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Numerical investigation of the hybrid ferrofluid flow in a heterogeneous porous channel with convectively heated and quadratically stretchable walls

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Abstract

The spatial heterogeneity in the porosity and permeability of the porous media is noticed in the catalytic reactors, composite materials, turbomachinery, groundwater remediation, filters, oil recovery, physiological processes, biological tissues and arteries. In the complex porous arrangement, the fluid follows a preferred path constituting a channelling effect. This channelling phenomenon occurring in such porous beds is considered here for the investigation with the novel inclusion of quadratically stretchable but convectively heated walls of the channel. In this study, the flow and temperature modulations in the hybrid ferrofluid flow due to the combined influence of Kelvin and Lorentz forces are examined. The heterogeneous porous channel is stretched quadratically in the fluid flow direction. The Carman–Kozeny correlation is used to estimate the permeability of the medium by taking the exponential variations in the porosity across the width of the channel. The mathematical model for the problem is developed and is further reduced to a self-similar form with the help of proper similarity transformations. The Chebyshev pseudospectral quasi-linearization scheme is utilized to obtain the optimal numerical information. A numerical survey is performed in the form of a comparative analysis between the heterogeneous porous medium (HePM) and homogeneous porous medium (HoPM). The interesting convection transfer mode is found to be dominant for HePM, whereas the substantial conduction transfer mode is noted for HoPM. The Nusselt number is pronounced with the uplifted values of the ferromagnetic number and nonlinear stretching constant. However, it is hampered with the Hartmann number and bead diameter number.

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Data Availability Statement

The manuscript has no associated data.

Abbreviations

a :

   Stretching rate (\(\mathrm s^{-1}\))

\(\mathrm Bi\) :

   Biot number

b :

   Quadratic constant (\(\mathrm (ms)^{-1}\))

\(c_1, c_2\) :

   Empirical constants

\(D_p\) :

   Dimensionless bead number

\(d^*\) :

   Dimensionless distance of magnetic source

\(\mathrm Ec\) :

   Eckert number

f, g :

   Dimensionless velocity components

Ha :

   Hartmann number

I :

   Dipole moment per unit length (\(\mathrm A\))

\(K'\) :

   Pyromagnetic coefficient (\(\mathrm K^{-1}\))

Mn:

   Ferromagnetic number

Pr:

   Prandtl number

p :

   Pressure (\(\mathrm kg m^{-1} s^{-2}\))

\(q_c\) :

   Convective heat transfer coefficient (\(\mathrm W m^{-2} K^{-1}\))

Re:

   Reynolds number

\(T_0\) :

   Mean temperature (\(\mathrm K\))

\(T_c\) :

   Curie temperature (\(\mathrm K\))

\(u_0\) :

   Mean velocity (\(\mathrm ms^{-1}\))

\(\varepsilon _0\) :

   Constant porosity

\(\delta _T\) :

   Temperature ratio

\(\delta _C\) :

   Curie temperature ratio

\(\varepsilon (\eta )\) :

   Dimensionless variable porosity

\(\theta\) :

   Dimensionless temperature

\(\eta\) :

   Similarity variable

\(\lambda _L\) :

   Stretching number

\(\lambda _Q\) :

   Quadratic stretching constant

\(\mu _e\) :

   Magnetic permeability (\(H m^{-1}\))

\(\phi\) :

  Nanoparticle concentration

bf :

   Base liquid

hnf :

   Hybrid nanofluid

p1:

   \({{\text{Fe}}_3 {\text{O}}_4}\) nanoparticle

p2:

   \({{\text{Co Fe}}_2 {\text{O}}_4}\) nanoparticle

References

  1. J. Warren, H. Price, Flow in heterogeneous porous media. Soc. Pet. Eng. J. 1(3), 153–169 (1961)

    Article  Google Scholar 

  2. M. Sahimi, Flow and transport in porous media and fractured rock: from classical methods to modern approaches (John Wiley & Sons, 2011)

    Book  MATH  Google Scholar 

  3. D.A. Nield, A. Bejan, Convection in porous media (Springer, 2006)

    MATH  Google Scholar 

  4. R. Benenati, C. Brosilow, Void fraction distribution in beds of spheres. AIChE J. 8(3), 359–361 (1962)

    Article  Google Scholar 

  5. B. Chandrasekhara, D. Vortmeyer, Flow model for velocity distribution in fixed porous beds under isothermal conditions. Wärme-und Stoffübertragung 12(2), 105–111 (1979)

