Applied Mathematics and Mechanics

, Volume 40, Issue 4, pp 481–498 | Cite as

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

  • K. Ramesh
  • O. OjjelaEmail author


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 





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

concentration at the lower plate

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

concentration at the upper plate


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


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

\({\dot n_A}\)

mass transfer rate


kinematic viscosity


distance between parallel plates


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


fluid pressure


Darcy parameter


permeability parameter


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


Hartmann number


ideal gas constant


molecular diffusion coefficient


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


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


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


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


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


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


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


Brinkman number, Pr · Ec


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


suction-injection ratio





i, j

unit vectors along X-and Y-directions, respectively


X-direction velocity component


Y-direction velocity component

Greek Letters


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


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


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


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


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


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


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


nondimensional frequency parameter, ωt

Chinese Library Classification


2010 Mathematics Subject Classification

65L06 74A15 76A05 76R10 82C35 76S05 


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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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