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Thermo-Diffusion and Diffusion-Thermo Effects on MHD Third-Grade Nanofluid Flow Driven by Peristaltic Transport

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Abstract

The current paper examines the thermo-diffusion and diffusion-thermo effects on MHD third-grade nanofluid driven by peristalsis. The governing equations are linearized by adopting the low Reynolds number and long wavelength approximations. The Adomian series expression for velocity, stream function, pressure, concentration and temperature are acquired. The effect of sundry variables are discussed and illustrated graphically. The results reveal that the temperature values are enhanced with increasing Dufour number and the Soret number diminishes the concentration. Further, it is found that this study can be used for fabrication of semiconductor devices, manipulation of DNA and separation of biomolecules, etc.

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Abbreviations

c :

Speed along the channel walls

t :

The time

a 1 :

Amplitude of the upper wall

a 2 :

Amplitude of the lower wall

d 1 :

Half-width of the upper wall

d 2 :

Half-width of the lower wall

\( \overrightarrow {J} \) :

Electric current density

\( \vec{E} \) :

Electric field density

\( \vec{v} \) :

Velocity vector

p :

Fluid pressure

\( X^{{\prime }} ,\;Y^{{\prime }} \) :

Cartesian coordinates in laboratory frame

\( U^{{\prime }} ,\;V^{{\prime }} \) :

Velocity components in laboratory frame

\( x^{{\prime }} ,\;y^{{\prime }} \) :

Cartesian coordinates in wave frame

\( u^{{\prime }} ,\;v^{{\prime }} \) :

Velocity components in wave frame

Re:

Reynolds’s number

Sc:

Schmidt’s number

Sr:

Soret’s number

Du:

Dufour number

Pr:

Prandtl number

D T :

Thermophoretic diffusion

l :

Thermal conductivity

g :

Acceleration due to gravity

D B :

Brownian diffusion coefficient

C 1 :

Volumetric expansion coefficient

\( C_{\text{s}} \) :

The concentration susceptibility

F :

Dimensionless mean flow

\( C_{p} \) :

The specific heat

Gt:

Local temperature Grashof number

Gc:

Mass Grashof number

N b :

Brownian motion parameter

N t :

Thermophoresis parameter

λ :

Wavelength

\( \phi \) :

Phase difference

\( \sigma \) :

Electrical conductivity

\( \mu_{e} \) :

Magnetic permeability

\( \alpha_{1} ,\alpha_{2} ,\beta_{1} ,\beta_{2} \;{\text{and}}\;\beta_{3} \) :

Material constants

\( \rho_{\text{f}} \) :

Density of the fluid

\( \rho_{\text{p}} \) :

Density of the nanoparticle

\( \mu \) :

Dynamic viscosity

\( \phi \) :

Dimensionless concentration

θ :

Temperature

\( \delta \) :

Wave number

\( \kappa \) :

Thermal diffusivity

References

  1. Latham, T. W.: Fluid motion in a peristaltic pump. M.S. Thesis. MIT. Cambridge, MA (1966)

  2. Fung, Y.C.; Yih, C.S.: Peristaltic transport. J. Appl. Mech. 35, 669–675 (1968)

    Article  MATH  Google Scholar 

  3. Shapiro, A.H.; Jaffrin, M.Y.; Weinberg, S.L.: Peristaltic pumping with long wavelength and low Reynolds number. J. Fluid Mech. 37, 799–825 (1969)

    Article  Google Scholar 

  4. El Shehawey, E.F.; Mekheimer, K.S.: Couple stresses in peristaltic transport of fluids. J. Phys. 27, 1163–1170 (1994)

    MathSciNet  MATH  Google Scholar 

  5. Ebaid, A.; Elshehawey, E.F.; Eldabe, N.T.; Elghazy, E.M.: Peristaltic transport in an asymmetric channel through a porous medium. Appl. Math. Comput. 182(1), 140–150 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Asha, S.K.; Deepa, C.K.: Influence of induced magnetic field and heat transfer on peristaltic transport of a micro polar fluid in a tapered asymmetric channel. Heat Transf. Asian Res. 48, 2714–2735 (2019)