    Article  ADS  Google Scholar 

  6. D. Poulikakos, K. Renken, Forced convection in a channel filled with porous medium, including the effects of flow inertia, variable porosity, and Brinkman friction. J. Heat Trans. 109(4), 880–888 (1987)

    Article  Google Scholar 

  7. B. Chandrasekhara, N. Radha, Effect of variable porosity on laminar convection in a uniformly heated vertical porous channel. Wärme-und Stoffübertragung 23(6), 371–377 (1988)

    Article  ADS  MATH  Google Scholar 

  8. S.M. Al-Weheibi, M. Rahman, M. Saghir, Impacts of variable porosity and variable permeability on the thermal augmentation of Cu-\(H_2 O\) nanofluid-drenched porous trapezoidal enclosure considering thermal nonequilibrium model. Arab. J. Sci. Eng. 45(2), 1237–1251 (2020)

    Article  Google Scholar 

  9. M.H. Park, P. Chhai, K. Rhee, Analysis of flow and wall deformation in a stenotic flexible channel containing a soft core, simulating atherosclerotic arteries. Int. J. Precis. Eng. Manuf. 20, 1047–1056 (2019)

    Article  Google Scholar 

  10. C. Park, B. Lee, J. Kim, H. Lee, J. Kang, J. Yoon, J. Ban, C. Song, S.J. Cho, Flexible sensory systems: structural approaches. Polymers 14(6), 1–32 (2022)

    Article  Google Scholar 

  11. K. Liu, M. Wiendels, H. Yuan, C. Ruan, P.H. Kouwer, Cell-matrix reciprocity in 3D culture models with nonlinear elasticity. Bioact. Mater. 9, 316–331 (2022)

    Article  Google Scholar 

  12. Y. Y. L. Wang, Y. H. Chen, D. J. Guo, C. C. Lin, W. K. Wang, The benefit of stretching along the artery, in 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 2451-2452 (2008)

  13. J. Misra, G. Shit, H.J. Rath, Flow and heat transfer of a MHD viscoelastic fluid in a channel with stretching walls: some applications to haemodynamics. Comput. Fluids 37(1), 1–11 (2008)

    Article  MATH  Google Scholar 

  14. T. Sharma, R. Kumar, K. Vajravelu, M. Sheikholeslami, Hybrid nanofluid flow in a deformable and permeable channel. Int. J. Mod. Phys. B 37(22), 1-26 (2023). https://doi.org/10.1142/S0217979223502168

    Article  Google Scholar 

  15. E. Elbashbeshy, Heat transfer over an exponentially stretching continuous surface with suction. Arch. Mech. 53(6), 643–651 (2001)

    MATH  Google Scholar 

  16. R. Kumar, S. Sood, Combined influence of fluctuations in the temperature and stretching velocity of the sheet on MHD flow of Cu-water nanofluid through rotating porous medium with cubic auto-catalysis chemical reaction. J. Mol. Liq. 237, 347–360 (2017)

    Article  Google Scholar 

  17. H. Vaidya, K. Prasad, I. Tlili, O. Makinde, C. Rajashekhar, S.U. Khan, R. Kumar, D. Mahendra, Mixed convective nanofluid flow over a non linearly stretched Riga plate,. Case Stud. Therm. Eng. 24, 1–19 (2021)

    Article  Google Scholar 

  18. N.A.A.M. Nasir, A. Ishak, I. Pop, Stagnation-point flow and heat transfer past a permeable quadratically stretching/shrinking sheet. Chin. J. Phys. 55(5), 2081–2091 (2017)

    Article  Google Scholar 

  19. N. Nasir, A. Ishak, I. Pop, N. Zainuddin, MHD stagnation point flow towards a quadratically stretching/shrinking surface. J. Phys. Conf. Series 1366(1), 1–9 (2019)

    Article  Google Scholar 

  20. M. Ferdows, G. Murtuza, E. Tzirtalakis, A duality of biomagnetic fluid flow and heat transfer over a quadratic stretched sheet. J. Power Technol. 101(3), 154–162 (2021)

    Google Scholar 

  21. M. Kole, S. Khandekar, Engineering applications of ferrofluids: a review. J. Magn. Magn. Mater. 537, 1–21 (2021)

    Article  Google Scholar 

  22. M. Pattanaik, V.B. Varma, S. Cheekati, V. Chaudhary, R.V. Ramanujan, Optimal ferrofluids for magnetic cooling devices. Sci. Rep. 11(1), 1–19 (2021)

