Abstract
The heat transfer phenomenon plays an imperative role in several biological and industrial processes, like the oil production industries, catalytic reactors, energy losses in several thermal systems, energy storage, papers manufacture and heat exchanger systems. Therefore, the present analysis investigates the heat transfer phenomena on peristaltic transportation of the hydromagnetic flow of non-Newtonian fourth-grade fluid in a tapered asymmetric channel filled with porous media. Moreover, the impacts of heat source/sink and thermal radiation in modeling are retained. The tapered asymmetry in the channel is generated by undertaking the peristaltic wave train inflicted on the non-uniform walls to have altered phases and amplitudes. The analysis is originated by adopting suppositions of long wavelength \(\left( {\delta < < 1} \right)\), small Deborah number \(\left( {\Gamma \to 0} \right)\) and low Reynolds number \(\left( {\text{R} \to 0} \right)\). A regular perturbation technique is utilized to acquire the series outcomes for the axial velocity, pressure gradient and streamlines distribution, while an analytical outcome has been acquired for the thermal profile. The pressure rise at each wavelength on the channel walls has been numerically computed. Impacts of arising parameters in the analysis are surveyed graphically. Outcomes divulge that magnitude of the axial velocity raises with a rise of Darcy number. In contrast, it diminishes with a raise of the magnetic parameter near the center of the channel. Furthermore, the calculated outcomes are found in admirable agreement with the outcomes acquired by the finite element technique and previously published results.
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Abbreviations
- \(A_{1}\) :
-
First Rivlin–Ericksen tensor \(\left[ - \right]\)
- \(\bar{X},\,\,\bar{Y}\) :
-
Space coordinates in laboratory frame \(\left[ m \right]\)
- \(\bar{P},p\) :
-
Pressure in the laboratory and wave frame \(\left[ {kg/s^{2} } \right],\left[ - \right]\)
- \(\bar{U},\bar{V}\) :
-
Velocity components in laboratory frame \(\left[ {ms^{{ - 1}} } \right]\)
- \(c\) :
-
Wave speed \(\left[ {ms^{{ - 1}} } \right]\)
- \(t\) :
-
Time \(\left[ s \right]\)
- \(u,v\) :
-
Velocity components in wave frame \(\left[ {ms^{{ - 1}} } \right]\)
- \(Q_{0}\) :
-
Heat generation parameter \(\left[ {m^{2} s^{{ - 3}} } \right]\)
- \(T_{0} ,\,\,T_{1}\) :
-
Temperature at upper and lower walls \(\left( K \right)\)
- \(d\) :
-
Channel half width \(\left[ m \right]\)
- \(a_{1} ,\,\,a_{2}\) :
-
Wave amplitude at upper and lower walls \(\left[ m \right]\)
- \(B\) :
-
Heat source/sink parameter \(\left[ { = {{Q_{0} d^{2} } \mathord{\left/ {\vphantom {{Q_{0} d^{2} } {k\left( {T_{1} - T_{0} } \right)}}} \right. \kern-\nulldelimiterspace} {k\left( {T_{1} - T_{0} } \right)}}} \right],\left[ - \right]\)
- \(Da\) :
-
Darcy number \(\left[ { = {K \mathord{\left/ {\vphantom {K {d^{2} }}} \right. \kern-\nulldelimiterspace} {d^{2} }}} \right],\left[ - \right]\)
- \(\Pr\) :
-
Prandtl number \(\left[ { = {{\mu C_{p} } \mathord{\left/ {\vphantom {{\mu C_{p} } k}} \right. \kern-\nulldelimiterspace} k}} \right],\left[ - \right]\)
- \(m\) :
-
Non-uniform parameter of the channel \(\left[ - \right]\)
- \(\bar{T}\) :
-
Temperature in laboratory frame \(\left( K \right)\)
