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Entropy generation analysis of natural convective radiative second grade nanofluid flow between parallel plates in a porous medium

  • K. Ramesh
  • O. OjjelaEmail author
Article
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Abstract

The present article explores the entropy generation of radiating viscoelastic second grade nanofluid in a porous channel confined between two parallel plates. The boundaries of the plates are maintained at distinct temperatures and concentrations while the fluid is being sucked and injected periodically through upper and lower plates. The buoyancy forces, thermophoresis and Brownian motion are also considered due to the temperature and concentration differences across the channel. The system of governing partial differential equations has been transferred into a system of ordinary differential equations (ODEs) by appropriate similarity relations, and a shooting method with the fourth-order Runge-Kutta scheme is used for the solutions. The results are analyzed in detail for dimensionless velocity components. The temperature, concentration distributions, the entropy generation number, and the Bejan number corresponding to various fluid and geometric parameters are shown graphically. The skin friction, heat and mass transfer rates are presented in the form of tables. It is noticed that the temperature profile of the fluid is enhanced with the Brownian motion, whereas the concentration profile of the fluid is decreased with the thermophoresis parameter, and the entropy and Bejan numbers exhibit the opposite trend for the suction and injection ratio.

Key words

nanofluid thermophoresis Brownian motion shooting method thermal radiation 

Nomenclature

Nomenclature

C

concentration

\({C_1}{e^{{\text{i}\omega \text{t}}}}\)

concentration at the lower plate

\({C_2}{e^{i\omega t}}\)

concentration at the upper plate

T*

dimensionless temperature \(\frac{{T - {T_1}{e^{i\omega t}}}}{{({T_2} - {T_1}){e^{i\omega t}}}}\)

\({T_1}{e^{i\omega t}}\)

temperature at the lower plate

\({T_2}{e^{i\omega t}}\)

temperature at the upper plate

C*

dimensionless concentration, \(\frac{{C - {C_1}{e^{i\omega t}}}}{{({C_2} - {C_1}){e^{i\omega t}}}}\)

\({\dot n_A}\)

mass transfer rate

v

kinematic viscosity

h

distance between parallel plates

Ec

Eckert number \(\frac{{\mu {V_2}}}{{\rho hC({T_1} - {T_2})}}\)

p

fluid pressure

Da

Darcy parameter

k1

permeability parameter

Re

Reynolds number, \(\frac{{\rho {V_2h}}}{\mu }\)

Ha

Hartmann number

R

ideal gas constant

D

molecular diffusion coefficient

Sh

Sherwood number, \(\frac{{{{\dot n}_A}}}{{hv({C_1} - {C_2})}}\)

Pr

Prandtl number, \(\frac{{\mu c}}{k}\)

Grs

solutal Grashof number, \({{\rho g{\beta _C}({C_2} - {C_1}){h^2}} \over {\mu {V_2}}}\)

Grt

thermal Grashof number, \({{\rho g{\beta _T}({T_2} - {T_1}){h^2}} \over {\mu {V_2}}}\)

Rd

radiation parameter, \(\frac{{16\sigma T_1^3}}{{3{K_3}K}}\)

Nb

Brownian motion parameter, \(\frac{{(\rho c)p{D_B}({C_2} - {C_1})}}{{{{(\rho c)}_f}{a_1}}}\)

Nt

thermophoresis parameter, \(\frac{{(\rho c)p{D_T}({C_2} - {C_1})}}{{{{(\rho c)}_f}{a_1}}}\)

Br

Brinkman number, Pr · Ec

Sc

Schmidt number, \(\frac{v}{{{D_1}}}\)

\({V_1}{e^{i\omega t}}\)

injection velocity at the lower plate

\({V_2}{e^{i\omega t}}\)

suction velocity at the upper plate

a

suction-injection ratio

T

temperature

t

time

i, j

unit vectors along X-and Y-directions, respectively

u

X-direction velocity component

v

Y -direction velocity component

Greek Letters

β1

second grade fluid parameter, \(\frac{{{a_1}{v_2}}}{{\mu h}}\)