    Article  Google Scholar 

  23. R. E. Rosensweig, Ferrohydrodynamics, Courier Corporation, 2013

  24. H. Andersson, O. Valnes, Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole. Acta Mech. 128(1), 39–47 (1998)

    Article  MATH  Google Scholar 

  25. E. Tzirtzilakis, V. Loukopoulos, Biofluid flow in a channel under the action of a uniform localized magnetic field. Comput. Mech. 36(5), 360–374 (2005)

    Article  MATH  Google Scholar 

  26. A. Malekzadeh, A. Heydarinasab, B. Dabir, Magnetic field effect on fluid flow characteristics in a pipe for laminar flow. J. Mech. Sci. Technol. 25, 333–339 (2011)

    Article  Google Scholar 

  27. Z. Mehrez, A. El Cafsi, A. Belghith, P. Le Quere, MHD effects on heat transfer and entropy generation of nanofluid flow in an open cavity. J. Magn. Magn. Mater. 374, 214–224 (2015)

    Article  ADS  Google Scholar 

  28. S.O. Giwa, M. Sharifpur, J.P. Meyer, Effects of uniform magnetic induction on heat transfer performance of aqueous hybrid ferrofluid in a rectangular cavity. Appl. Therm. Eng. 170, 1–12 (2020)

    Article  Google Scholar 

  29. M. Bezaatpour, M. Goharkhah, Effect of magnetic field on the hydrodynamic and heat transfer of magnetite ferrofluid flow in a porous fin heat sink. J. Magn. Magn. Mater. 476, 506–515 (2019)

    Article  ADS  Google Scholar 

  30. Z. Mehrez, A. El Cafsi, Heat exchange enhancement of ferrofluid flow into rectangular channel in the presence of a magnetic field. Appl. Math. Comput. 391, 1–14 (2021)

    MathSciNet  MATH  Google Scholar 

  31. M. Ghasemian, Z.N. Ashrafi, M. Goharkhah, M. Ashjaee, Heat transfer characteristics of \(Fe_3 O_4\) ferrofluid flowing in a mini channel under constant and alternating magnetic fields. J. Magn. Magn. Mater. 381, 158–167 (2015)

    Article  ADS  Google Scholar 

  32. F. Saba, N. Ahmed, U. Khan, S.T. Mohyud-Din, A novel coupling of (\(CNT\)-\(Fe_3 O_4\)/\(H_2 O\)) hybrid nanofluid for improvements in heat transfer for flow in an asymmetric channel with dilating/squeezing walls. Int. J. Heat Mass Trans. 136, 186–195 (2019)

    Article  Google Scholar 

  33. S. Saranya, L. Baranyi, Q.M. Al-Mdallal, Free convection flow of hybrid ferrofluid past a heated spinning cone. Therm. Sci. Eng. Prog. 32, 1–14 (2022)

    Google Scholar 

  34. R. Ningthoujam, R. Vatsa, A. Kumar, B. Pandey, S. Banerjee, A. Tyagi, Functionalized magnetic nanoparticles: concepts, synthesis and application in cancer hyperthermia. Funct. Mater. Prep. Process. Appl. 230-260 (2012)

  35. R. Zhang, L. Sun, Z. Wang, W. Hao, E. Cao, Y. Zhang, Dielectric and magnetic properties of \(Co Fe_2 O_4\) prepared by sol-gel auto-combustion method. Mater. Res. Bull. 98, 133–138 (2018)

    Article  Google Scholar 

  36. M. Sheikholeslami, D.D. Ganji, Ferrohydrodynamic and magnetohydrodynamic effects on ferrofluid flow and convective heat transfer. Energy 75, 400–410 (2014)

    Article  Google Scholar 

  37. T. Sharma, R. Kumar, K.R. Pardasani, K. Vajravelu, Linear stability analysis of asymmetrically heated hybrid nanofluid with variable viscosity and thermal conductivity. Eur. Phys. J. Plus 137(12), 1–20 (2022)

    Article  Google Scholar 

  38. C. Canuto, M.Y. Hussaini, A. Quarteroni, A. Thomas Jr., Spectral methods in fluid dynamics (Springer, 2012)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the constructive suggestions received from the learned Reviewers which led to definite improvement in the paper. The first author acknowledges the support of Central University of Himachal Pradesh for providing all the necessary sources to conduct the research.