    Article  Google Scholar 

  7. Gireesha, B.J.; Reddy Gorla, R.S.; Mahanthesh, B.: Effect of suspended nanoparticles on three-dimensional MHD flow, heat and mass transfer of radiating Eyring–Powell fluid over a stretching sheet. J. Nanofluids. 4(4), 1–11 (2015)

    Article  Google Scholar 

  8. Lakshmi, K.L.; Gireesha, B.J.; Mahanthesh, B.; Reddy Gorla, R.S.: Influence of nonlinear thermal radiation and magnetic field on upper-convected Maxwell fluid flow due to a convectively heated stretching sheet in the presence of dust particles. Int. Sci. Publ. Consult. Serv. 2016, 57–73 (2016)

    MathSciNet  Google Scholar 

  9. Kumar, B.; Seth, G.S.; Nandkeolyar, R.; Chamkha, A.J.: Outlining the impact of induced magnetic field and thermal radiation on magneto-convection flow of dissipative fluid. Int. J. Therm. Sci. 149, 106101 (2019)

    Article  Google Scholar 

  10. Shashikumar, N.S.; Gireesha, B.J.; Mahanthesh, B.; Prasannakumara, B.C.; Chamkha, A.J.: Entropy generation analysis of magneto-nanoliquids embedded with aluminium and titanium alloy nanoparticles in microchannel with partial slips and convective conditions. Int. J. Numer. Methods Heat Fluid Flow 29(10), 3638–3658 (2019)

    Article  Google Scholar 

  11. Taebi, T.; Chamkha, A.J.: Entropy generation analysis during MHD natural convection flow of hybrid nanofluid in a square cavity containing a corrugated conducting block. Int. J. Numer. Methods Heat Fluid Flow 30(3), 1115–1136 (2019)

    Article  Google Scholar 

  12. Alsabery, A.I.; Armaghani, T.; Chamkha, A.J.; Hashim, I.: Two-phase nanofluid model and magnetic field effects on mixed convection in a lid-driven cavity containing heated triangular wall. Alex. Eng. J. 59(1), 129–148 (2020)

    Article  Google Scholar 

  13. Dogonchi, A.S.; Tayebi, T.; Chamkha, A.J.; Ganji, D.D.: Natural convection analysis in a square enclosure with a wavy circular heater under magnetic field and nanoparticles. J. Therm. Anal. Calorim. 139, 661–671 (2020)

    Article  Google Scholar 

  14. Ghalambaz, M.; Mehryan, S.A.M.; Izadpanahi, E.; Chamkha, A.J.; Wen, D.: MHD natural convection of Cu-Al2O3 water hybrid nanofluids in a cavity equally divided into twoparts by a vertical flexible partition membrane. J. Therm. Anal. Calorim. 138, 1723–1743 (2020)

    Article  Google Scholar 

  15. Ayoubloo, K.A.; Ghalambaz, M.; Armaghani, T.; Aminreza, N.; Chamkha, A.J.: Pseudoplastic natural convection flow and heat transfer in a cylindrical vertical cavity partially filled with a porous layer. Int. J. Numer. Methods Heat Fluid Flow 30(3), 1096–1114 (2019)

    Article  Google Scholar 

  16. Amrouche, C.; Cioranescu, D.: On a class of fluids of grade 3. Int. J. Non-Linear Mech. 32, 73–88 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Noreen, S.: Mixed convection peristaltic flow of third order nanofluid with an induced magnetic field. PLoS One 8(11), e78770 (2013)

    Article  Google Scholar 

  18. Chamkha, A.J.: MHD flow of a micropolar fluid past a stretched permeable surface with heat generation or absorption. Non-linear Anal. Model. Control 14(1), 27–40 (2009)

    Article  MATH  Google Scholar 

  19. Mehryan, S.A.M.; Heidarshenas, M.H.; Hajjar, A.; Ghalambaz, M.: Numerical study of melting-process of a non-Newtonian fluid inside a metal foam. Alex. Eng. J. 59(1), 191–207 (2020)

    Article  Google Scholar 

  20. Choi, S.U.S.: Enhancing thermal conductivity of fluid with nanofluid: developments and applications of non-Newtonian flows. ASME J. Heat Transf. 66, 99–105 (1995)

