# Nonlinear convection flow of micropolar liquid: an application of improved Fourier’s expression

- 26 Downloads

## Abstract

Here, improved Fourier’s expression is utilized for heat transfer in nonlinear convection flow of micropolar liquid. The flow is generated due to a stretchable surface with variable thickness. Application of non-Fourier conduction phenomenon illustrates the characteristics of thermal relaxation factor. Temperature-dependent conductivity of liquid is also adopted. The set of partial differential equations governing the flow of micropolar liquid and heat transfer through non-Fourier heat conduction concept is established. The relevant transformations provide the highly nonlinear ordinary differential system. Homotopy theory is employed to acquire convergent solutions for nonlinear differential systems. Coefficient of skin friction is computed and examined for distinct embedded variables. Our presented analysis shows that temperature is decaying for larger thermal relaxation time.

### Keywords

Nonlinear convection flow Micropolar liquid Improved Fourier’s expression Variable thermal conductivity Stagnation point flow### List of symbols

*u*,*v*Velocity components

*x*,*y*Space coordinates

*U*_{w}(*x*)Stretching velocity

*U*_{e}(*x*)Free stream velocity

*U*_{0},*U*_{∞}Reference velocities

*N*Micro-rotation variable

*g*Gravitational accelerational

*β*_{1}Linear thermal expansion coefficient

*β*_{2}Nonlinear thermal expansion coefficient

*γ**Spin gradient viscosity

*j*Micro-inertia

*m*_{0}Boundary parameter

**q**Heat flux

*λ*Relaxation time of heat flux

*K*(*T*)Variable thermal conductivity

*K*_{∞}Thermal conductivity of ambient fluid

*μ*Dynamic viscosity

*ν*Kinematic viscosity

*ρ*Fluid density

*k*Vortex viscosity

*c*_{p}Specific heat

*b*Dimensional constant

*T*_{w}Variable temperature at the sheet

*T*_{∞}Variable temperature away from the sheet

*δ*Small parameter regarding the surface is sufficiently thin

*K*Micropolar parameter

*λ*Local buoyancy parameter

*Gr*_{x}Local Grashof number

*δ*Nonlinear convection

- Pr
Prandtl number

*γ*Thermal relaxation parameter

*α*Wall thickness parameter

*n*Power law index

- \(\in\)
Temperature dependent thermal conductivity parameter

*ψ*Stream function

*η*Dimensionless variable

*f*(*η*)Dimensionless velocity

*g*(*η*)Dimensionless micro-rotation velocity

*θ*(*η*)Dimensionless temperature

*C*_{f}Skin friction coefficient

*τ*_{w}Surface shear stress

- Re
_{x} Local Reynolds number

*f*_{0}(*η*),*g*_{0}(*η*)Initial approximations for velocity

*θ*_{0}(*η*)Initial approximations for temperature

*L*_{f},*L*_{g},*L*_{θ}Linear operators

*C*_{i}Arbitrary constants

- ℏ
_{f}, ℏ_{g}, ℏ_{θ} Auxiliary variables

### References

- 1.Eringen AC (1966) Theory of micropolar fluids. J Math Mech 16:1–18MathSciNetGoogle Scholar
- 2.Lukaszewicz G (1999) Micropolar fluids: theory and applications. Brikhauser, BaselCrossRefMATHGoogle Scholar
- 3.Eringen AC (2001) Microcontinuum field theories II: fluent media. Springer, New YorkMATHGoogle Scholar
- 4.Rashad AM, Abbasbandy S, Chamkha AJ (2014) Mixed convection flow of a micropolar fluid over a continuously moving vertical surface immersed in a thermally and solutally stratified medium with chemical reaction. J Taiwan Inst Chem Eng 45:2163–2169CrossRefGoogle Scholar
- 5.Sandeep N, Sulochana C (2015) Dual solutions for unsteady mixed convection flow of MHD micropolar fluid over a stretching/shrinking sheet with non-uniform heat source/sink. Eng Sci Technol Int J 18:738–745CrossRefGoogle Scholar
- 6.Shehzad SA, Waqas M, Alsaedi A, Hayat T (2016) Flow and heat transfer over an unsteady stretching sheet in a micropolar fluid with convective boundary condition. J Appl Fluid Mech 9:1437–1445CrossRefGoogle Scholar
- 7.Hayat T, Farooq S, Ahmad B, Alsaedi A (2016) Homogeneous-heterogeneous reactions and heat source/sink effects in MHD peristaltic flow of micropolar fluid with Newtonian heating in a curved channel. J Mol Liq 223:469–488CrossRefGoogle Scholar
- 8.Waqas M, Farooq M, Khan MI, Alsaedi A, Hayat T, Yasmeen T (2016) Magnetohydrodynamic (MHD) mixed convection flow of micropolar liquid due to nonlinear stretched sheet with convective condition. Int J Heat Mass Transfer 102:766–772CrossRefGoogle Scholar
- 9.Turkyilmazoglu M (2017) Mixed convection flow of magnetohydrodynamic micropolar fluid due to a porous heated/cooled deformable plate: exact solutions. Int J Heat Mass Transfer 106:127–134CrossRefGoogle Scholar
- 10.Crane L (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647CrossRefGoogle Scholar
- 11.Turkyilmazoglu M (2015) An analytical treatment for the exact solutions of MHD flow and heat over two–three dimensional deforming bodies. Int J Heat Mass Transfer 90:781–789CrossRefGoogle Scholar