Author information

Authors and Affiliations

  1. Srinivasa Ramanujan Department of Mathematics, Central University of Himachal Pradesh, Shahpur Campus, Shahpur, 176206, India

    Tanya Sharma & Rakesh Kumar

  2. Department of Mathematics, Vijayanagara Sri Krishnadevaraya University, Ballari, Karnataka, 583105, India

    Hanumesh Vaidya

  3. School of Mechanical Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul, 04763, Republic of Korea

    C. S. K. Raju

  4. Department of Mathematics, University of Central Florida, Orlando, FL, 32186, USA

    Kuppalapalle Vajravelu

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  1. Tanya Sharma
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Correspondence to Rakesh Kumar.

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Appendix

Appendix

$$\begin{aligned}&A_1=\dfrac{\nu_{\text{bf}}}{\nu_{\text{hnf}}}\dfrac{Re}{\varepsilon(\eta)}, \quad &n_{0,j}&=A_1\left(g_j'''-A_2A_3g_j''\right),\\ &A_2=\dfrac{\varepsilon_0c_1c_2D_p}{\varepsilon(\eta)},\quad &n_{1,j}&=-A_1\left(2g_j''-3A_2A_3g_j'\right),\\ &A_3=\exp\left(c_2D_p(\eta-1)\right),\quad &n_{2,j}&=-A_1\left(g_j'+2A_2A_3g_j\right)\\ &A_4=\dfrac{150\varepsilon_0c_1c_2(\varepsilon(\eta)-1)D_p^3}{\varepsilon(\eta)^3}, \quad &n_{3,j}&=2A_1g_j,\\ &A_5=\dfrac{\sigma_{\text{hnf}}}{\sigma_{\text{bf}}}\dfrac{\mu_{\text{bf}}}{\mu_{\text{hnf}}} \varepsilon(\eta)Ha,\quad &p_{0,j}&=2A_9f_j,\\ &A_6=\dfrac{150(1-\varepsilon(\eta))^2D_p^2}{\varepsilon(\eta)^2}, \quad &p_{1,j}&=A_{10}f_j,\\ &A_7=\dfrac{\mu_{\text{bf}}}{\mu_{\text{hnf}}} \dfrac{\varepsilon_0c_1c_2MnD_p}{(\eta+d^*)^6},\quad &q_{0,j}&=2A_9(\theta_j+\delta_T)+A_{10}\theta_j',\\ &A_8=\dfrac{\mu_{\text{bf}}}{\mu_{\text{hnf}}}Mn,\quad &H_{1,j}&=A_1(4g_jg_j'-f_j'f_j''+f_jf_j'''-\\ &A_9=\dfrac{k_{\text{bf}}}{k_{\text{hnf}}}\dfrac{MnEcPr}{(\eta+d^*)^5},\quad {} {}&&A_2A_3(f_jf_j''-f_j^{\prime 2}))+2A_3A_7(\delta_C-\delta_T)\\ &A_{10}=\dfrac{\alpha_{\text{bf}}}{\alpha_{\text{hnf}}}RePr,\quad &H_{2,j}&=-A_1((2f_j'g_j''-2f_j''g_j+f_j''g_j'-\\ &a_{0,j}=-A_1\left(A_2A_3f_j''-f_j'''\right),\quad {}&&f_jg_j''')-A_2(3f_j'g_j'-2f_j''g_j-f_jg_j''))\\ &a_{1,j}=-A_3\left(A_4+A_2A_5\right)-A_1\left(f_j''-2A_2A_3f_j'\right),\quad &H_{3,j}&=A_{10}f_j\theta_j'+2A_9f_j\theta_j.\\ & a_{2,j}=-A_5-A_6-A_1\left(A_2A_3f_j+f_j'\right),\\ & a_{3,j}=A_1f_j,\\ &b_{0,j}=4A_1g_j',\\ &b_{1,j}=4A_1g_j,\\&c_{0,j}=2A_3A_7,\\&c_{1,j}=2A_2A_8,\\&m_{0,j}=-2A_1\left(A_2A_3f_j''-f_j'''\right),\\&m_{1,j}=-A_3(A_4+A_2A_5)-A_1\left(f_j''-3A_2A_3f_j'\right),\\ &m_{2,j}=-A_5-A_6-A_1\left(A_2A_3f_j+f_j'\right),\\ &m_{3,j}=A_1f_j,\end{aligned}$$

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Sharma, T., Kumar, R., Vaidya, H. et al. Numerical investigation of the hybrid ferrofluid flow in a heterogeneous porous channel with convectively heated and quadratically stretchable walls. Eur. Phys. J. Plus 138, 745 (2023). https://doi.org/10.1140/epjp/s13360-023-04371-w

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