    Google Scholar 

  21. Asha, S.K.; Deepa, C.K.: Effect of inclined magnetic field and mixed convection peristaltic flow of nanofluids in an asymmetric channel by Adomian Decomposition Method. J. Appl. Sci. Comput. 5(9), 135–147 (2018)

    Google Scholar 

  22. Abbasi, F.M.; Hayat, T.; Ahmad, B.; Chen, G.Q.: Peristaltic motion of non-Newtonian nanofluid in an asymmetric channel. Z. Naturforsch. 69, 451–461 (2014)

    Article  Google Scholar 

  23. Gireesha, B.J.; Mahanthesh, B.; Shivakumara, I.S.; Eshwarappa, K.M.: Melting heat transfer in boundary layer stagnation-point flow of nanofluid toward a stretching sheet with induced magnetic field. Eng. Sci. Technol. Int. J. 19(1), 313–321 (2016)

    Google Scholar 

  24. Mahanthesh, B.; Gireesha, B.J.; Reddy Gorla, R.S.; Abbasi, F.M.; Shehzad, S.A.: Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary. J. Magn. Magn. Mater. 417, 189–196 (2016)

    Article  Google Scholar 

  25. Asha, S.K.; Deepa, C.K.: Mixed convection of MHD peristaltic flow of a pseudoplastic nanofluid in asymmetric channel. J. Comput. Math. Sci. 9(7), 736–747 (2018)

    Google Scholar 

  26. Amala, S.; Mahanthesh, B.: Hybrid nanofluid flow over a vertical rotating plate in the presence of Hall current. Nonlinear convection and heat absorption. J. Nanofluids 7(6), 1138–1148 (2018)

    Article  Google Scholar 

  27. Mohammad, G.; Chamka, A.J.; Wen, D.: Natural convective flow and heat transfer of nano-encapsulated phase change materials (NEPCMs) in a cavity. Int. J. Heat Mass Transf. 138, 738–749 (2019)

    Article  Google Scholar 

  28. Chamkha, A.J.; Sazegar, S.; Jamesahar, E.; Ghalambaz, M.: Thermal non-equilibrium heat transfer modelling of hybrid nanofluids in a structure composed of the layers of solid and porous media and free nanofluids. Energies 12(3), 541 (2019)

    Article  Google Scholar 

  29. Mehryan, S.A.M.; Izadpanahi, E.; Ghalambaz, M.; Chamkha, A.J.: Mixed convection flow caused by an oscillating cylinder in a square cavity filled with Cu–Al2O3/water hybrid nanofluid. J. Therm. Anal. Calorim. 137, 965–982 (2019)

    Article  Google Scholar 

  30. Mehryan, S.A.M.; Ghalambaz, M.; Gargari, L.S.; Hajjar, A.; Sheremet, M.: Natural convection flow of a suspension containing nano-encapsulated phase change particles in an eccentric annulus. J. Energy Storage 28, 101236 (2020)

    Article  Google Scholar 

  31. Ghalambaz, M.; Doostani, A.; Izadpanahi, E.; Chamkha, A.J.: Conjugate natural convection flow of Ag–MgO/water hybrid nanofluid in a square cavity. J. Therm. Anal. Calorim. 139, 2321–2336 (2020)

    Article  Google Scholar 

  32. Platten, J.K.: The Soret effect: a review of recent experimental results. J. Appl. Mech. 73, 5–15 (2006)

    Article  MATH  Google Scholar 

  33. Hayat, T.; Rafiq, M.; Bashir, A.: Soret and Dufour effects on MHD peristaltic flow of a Jeffrey fluid in a rotating system with porous medium. PLoS One 11(1), e0145525 (2016)

    Article  Google Scholar 

  34. Bilal Ashraf, M.; Hayat, T.; Alsaedi, A.; Shehzad, S.A.: Soret and Dufour effects on the mixed convection flow of an Oldroyd-B fluid with convective boundary conditions. Res. Phys. 6, 917–924 (2016)