λ

nondimensional coordinate, \(\frac{y}{h}\)

Ω

nondimensional temperature difference parameter, \(\frac{{\Delta T}}{{{T_0}}}\)

α

nondimensional concentration difference parameter, \(\frac{{\Delta C}}{{{C_0}}}\)

α1

thermal diffusivity, \(\frac{k}{{\rho c}}\)

λ1

diffusive constant parameter, \(\frac{{RD\Delta C}}{K}\)

ζ

dimensionless axial variable, \((\frac{{{U_0}}}{{a{V_2}}} - \frac{x}{h})\)

ψ

nondimensional frequency parameter, ωt

Chinese Library Classification

O357.5+

2010 Mathematics Subject Classification

65L06 74A15 76A05 76R10 82C35 76S05 

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References

  1. [1]
    RIVLIN, R. S. and ERICKSEN, J. L. Stress deformation relations for isotropic materials. Journal of Rational Mechanics and Analysis, 4, 323–425 (1955)MathSciNetzbMATHGoogle Scholar
  2. [2]
    RAJAGOPAL, K. R. On the creeping flow of the second–order fluid. Journal of Non–Newtonian Fluid Mechanics, 15(2), 239–246 (1984)zbMATHGoogle Scholar
  3. [3]
    MASSOUDI, M. and PHUOC, T. X. Fully developed flow of a modified second grade fluid with temperature dependent viscosity. Acta Mechanica, 150(1), 23–37 (2001)zbMATHGoogle Scholar
  4. [4]
    DONALD–ARIEL, P. On exact solutions of flow problems of a second grade fluid through two parallel porous walls. International Journal of Engineering Science, 40(8), 913–941 (2002)MathSciNetzbMATHGoogle Scholar
  5. [5]
    EMIN–ERDOĞAN, M. and ERDEM–IMRAK, C. On some unsteady flows of a non–Newtonian fluid. Applied Mathematical Modelling, 31(2), 170–180 (2007)zbMATHGoogle Scholar
  6. [6]
    HAYAT, T., IQBAL, Z., and MUSTAFA, M. Flow of a second grade fluid over a stretching surface with Newtonian heating. Journal of Mechanics, 28(1), 209–216 (2012)Google Scholar
  7. [7]
    SHRESTHA, G. M. Singular perturbation problems of laminar flow in a uniformly porous channel in the presence of a transverse magnetic field. The Quarterly Journal of Mechanics and Applied Mathematics, 20(2), 233–246 (1967)zbMATHGoogle Scholar
  8. [8]
    RAFTARI, B., PARVANEH, F., and VAJRAVELU, K. Homotopy analysis of the magneto hydro–dynamic flow and heat transfer of a second grade fluid in a porous channel. Energy, 59, 625–632 (2013)Google Scholar
  9. [9]
    RAMZAN, M. and BILAL, M. Time dependent MHD nano–second grade fluid flow induced by permeable vertical sheet with mixed convection and thermal radiation. PLoS One, 10(5), e0124929 (2015)Google Scholar
  10. [10]
    LABROPULU, F., XU, X., and CHINICHIAN, M. Unsteady stagnation point flow of a non–Newtonian second–grade fluid. International Journal of Mathematics and Mathematical Sciences, 60, 3797–3807 (2003)MathSciNetzbMATHGoogle Scholar
  11. [11]
    HAYAT, T., KHAN, M. W. A., ALSAEDI, A., and KHAN, M. I. Squeezing flow of second grade liquid subject to non–Fourier heat flux and heat generation/absorption. Colloid and Polymer Science, 295(6), 967–975 (2017)Google Scholar
  12. [12]
    CHENG, C. Y. Fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel with asymmetric wall temperatures and concentrations. International Com–munications in Heat and Mass Transfer, 33(5), 627–635 (2006)Google Scholar
  13. [13]
    ABDULAZIZ, O. and HASHIM, I. Fully developed free convection heat and mass transfer of a micropolar fluid between porous vertical plates. Numerical Heat Transfer, 55(3), 270–288 (2009)Google Scholar
  14. [14]
