# Computational Analysis of Third-Grade Liquid Flow with Cross Diffusion Effects: Application to Entropy Modeling

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## Abstract

The key goal of this current study is to analyze the entropy generation with cross diffusion effects. The third-grade type non-Newtonian fluid model is used in this study. The current flow problem is modelled with stretching plate. Modified Fourier heat flux is replaced the classical heat flux. The appropriate transformation is availed to convert the basic boundary layers equations into ODEs and then verified by homotopy algorithm. The consequences of various physical quantities on temperature, velocity, entropy and concentration profile are illustrated graphically.

## Keywords

Third grade fluid Linear stretching sheet Homotopy Analysis Method (HAM) Soret and Dufour effects Entropy generation## 1 Introduction

Third grade fluid is one of the notable sub kinds of non-Newtonian fluids. The non-Newtonian fluid flow due to the stretching surface is the important area of research due to its broad applications in many industrial and production domains such as, rolling of polymer films, extrusion of metallic sheets, etc. The study on 2nd grade fluid which passed through the stretching sheet is numerically discussed including the variations in thermophysical properties like thermal conductivity, viscosity [1]. It is shown that Eckert number increases the heat transport rate. Hydromagnetic mixed convective heat transfer of 3rd-grade fluid with gyrotactic microorganism is examined [2]. Unsteady flow of power law fluid with uniform velocity is evaluated [3]. With the consideration of heat source and heat sink of MHD flow over a oscillatory stretching sheet is numerically studied [4]. With the impact of chemical reaction, the fourth grade fluid through porous plate of MHD radiative fluid is investigated [5]. In addition to MHD nanofluid, the electrically conductive fluid that of second grade with suction parameter is developed [6]. For the application of bio magnetic the third grade fluid is correlated numerically [7]. The modified Fourier heat flux model for the study of carreau fluid is explored numerically [8]. The various features and applications of non-Newtonian fluids are studied in ref’s [9, 10, 11, 12, 13, 14, 15].

There are several techniques available to solve nonlinear problems. The homotopy analysis method (HAM) is initially constructed by Liao [16] in 1992. Moreover, he altered with a non-zero auxiliary parameter [17] . This parameter shows the way to calculate the convergence rate. It also offers great independence to choose the base functions of the solutions. A few more studies about this technique was seen in previous works [18, 19].

Inspired by the above literature surveys, we are constructing a steady 3rd-grade liquid flow with considering radiation, and convective heating effects. Dufour and Soret effects are examined. The system of entropy is discussed briefly for various parameters.

## 2 Mathematical Formation

*u*&

*v*(\(=\)velocity components along the

*x*&

*y*-direction), \(\mu \) (\(=\)kinematic viscosity), (\(\alpha _{1}^*\),\(\alpha _{2}^*\) & \(\beta _1^*\))(\(=\)material parameters), \(\rho \) (\(=\)fluid density), \((\beta _T\), \(\beta _C\)) (\(=\)coefficient of thermal and concentration expansions), \(c_p\) (\(=\)specific heat), \(c_s\) (\(=\)concentration susceptibility),

*Q*(\(=\)heat capacity of ordinary fluid), \(q_r\) (\(=\) radiative heat flux),

*C*(\(=\)concentration), \(C_w\) (\(=\) fluid wall concentration), \(D_m\) (\(=\)mass diffusion coefficient), \(k_m\) (\(=\)first order chemical reaction parameter).

## 3 Analytical Procedure and Convergence Study

*h*values as \(h_F=h_{\theta } =h_{\phi }=-0.7\) (Table 1).

Displays the convergence solutions of HAM in order of approximation when \(Pr=0.9,\alpha _1=0.1,Sc=0.9,D_F=0.5,Re=0.1,\beta =0.1,\alpha _2=0.1,\gamma =0.1, Bi=0.5,Rd=0.3,\lambda _1=0.2,N=0.1,Sr=0.3,Q_H=-0.3,h=-0.7,C_r=0.1.\)

Order | \(-F''(0)\) | \(\theta '(0)\) | \(-\phi '(0)\) |
---|---|---|---|

1 | 0.72116 | 0.23913 | 0.72777 |

5 | 0.67605 | 0.23555 | 0.63669 |

10 | 0.67817 | 0.23648 | 0.63945 |

15 | 0.67803 | 0.23643 | 0.63897 |

20 | 0.67801 | 0.23644 | 0.63901 |

25 | 0.67803 | 0.23644 | 0.63903 |

30 | 0.67803 | 0.23644 | 0.63903 |

40 | 0.67803 | 0.23644 | 0.63903 |

50 | 0.67803 | 0.23644 | 0.63903 |

## 4 Entropy Optimization

## 5 Validation

Comparison in absence of \(D_F=0,SR=0,\omega =0,Rd=0,C_r=0, \lambda _1=0,N=0,Q_H=0.\)

Order | \( -f''(0)\) | \(-\theta '(0)\) | \(-\phi '(0)\) | |||
---|---|---|---|---|---|---|