- \(R\) :
-
Radiation parameter \(\left[ { = {{16T_{\infty }^{3} \sigma ^{ * } } \mathord{\left/ {\vphantom {{16T_{\infty }^{3} \sigma ^{ * } } {3k^{ * } k}}} \right. \kern-\nulldelimiterspace} {3k^{ * } k}}} \right],\left[ - \right]\)
- \(Q\) :
-
Time average of the flow
- \(M\) :
-
Magnetic parameter \(\left[ { = \sqrt {{\sigma \mathord{\left/ {\vphantom {\sigma \mu }} \right. \kern-\nulldelimiterspace} \mu }} B_{0} d} \right],\left[ - \right]\)
- \(C_{p}\) :
-
Specific heat \(\left[ {Jkg^{{ - 1}} K^{{ - 1}} } \right]\)
- \(g\) :
-
Acceleration due to gravity \(\left[ {LT^{{ - 2}} } \right]\)
- \(\Delta p\) :
-
Pressure rise \(\left[ - \right]\)
- \(\text{R}\) :
-
Reynold’s number \(\left[ { = {{\rho cd_{1} } \mathord{\left/ {\vphantom {{\rho cd_{1} } \mu }} \right. \kern-\nulldelimiterspace} \mu }} \right]\)
- \(B_{0}\) :
-
Strength of magnetic field \(\left[ T \right]\)
- \(\bar{x},\,\,\bar{y}\) :
-
Space coordinates in wave frame \(\left[ m \right]\)
- \(A_{n}\) :
-
Rivlin–Ericksen tensor for \(n > 1\)\(\left[ - \right]\)
- \(G_{r}\) :
-
Grashof number \(\left[ { = {{\rho g\beta _{T} d^{2} \left( {T_{1} - T_{0} } \right)} \mathord{\left/ {\vphantom {{\rho g\beta _{T} d^{2} \left( {T_{1} - T_{0} } \right)} {\mu c}}} \right. \kern-\nulldelimiterspace} {\mu c}}} \right],\left[ - \right]\)
- \(q_{r}\) :
-
Radiative thermal heat flux \(\left[ {Wm^{{ - 2}} } \right]\)
- \(k^{ * }\) :
-
Mean absorption coefficient \(\left[ {KW^{{ - 1}} } \right]\)
- \(\beta _{T}\) :
-
Thermal expansion coefficient \(\left[ {K^{{ - 1}} } \right]\)
- \(\theta\) :
-
Time mean flow rate in fixed frame \(\left[ - \right]\)
- \(\phi\) :
-
Phase difference \(\left[ {{\text{rad}}} \right]\)
- \(\Gamma\) :
-
Deborah number \(\left[ - \right]\)
- \(\lambda\) :
-
Wavelength [m]
- \(\psi\) :
-
Stream function \(\left[ {m^{2} s^{{ - 1}} } \right]\)
- \(\sigma\) :
-
Electrical conductivity \(\left[ {sm^{{ - 1}} } \right]\)
- \(\mu\) :
-
Viscosity \(\left[ {kgm^{{ - 1}} s^{{ - 1}} } \right]\)
- \(\delta\) :
-
Wavenumber \(\left[ { = {{2\pi d} \mathord{\left/ {\vphantom {{2\pi d} \lambda }} \right. \kern-\nulldelimiterspace} \lambda }\,} \right],\left[ - \right]\)
- \(\rho\) :
-
Fluid density \(\left[ {kgm^{{ - 3}} } \right]\)
- \(k\) :
-
Thermal conductivity \(\left[ {Wm^{{ - 1}} K^{{ - 1}} } \right]\)
- \(K\) :
-
Permeability parameter \(\left[ - \right]\)
- \(\gamma\) :
-
Dimensionless temperature \(= \left[ {{{\bar{T} - T_{0} } \mathord{\left/ {\vphantom {{\bar{T} - T_{0} } {T_{1} - T_{0} }}} \right. \kern-\nulldelimiterspace} {T_{1} - T_{0} }}} \right],\left[ - \right]\)
- \(\sigma ^{ * }\) :
-
Stephen Boltzmann constant \(\left[ {Wm^{{ - 2}} K^{{ - 4}} } \right]\)
- NN:
-
Non-Newtonian
- FGF:
-
Fourth-grade fluid
- MHD:
-
Magnetohydrodynamics
- FEM:
-
Finite element method
References
Haynes, R.H.: Physical basis of the dependence of blood viscosity on tube radius. Am. J. Physiol.-Legacy Content 198(6), 1193–1200 (1960)
Haroun, M.H.: Non-linear peristaltic flow of a fourth grade fluid in an inclined asymmetric channel. Comput. Mater. Sci. 39(2), 324–333 (2007)
Hayat, T.; Asghar, Z.; Asghar, S.; Mesloub, S.: Influence of inclined magnetic field on peristaltic transport of fourth grade fluid in an inclined asymmetric channel. J. Taiwan Inst. Chem. Eng. 41(5), 553–563 (2010)