    CHAMKHA, A. J., MOHAMED, R. A., and AHMED, S. E. Unsteady MHD natural convection from a heated vertical porous plate in a micropolar fluid with Joule heating, chemical reaction and radiation effects. Meccanica, 46(2), 399–411 (2011)MathSciNetzbMATHGoogle Scholar
  15. [15]
    SINGH, A. K. and GORLA, R. S. R. Free convection heat and mass transfer with Hall current, Joule heating and thermal diffusion. Heat and Mass Transfer, 45(11), 1341–1349 (2009)Google Scholar
  16. [16]
    HAYAT, T., MUHAMMAD, T., ALSAEDI, A., and ALHUTHALI, M. S. Magneto hydrodynamic three–dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation. Journal of Magnetism and Magnetic Materials, 385, 222–229 (2015)Google Scholar
  17. [17]
    HAYAT, T., WAQAS, M., SHEHZAD, S. A., and ALSAEDI, A. Chemically reactive flow of third grade fluid by an exponentially convected stretching sheet. Journal of Molecular Liquids, 223, 853–860 (2016)Google Scholar
  18. [18]
    AHMED, N., KHAN, U., and MOHYUD–DIN, S. T. Influence of nonlinear thermal radiation on the viscous flow through a deformable asymmetric porous channel: a numerical study. Journal of Molecular Liquids, 225, 167–173 (2017)Google Scholar
  19. [19]
    SUDARSANA–REDDY, P., CHAMKHA, A. J., and AL–MUDHAF, A. MHD heat and mass transfer flow of a nanofluid over an inclined vertical porous plate with radiation and heat generation/absorption. Advanced Powder Technology, 28(3), 1008–1017 (2017)Google Scholar
  20. [20]
    DOGONCHI, A. S., ALIZADEH, M., and GANJI, D. D. Investigation of MHD Go–water nanofluid flow and heat transfer in a porous channel in the presence of thermal radiation effect. Advanced Powder Technology, 28(7), 1815–1825 (2017)Google Scholar
  21. [21]
    EEGUNJOBI, A. S., MAKINDE, O. D., and JANGILI, S. Unsteady MHD chemically reacting and radiating mixed convection slip flow past a stretching surface in a porous medium. Defect and Diffusion Forum, 377, 200–210 (2017)Google Scholar
  22. [22]
    BHATTI, M. M., ABBAS, M. A., and RASHIDI, M. M. A robust numerical method for solving stagnation point flow over a permeable shrinking sheet under the influence of MHD. Applied Mathematics and Computation, 316, 381–389 (2018)MathSciNetGoogle Scholar
  23. [23]
    ZHU, J., ZHENG, L., ZHENG, L., and ZHANG, X. Second–order slip MHD flow and heat transfer of nanofluids with thermal radiation and chemical reaction. Applied Mathematics and Mechanics (English Edition), 36(9), 1131–1146 (2015) https://doi.org/10.1007/s10483-015-1977-6 MathSciNetzbMATHGoogle Scholar
  24. [24]
    ABOLBASHARI, M. H., FREIDOONIMEHR, N., NAZARI, F., and RASHIDI, M. M. Entropy analysis for an unsteady MHD flow past a stretching permeable surface in nano–fluid. Powder Technology, 267, 256–267 (2014)Google Scholar
  25. [25]
    RASHIDI, M. M., MAHMUD, S., FREIDOONIMEHR, N., and ROSTAMI, B. Analysis of en–tropy generation in an MHD flow over a rotating porous disk with variable physical properties. International Journal of Exergy, 16(4), 481–503 (2015)Google Scholar
  26. [26]
    MAHMOODI, M. and KANDELOUSI, S. Effects of thermophoresis and Brownian motion on nanofluid heat transfer and entropy generation. Journal of Molecular Liquids, 211, 15–24 (2015)Google Scholar
  27. [27]
    SRINIVAS, J., RAMANA–MURTHY, J. V., and ANWAR–B´ EG, O. Entropy generation analysis of radiative heat transfer effects on channel flow of two immiscible couple stress fluids. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 9(6), 2191–2202 (2017)Google Scholar