    Google Scholar 

  35. Hayat, T.; Nawaz, M.; Asghar, S.; Mesloub, S.: Thermal diffusion and diffuision thermo effects on axisymmetric flow of a second grade fluid. Int. J. Heat Mass Transf. 54, 3031–3041 (2011)

    Article  MATH  Google Scholar 

  36. Hayat, T.; Rija, I.; Tanveer, A.; Alsaedi, A.: Soret and Dufour effects in MHD peristalsis of pseudoplastic nanofluid with chemical reaction. J. Mol. Liq. 220, 693–706 (2016)

    Article  Google Scholar 

  37. Asha, S.K.; Deepa, C.K.: Entropy generation for peristaltic blood flow of a magneto-micropolar fluid with thermal radiation in a tapered asymmetric channel. Res. Eng. 3, 100024 (2019)

    Article  Google Scholar 

  38. Reddy, P.S.; Chamkha, A.J.: Soret and Dufour effects on MHD convective flow of Al2O3-water and TiO2-water nanofluids past a stretching sheet in porous media with heat generation/absorption. Adv. Powder Technol. 27(4), 1207–1218 (2016)

    Article  Google Scholar 

  39. Adomain, G.: A review of the decomposition method and some recent results for nonlinear equation. Math. Comput. Mod. 13, 17–43 (1992)

    Article  MathSciNet  Google Scholar 

  40. Mekheimer, K.S.; Hemada, K.A.; Raslan, K.R.; Abo-Elkhair, R.E.; Moawad, A.M.A.: Numerical study of a non-linear peristaltic transport Application of Adomian decomposition method. Gen. Math. Notes. 20(2), 22–49 (2014)

    Google Scholar 

  41. Siddiqui, A.M.; Hameed, M.; Siddiqui, B.M.; Babcock, B.S.: Adomian decomposition method applied to study non-linear equations arising in non-Newtonian flows. Appl. Math. Sci. 6(98), 4889–4909 (2012)

    MathSciNet  MATH  Google Scholar 

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Funding

Funding was provided by University Grants Commission (Grant No. F510/3/DRS-III/2016(SAP-I)).

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Authors and Affiliations

  1. Department of Mathematics, Karnatak University, Dharwad, Karnataka, India

    Asha S. Kotnurkar & Deepa C. Katagi

Authors
  1. Asha S. Kotnurkar
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  2. Deepa C. Katagi
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Correspondence to Deepa C. Katagi.