Mustafa, M.; Abbasbandy, S.; Hina, S.; Hayat, T.: Numerical investigation on mixed convective peristaltic flow of fourth grade fluid with Dufour and Soret effects. J. Taiwan Inst. Chem. Eng. 45(2), 308–316 (2014)
Abd-Alla, A.M.; Abo-Dahab, S.M.; El-Shahrany, H.D.: Effects of rotation and initial stress on peristaltic transport of fourth grade fluid with heat transfer and induced magnetic field. J. Magn. Magn. Mater. 349, 268–280 (2014)
Kothandapani, M.; Pushparaj, V.; Prakash, J.: Effect of magnetic field on peristaltic flow of a fourth grade fluid in a tapered asymmetric channel. J. King Saud Univ.-Eng. Sci. 30(1), 86–95 (2018)
Arifuzzaman, S.M.; Khan, M.S.; Al-Mamun, A.; Reza-E-Rabbi, S.; Biswas, P.; Karim, I.: Hydrodynamic stability and heat and mass transfer flow analysis of MHD radiative fourth-grade fluid through porous plate with chemical reaction. J. King Saud Univ.-Sci. 31(4), 1388–1398 (2019)
Latham, T.W.: Fluid motions in peristaltic pump (MS thesis). MIT, Cambridge, MA (1966)
Shapiro, A.H.; Jaffrin, M.Y.; Weinberg, S.L.: Peristaltic pumping with long wavelengths at low Reynolds number. J. Fluid Mech. 37(4), 799–825 (1969)
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. Results Eng. 3, 100024 (2019)
Javed, M.; Naz, R.: Peristaltic flow of a realistic fluid in a compliant channel. Phys. A: Stat. Mech. Appl. 551, 123895 (2020)
Rajashekhar, C.; Vaidya, H.; Prasad, K.V.; Tlili, I.; Patil, A.; Nagathan, P.: Unsteady flow of Rabinowitsch fluid peristaltic transport in a non-uniform channel with temperature-dependent properties. Alex. Eng. J. 59(2), 693–706 (2020)
Vaidya, H.; Rajashekhar, C.; Divya, B.B.; Manjunatha, G.; Prasad, K.V.; Animasaun, I.L.: Influence of transport properties on the peristaltic MHD Jeffrey fluid flow through a porous asymmetric tapered channel. Results Phys. 18, 103295 (2020)
Saleem, A.; Akhtar, S.; Alharbi, F.M.; Nadeem, S.; Ghalambaz, M.; Issakhov, A.: Physical aspects of peristaltic flow of hybrid nano fluid inside a curved tube having ciliated wall. Results Phys. 19, 103431 (2020)
Abo-Elkhair, R.E.; Bhatti, M.M.; Mekheimer, K.S.: Magnetic force effects on peristaltic transport of hybrid bio-nanofluid (AuCu nanoparticles) with moderate Reynolds number: An expanding horizon. Int. Commun. Heat Mass Transf. 123, 105228 (2021)
Imran, N.; Javed, M.; Sohail, M.; Tlili, I.: Simultaneous effects of heterogeneous-homogeneous reactions in peristaltic flow comprising thermal radiation: Rabinowitsch fluid model. J. Mater. Res. Technol. 9, 3520–3529 (2020)
Noreen, S.; Kausar, T.; Tripathi, D.; Ain, Q.U.; Lu, D.C.: Heat transfer analysis on creeping flow Carreau fluid driven by peristaltic pumping in an inclined asymmetric channel. Thermal Sci. Eng. Progress 17, 100486 (2020)
Rashid, M.; Ansar, K.; Nadeem, S.: Effects of induced magnetic field for peristaltic flow of Williamson fluid in a curved channel. Phys. A: Stat. Mech. Appl. 553, 123979 (2020)
Saleem, S.; Akhtar, S.; Nadeem, S.; Saleem, A.; Ghalambaz, M.; Issakhov, A.: Mathematical study of electroosmotically driven peristaltic flow of Casson fluid inside a tube having systematically contracting and relaxing sinusoidal heated walls. Chin. J. Phys. 71, 300–311 (2021)