  28. [28]
    MAKINDE, O. D. Entropy analysis for MHD boundary layer flow and heat transfer over a flat plate with a convective surface boundary condition. International Journal of Exergy, 10(2), 142–154 (2012)MathSciNetGoogle Scholar
  29. [29]
    NOGHREHABADI, A., SAFFARIAN, M., POURRAJAB, R., and GHALAMBAZ, M. Entropy analysis for nanofluid flows over a stretching sheet in the presence of heat generation/absorption and partial slip. Journal of Mechanical Science and Technology, 27(3), 927–937 (2013)Google Scholar
  30. [30]
    SHIT, G. C., HALDAR, R., and MANDAL, S. Entropy generation on MHD flow and convective heat transfer in a porous medium of exponentially stretching surface saturated by nanofluids. Advanced Powder Technology, 28(6), 1519–1530 (2017)Google Scholar
  31. [31]
    ABOLBASHARI, M. H., FREIDOONIMEHR, N., NAZARI, F., and RASHIDI, M. M. Analyt–ical modeling of entropy generation for Casson nano–fluid flow induced by a stretching surface. Advanced Powder Technology, 26(2), 542–552 (2015)Google Scholar
  32. [32]
    JBARA, A., SLIMI, K., and MHIMID, A. Entropy generation for unsteady natural convection and thermal radiation inside a porous enclosure. International Journal of Exergy, 12(4), 522–551 (2013)Google Scholar
  33. [33]
    DAS, S., CHAKRABORTY, S., JANA, R. N., and MAKINDE, O. D. Entropy analysis of unsteady magneto–nanofluid flow past accelerating stretching sheet with convective boundary condition. Applied Mathematics and Mechanics (English Edition), 36(12), 1593–1610 (2015) https://doi.org/10.1007/s10483-015-2003-6 MathSciNetGoogle Scholar
  34. [34]
    BHATTI, M. M., ABBAS, T., RASHIDI, M. M., and ALI, M. E. S. Numerical simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet. En–tropy, 18(6), 200 (2016)MathSciNetGoogle Scholar
  35. [35]
    BHATTI, M. M., ABBAS, T., RASHIDI, M. M., ALI, M. E. S., and YANG, Z. Entropy generation on MHD Eyring–Powell nanofluid through a permeable stretching surface. Entropy, 18(6), 224 (2016)MathSciNetGoogle Scholar
  36. [36]
    BHATTI, M. M. and RASHIDI, M. M. Numerical simulation of entropy generation on MHD nanofluid towards a stagnation point flow over a stretching surface. International Journal of Applied and Computational Mathematics, 3(3), 2275–2289 (2017)MathSciNetzbMATHGoogle Scholar
  37. [37]
    ABBAS, M. A., BAI, Y., RASHIDI, M. M., and BHATTI, M. M. Analysis of entropy generation in the flow of peristaltic nanofluids in channels with compliant walls. Entropy, 18(3), 90 (2016)MathSciNetGoogle Scholar
  38. [38]
    BHATTI, M. M., ABBAS, T., and RASHIDI, M. M. Entropy generation as a practical tool of optimisation for non–Newtonian nanofluid flow through a permeable stretching surface using SLM. Journal of Computational Design and Engineering, 4(1), 21–28 (2017)Google Scholar
  39. [39]
    BHATTI, M. M., RASHIDI, M. M., and POP, I. Entropy generation with nonlinear heat and mass transfer on MHD boundary layer over a moving surface using SLM. Nonlinear Engineering, 6(1), 43–52 (2017)Google Scholar
  40. [40]
    BHATTI, M. M., SHEIKHOLESLAMI, M., and ZEESHAN, A. Entropy analysis on electro–kinetically modulated peristaltic propulsion of magnetized nanofluid flow through a microchannel. Entropy, 19(9), 481 (2017)Google Scholar
  41. [41]