Ref. [10] | Current | Ref. [10] | Current | Ref. [10] | Current | |

1 | 0.81450 | 0.81450 | 0.72778 | 0.72778 | 0.72778 | 0.72778 |

5 | 0.81221 | 0.81221 | 0.58070 | 0.58070 | 0.64933 | 0.64933 |

8 | 0.81235 | 0.81235 | 0.57779 | 0.57779 | 0.64835 | 0.64835 |

14 | 0.81235 | 0.81235 | 0.57871 | 0.57871 | 0.64873 | 0.64873 |

17 | 0.81235 | 0.81235 | 0.57878 | 0.57878 | 0.64873 | 0.64873 |

25 | 0.81235 | 0.81235 | 0.57878 | 0.57878 | 0.64873 | 0.64873 |

30 | 0.81235 | 0.81235 | 0.57878 | 0.57878 | 0.64873 | 0.64873 |

35 | 0.81235 | 0.81235 | 0.57878 | 0.57878 | 0.64873 | 0.64873 |

## 6 Results and Discussion

In this section, we examine the impacts of physical quantities on temperature(\(\theta (\eta )\)), velocity (\(F^{'}(\eta )\)), entropy(\(E_G\)) and concentration profiles(\(\varPhi (\eta )\)) with the fixed values \(Pr=0.9\), \(Sc=0.9\), \(Re=0.14\), \(\alpha _{1}=0.1\), \(\alpha _2=0.1\), \(\beta =0.1\), \(\gamma =0.1\), \(D_F=0.5\), \(Sr=0.3\), \(Bi=0.5\), \(Rd=0.3\), \(\lambda _1=0.2\), \(N=0.1\),\(Q_H=-0.3\), \(h=-0.7\) and \(Cr=0.1\).

Figure 1 describes the effect of velocity profile (\(F^{'}(\eta )\)) on material fluid parameter \((\alpha _1)\) and mixed convection parameter \((\lambda _1)\). From Fig. 1(a & b), we have seen an increase in \(\alpha _1\) and \(\lambda _1\) the velocity profile (\(F^{'}(\eta )\)) rises. Figure 2 revels the temperature profile (\(\theta (\eta )\)) for different parameters. In Fig. 2(a, b, d), we note that, the (\(\theta (\eta )\)) enhances for the augmentation in *Rd*, *Bi* and \(D_F\) and it diminishes for the higher values of Prandtl number *Pr* as shown in Fig. 2c. Figure 3 depicts the different effects of *Cr*, *Sr*, and *Sc* on concentration profile. From Fig. 3(a) and Fig. 3(c) concentration profile (\(\varPhi (\eta )\)) is inversely proportional to the higher *Cr* and *Sc*. Whereas in Fig. 3(b), it is found that (\(\varPhi (\eta )\)) increases with augments in *Sr*. Figures 4(a–c) shows the effects of \(D_F\), *Sr* and *Rd* on \(E_G\) (entropy generation profile). From these plots we obtain that the system of entropy enhances for the larger values of \(D_F\), *Sr* and *Rd*.

From Fig. 5(a) we note that the Nusselt number \((Nu_x)\) decreases with increases in \(D_F\). Also, in Fig. 5(b) and Fig. 5(c), it is noted that \(Nu_x\) decrease with upsurge in *Rd* and \(\gamma \). By increasing the \(D_F\), fluid resits to move the hotter side of the sheet that subsequently \(Nu_x\) decreases. In addition, it is noted that as we increase the *Rd* and \(\gamma \) the \(Nu_x\) enhances. Figure 6 exposes the mass transfer rate for the combined parameters *Sr* and *Cr*. We noted that decreasing trend in mass transfer for larger *Sr* and mass transfer rate enhance for *Cr*.

## 7 Conclusion

- 1
Higher range of mixed convection parameter\((\lambda _1)\) and fluid parameter\((\alpha _1)\) intensifying the velocity profile.

- 2
Entropy of the system enhances with radiation, Sored and Dufour numbers.

- 3
Mass transfer rate rises with chemical reaction and reduces with Soret number.