Hayat, T.; Asghar, S.; Tanveer, A.; Alsaedi, A.: Chemical reaction in peristaltic motion of MHD couple stress fluid in channel with Soret and Dufour effects. Results Phys. 10, 69–80 (2018)
Bhatti, M.M.; Riaz, A.; Zhang, L.; Sait, S.M.; Ellahi, R.: Biologically inspired thermal transport on the rheology of Williamson hydromagnetic nanofluid flow with convection: an entropy analysis. J. Thermal Anal. Calorim. 144, 1–16 (2020)
Divya, B.B.; Manjunatha, G.; Rajashekhar, C.; Vaidya, H.; Prasad, K.V.: The hemodynamics of variable liquid properties on the MHD peristaltic mechanism of Jeffrey fluid with heat and mass transfer. Alex. Eng. J. 59(2), 693–706 (2020)
Iqbal, J.; Abbasi, F.M.; Shehzad, S.A.: Heat transportation in peristalsis of Carreau-Yasuda nanofluid through a curved geometry with radial magnetic field. Int. Commun. Heat Mass Transf. 117, 104774 (2020)
Abbas, Z.; Rafiq, M.Y.; Hasnain, J.; Umer, H.: Impacts of lorentz force and chemical reaction on peristaltic transport of Jeffrey fluid in a penetrable channel with injection/suction at walls. Alex. Eng. J. 60(1), 1113–1122 (2021)
Abbas, Z.; Rafiq, M.Y.; Alshomrani, A.S.; Ullah, M.Z.: Analysis of entropy generation on peristaltic phenomena of MHD slip flow of viscous fluid in a diverging tube. Case Studies Thermal Eng. 23, 100817 (2021)
Arain, M.B.; Bhatti, M.M.; Zeeshan, A.; Saeed, T.; Hobiny, A.: Analysis of arrhenius kinetics on multiphase flow between a pair of rotating circular plates. Math. Problems Eng. 2020, 1–17 (2020)
Khazayinejad, M.; Hafezi, M.; Dabir, B.: Peristaltic transport of biological graphene-blood nanofluid considering inclined magnetic field and thermal radiation in a porous media. Powder Technol. 384, 452–465 (2021)
Alsaedi, A.; Nisar, Z.; Hayat, T.; Ahmad, B.: Analysis of mixed convection and hall current for MHD peristaltic transport of nanofluid with compliant wall. Int. Commun. Heat Mass Transf. 121, 105121 (2021)
Qureshi, I.H.; Awais, M.; Awan, S.E.; Abrar, M.N.; Raja, M.A.Z.; Alharbi, S.O.; Khan, I.: Influence of radially magnetic field properties in a peristaltic flow with internal heat generation: numerical treatment. Case Studies Thermal Eng. 26, 101019 (2021)
Rosseland, S.: Astrophysik und Atom-Theoretische Grundlagen, 41–44. Springer-Verlag, Berlin (1931)
Bhatti, M.M.; Marin, M.; Zeeshan, A.; Ellahi, R.; Abdelsalam, S.I.: Swimming of motile gyrotactic microorganisms and nanoparticles in blood flow through anisotropically tapered arteries. Front. Phys. 8, 95 (2020)
Akram, S.; Athar, M.; Saeed, K.: Hybrid impact of thermal and concentration convection on peristaltic pumping of Prandtl nanofluids in non-uniform inclined channel and magnetic field. Case Studies Thermal Eng. 25, 100965 (2021)
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We are thankful to the reviewers for their encouraging comments and constructive suggestions to improve the quality of the manuscript. The authors appreciate the financial support from HEC Pakistan through NRPU-6295.
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Rafiq, M.Y., Abbas, Z. & Hasnain, J. Theoretical Exploration of Thermal Transportation with Lorentz Force for Fourth-Grade Fluid Model Obeying Peristaltic Mechanism. Arab J Sci Eng 46, 12391–12404 (2021). https://doi.org/10.1007/s13369-021-05877-0
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DOI: https://doi.org/10.1007/s13369-021-05877-0