholar 

  17. Hussain, A., Malik, M.Y., Salahuddin, T., Rubab, A., Khan, M.: Effects of viscous dissipation on MHD tangent hyperbolic fluid over a nonlinear stretching sheet with convective boundary conditions. Results Phys. 7, 3502–3509 (2017)

    Article  Google Scholar 

  18. Khan, I., Malik, M.Y., Salahuddin, T., Khan, M., Rehman, K.U.: Homogeneous-heterogeneous reactions in MHD flow of Powell-Eyring fluid over a stretching sheet with Newtonian heating. Neural Comput. Appl. 30, 3581–3588 (2018)

    Article  Google Scholar 

  19. Khan, M., Irfan, M., Khan, W.A.: Heat transfer enhancement for Maxwell nanofluid flow subject to convective heat transport. Pramana 92, 1–9 (2019)

    Article  Google Scholar 

  20. Khoshrouye Ghiasi, E., Saleh, R.: Unsteady shrinking embedded horizontal sheet subjected to inclined Lorentz force and Joule heating, an analytical solution. Results Phys. 11, 65–71 (2018)

    Article  Google Scholar 

  21. Khoshrouye Ghiasi, E., Saleh, R.: Non-dimensional optimization of magnetohydrodynamic Falkner-Skan fluid flow. INAE Lett. 3, 143–147 (2018)

    Article  Google Scholar 

  22. Khoshrouye Ghiasi, E., Saleh, R.: Optimal homotopy asymptotic method-based Galerkin approach for solving generalized Blasius boundary value problem. J. Adv. Phys. 7, 408–411 (2018)

    Article  Google Scholar 

  23. Khoshrouye Ghiasi, E., Saleh, R.: Constructing analytic solutions on the Tricomi equation. Open Phys. 16, 143–148 (2018)

    Article  Google Scholar 

  24. Khoshrouye Ghiasi, E., Saleh, R.: Nonlinear stability and thermomechanical analysis of hydromagnetic Falkner-Skan Casson conjugate fluid flow over an angular-geometric surface based on Buongiorno’s model using homotopy analysis method and its extension. Pramana 92, 1–12 (2019)

    Article  Google Scholar 

  25. Khoshrouye Ghiasi, E., Saleh, R.: Homotopy analysis method for the Sakiadis flow of a thixotropic fluid. Eur. Phys. J. Plus 134, 1–9 (2019)

    Article  Google Scholar 

  26. Khoshrouye Ghiasi, E., Saleh, R.: Analytical and numerical solutions to the 2D Sakiadis flow of Casson fluid with cross diffusion, inclined magnetic force, viscous dissipation and thermal radiation based on Buongiorno’s mathematical model. CFD Lett. 11, 40–54 (2019)

    Google Scholar 

  27. Khoshrouye Ghiasi, E., Saleh, R.: 2D flow of Casson fluid with non-uniform heat source/sink and Joule heating. Front. Heat Mass Transfer 12, 1–7 (2019)

    Google Scholar 

  28. Liao, S.J.: Beyond perturbation: Introduction to the homotopy analysis method. Chapman & Hall/CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  29. Mustafa, M., Khan, J.A., Hayat, T., Alsaedi, A.: Analytical and numerical solutions for axisymmetric flow of nanofluid due to non-linearly stretching sheet. Int. J. Nonlinear Mech. 71, 22–29 (2015)

    Article  Google Scholar 

  30. Nadeem, S., Akbar, N.S.: Numerical analysis of peristaltic transport of a tangent hyperbolic fluid in an endoscope. J. Aerospace Eng. 24, 309–317 (2011)

    Article  Google Scholar 

  31. Pop, A., Ingham, D.B.: Convective heat transfer: Mathematical and computational modeling of viscous fluids and porous media. Pergamon, Oxford (2001)

    Google Scholar 

  32. Qayyum, S., Khan, R., Habib, H.: Simultaneous effects of melting heat transfer and inclined magnetic field flow of tangent hyperbolic fluid over a nonlinear stretching surface with homogeneous-heterogeneous reactions. Int. J. Mech. Sci. 133, 1–10 (2017)