## References

- 1.Akinbobola, T.E., Okoya, S.S.: The flow of second grade fluid over a stretching sheet with variable thermal conductivity and viscosity in the presence of heat source/sink. J. Niger. Math. Soc.
**34**(3), 331–342 (2015). https://doi.org/10.1016/j.jnnms.2015.10.002MathSciNetCrossRefzbMATHGoogle Scholar - 2.Alzahrani, E.O., Shah, Z., Dawar, A., Malebary, S.J.: Hydromagnetic mixed convective third grade nanomaterial containing gyrotactic microorganisms toward a horizontal stretched surface. Alexandria Eng. J.
**58**(4), 1421–1429 (2019). https://doi.org/10.1016/j.aej.2019.11.013CrossRefGoogle Scholar - 3.Ahmed, J., Mahmood, T., Iqbal, Z., Shahzad, A., Ali, R.: NU SC. J. Mol. Liq. (2016). https://doi.org/10.1016/j.molliq.2016.06.022CrossRefGoogle Scholar
- 4.Ali, N., Ullah, S., Sajid, M., Abbas, Z.: MHD flow and heat transfer of couple stress fluid over an oscillatory stretching sheet with heat source/sink in porous medium. Alexandria Eng. J. (2016). https://doi.org/10.1016/j.aej.2016.02.018CrossRefGoogle Scholar
- 5.Arifuzzaman, S.M., Khan, S., Al-mamun, A., Rezae-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. (2018). https://doi.org/10.1016/j.jksus.2018.12.009
- 6.Cortell, R.: Flow and heat transfer of an electrically conducting fluid of second grade over a stretching sheet subject to suction and to a transverse magnetic field. Int. J. Heat Mass Transf.
**49**, 1851–1856 (2006). https://doi.org/10.1016/j.ijheatmasstransfer.2005.11.013CrossRefzbMATHGoogle Scholar - 7.Ghosh, S.K.: Unsteady non-Newtonian fluid flow and heat transfer: a bio-magnetic application Sushil Kumar Ghosh. J. Magn. Magn. Mater. (2017). https://doi.org/10.1016/j.jmmm.2017.07.050CrossRefGoogle Scholar
- 8.Nazir, U., Saleem, S., Nawaz, M., Sadiq, M.A., Alderremy, A.A.: Study of transport phenomenon in Carreau fluid using Cattaneo-Christov heat flux model with temperature dependent diffusion coefficients. Phys. A Stat. Mech. its Appl. 123921 (2020). https://doi.org/10.1016/j.physa.2019.123921
- 9.Waqas, M., Hayat, T., Farooq, M., Shehzad, S.A., Alsaedi, A.: Cattaneo-Christov heat flux model for flow of variable thermal conductivity generalized Burgers fluid. J. Mol. Liq.
**220**, 642–648 (2016). https://doi.org/10.1016/j.molliq.2016.04.086CrossRefGoogle Scholar - 10.Imtiaz, M., Alsaedi, A., Shafiq, A., Hayat, T.: Impact of chemical reaction on third grade fluid flow with Cattaneo-Christov heat flux. J. Mol. Liq.
**229**, 501–507 (2017). https://doi.org/10.1016/j.molliq.2016.12.103CrossRefGoogle Scholar - 11.Bhuvaneswari, M., Eswaramoorthi, S., Sivasankaran, S., Rajan, S., Saleh Alshomrani, A.: Effects of viscous dissipation and convective heating on convection flow of a second-grade liquid over a stretching surface: an analytical and numerical study. Sci. Iran. B
**26**(3), 1350–1357 (2019)Google Scholar - 12.Loganathan, K., Rajan, S.: An entropy approach of Williamson nanofluid flow with Joule heating and zero nanoparticle mass flux. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-09414-3
- 13.Eswaramoorthi, S., Sivasankaran, S., Bhuvaneswari, M., Rajan, S.: Soret and Dufour effects on viscoelastic boundary layer flow over a stretchy surface with convective boundary condition with radiation and chemical reaction. Sci. Iran. B.
**23**(6), 2575–2586 (2016)Google Scholar - 14.Loganathan. K, Mohana. K, Mohanraj, M., Sakthivel, P., Rajan, S.: Impact of 3rd-grade nanofluid flow across a convective surface in the presence of inclined Lorentz force: an approach to entropy optimization. J. Therm. Anal. Calorim. (2020), https://doi.org/10.1007/s10973-020-09751-3
- 15.Elanchezhian, E., Nirmalkumar, R., Balamurugan, M., Mohana, K., Prabu, K.M., Viloria, A.: Heat and mass transmission of an Oldroyd-B nanofluid flow through a stratified medium with swimming of motile gyrotactic microorganisms and nanoparticles. J. Therm. Anal. Calorim. (2020). https://doi.org/10.1007/s10973-020-09847-w
- 16.Liao, S., Tan, Y.A.: General approach to obtain series solutions of nonlinear differential. Stud. Appl. Math.
**119**(4), 297–354 (2007)MathSciNetCrossRefGoogle Scholar - 17.Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Non-Linear Mech.
**34**, 759–778 (1999)CrossRefGoogle Scholar - 18.Loganathan, K., Sivasankaran, S., Bhuvaneswari, M., Rajan, S.: Second-order slip, cross-diffusion and chemical reaction effects on magneto-convection of Oldroyd-B liquid using Cattaneo-Christovheat flux with convective heating. J. Therm. Anal. Calorim.
**136**(1), 401–409 (2019)CrossRefGoogle Scholar - 19.Loganathan, K., Prabu, K.M., Elanchezhian, E., Nirmalkumar, R., Manimekalai, K.: Computational analysis of thermally stratified mixed convective non-Newtonian fluid flow with radiation and chemical reaction impacts. J. Phys: Conf. Ser.
**1432**, 012048 (2020)Google Scholar