    AFRIDI, M. I., QASIM, M., and MAKINDE, O. D. Second law analysis of boundary layer flow with variable fluid properties. Journal of Heat Transfer, 39(10), 104505 (2017)Google Scholar
  42. [42]
    EEGUNJOBI, A. S. and MAKINDE, O. D. Irreversibility analysis of hydromagnetic flow of couple stress fluid with radiative heat in a channel filled with a porous medium. Results in Physics, 7, 459–469 (2017)Google Scholar
  43. [43]
    DAS, S., JANA, R. N., and MAKINDE, O. D. MHD flow o. Cu–Al2O3/water hybrid nanofluid in porous channel: analysis of entropy generation. Defect and Diffusion Forum, 377, 42–61 (2017)Google Scholar
  44. [44]
    EEGUNJOBI, A. S. and MAKINDE, O. D. MHD mixed convection slip flow of radiatin. Casson fluid with entropy generation in a channel filled with porous media. Defect and Diffusion Forum, 374, 47–66 (2017)Google Scholar
  45. [45]
    ALAM, M., KHAN, M., HAKIM, A., and MAKINDE, O. D. Magneto–nanofluid dynamics in convergent–divergent channel and its inherent irreversibility. Defect and Diffusion Forum, 377, 95–110 (2017)Google Scholar
  46. [46]
    EEGUNJOBI, A. S. and MAKINDE, O. D. Inherent irreversibility in a variable viscosity Hart–mann flow through a rotating permeable channel with Hall effects. Defect and Diffusion Forum, 377, 180–188 (2017)Google Scholar
  47. [47]
    CHOI, S. U. S. and ESTMAN, J. A. Enhancing thermal conductivity of fluids with nanoparticles. ASME International Mechanical Engineering Congress and Exposition, 231, 99–106 (1995)Google Scholar
  48. [48]
    BUONGIORNO, J. Convective transport in nanofluids. Journal of Heat Transfer, 128(3), 240–250 (2006)Google Scholar
  49. [49]
    SHEIKHOLESLAMI, M., RASHIDI, M. M., AL–SAAD, D. M., FIROUZI, F., ROKNI, H. B., and DOMAIRRY, G. Steady nanofluid flow between parallel plates considering thermophoresis and Brownian effects. Journal of King Saud University—Science, 28(4), 380–389 (2016)Google Scholar
  50. [50]
    HAYAT, T., MUHAMMAD, T., QAYYUM, A., ALSAEDI, A., and MUSTAFA, M. On squeezing flow of nanofluid in the presence of magnetic field effects. Journal of Molecular Liquids, 213, 179–185 (2016)Google Scholar
  51. [51]
    RAMANA–REDDY, J. V., SUGUNAMMA, V., and SANDEEP, N. Thermophoresis and Brownian motion effects on unsteady MHD nano fluid flow over a slendering stretching surface with slip effects. Alexandria Engineering Journal, 57, 2465–2473 (2017)Google Scholar
  52. [52]
    GUHA, A. and SAMANTA, S. Effect of thermophoresis on the motion of aerosol particles in natural convective flow on horizontal plates. International Journal of Heat and Mass Transfer, 68, 42–50 (2014)Google Scholar
  53. [53]
    AWAD, F. G., AHAMED, S.M. S., SIBANDA, P., and KHUMALO,M. The effect of thermophoresis on unstead. Oldroyd–B nanofluid flow over stretching surface. PLoS One, 10(8), e0135914 (2015)Google Scholar
  54. [54]
    QAYYUM, S., HAYAT, T., ALSAEDI, A., and AHMAD, B. Magnetohydrodynamic (MHD) non–linear convective flow of Jeffrey nanofluid over a nonlinear stretching surface with variable thick–ness and chemical reaction. International Journal of Mechanical Sciences, 134, 306–314 (2017)Google Scholar
  55. [55]
    KIUSALAAS, J. Numerical Methods in Engineering with MATLAB, Cambridge University Press, New York (2005)zbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsDefence Institute of Advanced TechnologyPuneIndia

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