Appendix

Appendix

$$ \begin{aligned} c_{1} &= \frac{{1 - \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right)}}{{\left( {h_{1} - h_{2} } \right)}},\;c_{2} = \frac{{ - 1 + \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right)}}{{\left( {h_{1} - h_{2} } \right)}},\\ c_{3} &= \frac{{ - e^{{Ch_{1} }} }}{{\left( {e^{{Ch_{1} }} - e^{{Ch_{2} }} } \right)}}\;{\text{and}}\;c_{4} = \frac{1}{{\left( {e^{{Ch_{1} }} - e^{{Ch_{2} }} } \right)}} \end{aligned}$$
$$ \begin{aligned} F_{1} & = \frac{F}{4\varGamma } - \left[ {\frac{{A_{2} }}{k}\sinh \left( {kh_{1} } \right) + \frac{{A_{3} }}{{k^{2} }}\left( {\cosh \left( {kh_{1} } \right) - 1} \right)\left\{ {\left( {\frac{\partial p}{\partial x} + n^{2} } \right) - {\text{Gt}}c_{3} + {\text{Gc}}c_{3} \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right) + {\text{Gc}}c_{2} } \right\}\frac{1}{{k^{3} }}} \right.\left( {\sinh \left( {kh_{1} } \right) - kh_{1} } \right) \\ & \quad + \;\left. {\left\{ {{\text{Gc}}c_{4} \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right) - {\text{Gt}}c_{4} } \right\}\frac{{e^{{Ch_{1} }} }}{{Ck^{2} }} + \frac{{{\text{Gc}}c_{1} }}{{k^{4} }}\left( {\cosh \left( {kh_{1} } \right) - 1 - \frac{{\left( {kh_{1} } \right)^{2} }}{2!}} \right)} \right] \\ \end{aligned} $$
$$ \begin{aligned} F_{2} & = \frac{{ - \tanh \left( {kh_{2} } \right)}}{{\sinh \left[ {k\left( {h_{1} + h_{2} } \right)} \right]}} + \left[ {\left\{ {\left( {\frac{\partial p}{\partial x} + n^{2} } \right) - {\text{Gt}}c_{3} + {\text{Gc}}c_{3} \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right) + {\text{Gc}}c_{2} } \right\}} \right.\left( {\frac{{\cosh \left( {kh_{1} - 1} \right)\cosh \left( {kh_{2} } \right)}}{{k^{2} }}} \right. \\ & \quad - \;\left. {\frac{{\cosh \left( {kh_{1} - 1} \right)\cosh \left( {kh_{2} } \right)}}{{k^{2} }}} \right) + \left\{ {{\text{Gc}}c_{4} \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right) - {\text{Gt}}c_{4} } \right\}\left( {\frac{{e^{{Ch_{1} }} \cosh \left( {kh_{2} } \right)}}{{k^{2} }} - \frac{{e^{{Ch_{2} }} \cosh \left( {kh_{1} } \right)}}{{k^{2} }}} \right) \\ & \quad + \;\frac{{{\text{Gc}}c_{1} }}{{k^{3} }}\left( {\left\{ {\sinh \left( {kh_{2} } \right) + kh_{2} } \right\}\cosh \left( {kh_{1} } \right) - \left\{ {\sinh \left( {kh_{1} } \right) + kh_{1} } \right\}\cosh \left( {kh_{2} } \right)} \right) + \left\{ {\left( {\frac{\partial p}{\partial x} + n^{2} } \right) - {\text{Gt}}c_{3} } \right. \\ & \quad + \;\left. {{\text{Gc}}c_{3} \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right) + {\text{Gc}}c_{2} } \right\}\frac{{\cosh \left( {kh_{2} - 1} \right)\cosh \left( {kh_{1} } \right)}}{{k^{2} }} - \frac{{{\text{Gc}}c_{1} }}{{k^{3} }}\left\{ {\sinh \left( {kh_{2} } \right) + kh_{2} } \right\}\cosh \left( {kh_{1} } \right) \\ \end{aligned} $$
$$ \begin{aligned} F_{3} & = \frac{ - k}{{\sinh \left[ {k\left( {h_{1} + h_{2} } \right)} \right]}}\left\{ {\left( {\frac{\partial p}{\partial x} + n^{2} } \right) - {\text{Gt}}c_{3} + {\text{Gc}}c_{3} \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right) + {\text{Gc}}c_{2} } \right\}\left( {\frac{{\cosh \left( {kh_{1} - 1} \right)\cosh \left( {kh_{2} } \right)}}{{k^{2} }}} \right. \\ & \quad - \;\left. {\frac{{\cosh \left( {kh_{2} - 1} \right)\cosh \left( {kh_{1} } \right)}}{{k^{2} }}} \right) + \left\{ {{\text{Gc}}c_{4} \left( {{\text{Sr}}\;{\text{Sc}} + \frac{{N_{\text{t}} }}{{N_{\text{b}} }}} \right) - {\text{Gt}}c_{4} } \right\}\left( {\frac{{e^{{Ch_{1} }} \cosh \left( {kh_{2} } \right)}}{{k^{2} }} - \frac{{e^{{Ch_{2} }} \cosh \left( {kh_{1} } \right)}}{{k^{2} }}} \right) \\ & \quad + \;\frac{{{\text{Gc}}c_{1} }}{{k^{3} }}\left( {\left\{ {\sinh \left( {kh_{2} } \right) + kh_{2} } \right\}\cosh \left( {kh_{1} } \right) - \left\{ {\sinh \left( {kh_{1} } \right) + kh_{1} } \right\}\cosh \left( {kh_{2} } \right)} \right) \\ \end{aligned} $$

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Kotnurkar, A.S., Katagi, D.C. Thermo-Diffusion and Diffusion-Thermo Effects on MHD Third-Grade Nanofluid Flow Driven by Peristaltic Transport. Arab J Sci Eng 45, 4995–5008 (2020). https://doi.org/10.1007/s13369-020-04590-8

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