    Article  Google Scholar 

  33. Qayyum, S., Hayat, T., Shehzad, S.A., Alsaedi, A.: Mixed convection and heat generation/absorption aspects in MHD flow of tangent-hyperbolic nanoliquid with Newtonian hast/mass transfer. Rad. Phys. Chem. 144, 396–404 (2018)

    Article  Google Scholar 

  34. Shafiq, A., Hammouch, Z., Sindhu, T.N.: Bioconvective MHD flow of tangent hyperbolic nanofluid with newtonian heating. Int. J. Mech. Sci. 133, 759–766 (2017)

    Article  Google Scholar 

  35. Shehzad, S.A., Hayat, T., Alsaedi, A.: MHD flow of a Casson fluid with power law heat flux and heat source. Comput. Appl. Math. 37, 2932–2942 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tanner, R.I.: Engineering rheology, 2nd edn. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  37. Tropea, C., Yarin, A.L., Foss, J.F.: Springer handbook of experimental fluid mechanics. Springer, Berlin (2007)

    Book  Google Scholar 

  38. Vajravelu, K., Sarojamma, G., Sreelakshmi, K., Kalyani, C.: Dual solutions of an unsteady flow, heat and mass transfer of an electrically conducting fluid over a shrinking sheet in the presence of radiation and viscous dissipation. Int. J. Mech. Sci. 130, 119–132 (2017)

    Article  Google Scholar 

  39. Zhu, J., Yang, D., Zheng, L., Zhang, X.: Effects of second order velocity slip and nanoparticles migration on flow of Buongiorno nanofluid. Appl. Math. Lett. 52, 183–191 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, College of Engineering, Mashhad Branch, Islamic Azad University, Mashhad, Iran

    Emran Khoshrouye Ghiasi & Reza Saleh

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  1. Emran Khoshrouye Ghiasi
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  2. Reza Saleh
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Correspondence to Emran Khoshrouye Ghiasi or Reza Saleh.

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Appendix 1

Appendix 1

The left-hand side of Eq. (15) can be derived in the following way:

$$ \begin{aligned} \left( {1 - q} \right)L\left[ {\bar{f}\left( {\eta ;q} \right) - f_{0} \left( \eta \right)} \right] & = \left( {1 - q} \right)L\left[ {\mathop \sum \limits_{m = 0}^{\infty } f_{m} \left( \eta \right)q^{m} - f_{0} \left( \eta \right)} \right] = \left( {1 - q} \right)L\left[ {\mathop \sum \limits_{m = 1}^{\infty } f_{m} \left( \eta \right)q^{m} } \right] \\ & = \mathop \sum \limits_{m = 1}^{\infty } L\left[ {f_{m} \left( \eta \right)} \right]q^{m} - \mathop \sum \limits_{m = 1}^{\infty } L\left[ {f_{m} \left( \eta \right)} \right]q^{m + 1} \\ & = \mathop \sum \limits_{m = 1}^{\infty } L\left[ {f_{m} \left( \eta \right)} \right]q^{m} - \mathop \sum \limits_{m = 2}^{\infty } L\left[ {f_{m - 1} \left( \eta \right)} \right]q^{m} \\ & = L\left[ {f_{1} \left( \eta \right)} \right]q + \mathop \sum \limits_{m = 2}^{\infty } L\left[ {f_{m} \left( \eta \right) - f_{m - 1} \left( \eta \right)} \right]q^{m} \\ & = \mathop \sum \limits_{m = 1}^{\infty } L\left[ {f_{m} \left( \eta \right) - \chi_{m} f_{m - 1} \left( \eta \right)} \right]q^{m} , \\ \end{aligned} $$
(27)

where \( \chi_{m} \) is defined by Eq. (16).

As it can be seen from Eq. (27), employing a Taylor’s series due to the fundamental theorem of calculus [8] provides the \( m \)th-order deformation equation from the same homotopy function. In this way, it is to be noted that similar procedures regarding two dimensional laminar flows have also been reported by Hayat and Sajid [10] and Abbasbandy [1].

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Khoshrouye Ghiasi, E., Saleh, R. A convergence criterion for tangent hyperbolic fluid along a stretching wall subjected to inclined electromagnetic field. SeMA 76, 521–531 (2019). https://doi.org/10.1007/s40324-019-00190-1

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