- 12.Turkyilmazoglu M (2016) Flow of a micropolar fluid due to a porous stretching sheet and heat transfer. Int J Non-Linear Mech 83:59–64CrossRefGoogle Scholar
- 13.Waqas M, Hayat T, Farooq M, Shehzad SA, Alsaedi A (2016) Cattaneo–Christov heat flux model for flow of variable thermal conductivity generalized Burgers fluid. J Mol Liq 220:642–648CrossRefGoogle Scholar
- 14.Turkyilmazoglu M (2016) Equivalences and correspondences between the deforming body induced flow and heat in two-three dimensions. Phys Fluids 28:043102CrossRefGoogle Scholar
- 15.Babu MJ, Sandeep N (2016) 3D MHD slip flow of a nanofluid over a slendering stretching sheet with thermophoresis and Brownian motion effects. J Mol Liq 222:1003–1009CrossRefGoogle Scholar
- 16.Khan MI, Waqas M, Hayat T, Alsaedi A (2017) A comparative study of Casson fluid with homogeneous–heterogeneous reactions. J Colloid Interface Sci 498:85–90CrossRefGoogle Scholar
- 17.Khan WA, Irfan M, Khan M, Alshomrani AS, Alzahrani AK, Alghamdi MS (2017) Impact of chemical processes on magneto nanoparticle for the generalized Burgers fluid. J Mol Liq 234:201–208CrossRefGoogle Scholar
- 18.Hayat T, Khan MI, Waqas M, Alsaedi A, Farooq M (2017) Numerical simulation for melting heat transfer and radiation effects in stagnation point flow of carbon-water nanofluid. Comp Methods Appl Mech Eng 315:1011–1024MathSciNetCrossRefGoogle Scholar
- 19.Bilal S, Rehman KU, Malik MY, Hussain A, Khan M (2017) Effects of temperature dependent conductivity and absorptive/generative heat transfer on MHD three dimensional flow of Williamson fluid due to bidirectional non-linear stretching surface. Results Phys 7:204–212CrossRefGoogle Scholar
- 20.Hayat T, Waqas M, Khan MI, Alsaedi A (2017) Impacts of constructive and destructive chemical reactions in magnetohydrodynamic (MHD) flow of Jeffrey liquid due to nonlinear radially stretched surface. J Mol Liq 225:302–310CrossRefGoogle Scholar
- 21.Devi SPA, Prakash M (2015) Temperature dependent viscosity and thermal conductivity effects on hydromagnetic flow over a slendering stretching sheet. J Niger Math Soc 34:318–330MathSciNetCrossRefMATHGoogle Scholar
- 22.Hayat T, Hussain Z, Alsaedi A, Ahmad B (2017) Numerical study for slip flow of carbon–water nanofluids. Computer Methods Appl Mech Eng 319:366–378MathSciNetCrossRefGoogle Scholar
- 23.Khan MI, Hayat T, Waqas M, Alsaedi A (2017) Outcome for chemically reactive aspect in flow of tangent hyperbolic material. J Mol Liq 230:143–151CrossRefGoogle Scholar
- 24.Hayat T, Khan MI, Waqas M, Alsaedi A (2017) Mathematical modeling of non-Newtonian fluid with chemical aspects: a new formulation and results by numerical technique. Colloids Surf A Physicochem Eng Asp 518:263–272CrossRefGoogle Scholar
- 25.Imran Khan M, Hayat T, Ijaz Khan M, Alsaedi A (2017) A modified homogeneous-heterogeneous reactions for MHD stagnation flow with viscous dissipation and Joule heating. Int J Heat Mass Transfer 113:310–317CrossRefGoogle Scholar
- 26.Hayat T, Ullah I, Alsaedi A, Farooq M (2017) MHD flow of Powell–Eyring nanofluid over a non-linear stretching sheet with variable thickness. Results Phys 7:189–196CrossRefGoogle Scholar
- 27.Hayat T, Khan MI, Farooq M, Alsaedi A, Waqas M, Yasmeen T (2016) Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface. Int J Heat Mass Transfer 99:702–710CrossRefGoogle Scholar
- 28.Ijaz Khan M, Hayat T, Waqas M, Imran Khan M, Alsaedi A (2017) Impact of heat generation/absorption and homogeneous–heterogeneous reactions on flow of Maxwell fluid. J Mol Liq 233:465–470CrossRefGoogle Scholar
- 29.Hayat T, Qayyum S, Imtiaz M, Alsaedi A (2017) Radiative flow due to stretchable rotating disk with variable thickness. Results Phys 7:156–165CrossRefGoogle Scholar
- 30.Hayat T, Qayyum S, Alsaedi A, Ahmad B (2017) Magnetohydrodynamic (MHD) nonlinear convective flow of Walters-B nanofluid over a nonlinear stretching sheet with variable thickness. Int J Heat Mass Transfer 110:506–514CrossRefGoogle Scholar
- 31.Fourier J B J (1822) Théorie Analytique De La Chaleur, ParisGoogle Scholar
- 32.Catteneo C (1958) A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Comput Rendus 247:431–433Google Scholar
- 33.Vernotte P (1958) Les paradoxes de la theorie continue de L’equation de lachaleur. Comput Rendus 246:3154–3155MathSciNetMATHGoogle Scholar
- 34.Christov CI (2009) On frame indifferent formulation of the Maxwell-Cattaneo model of finite speed heat conduction. Mech Res Commun 36:481–486MathSciNetCrossRefMATHGoogle Scholar
- 35.Straughan B (2011) Thermal convection with the Cattaneo–Christov model. Int J Heat Mass Transfer 53:95–98CrossRefMATHGoogle Scholar
- 36.Tibullo V, Zampoli V (2011) A uniqueness result for the Cattaneo–Christov heat conduction model applied to incompressible fluids. Mech Res Commun 38:77–99CrossRefMATHGoogle Scholar
- 37.Han S, Zheng L, Li C, Zhang X (2014) Coupled flow and heat transfer in viscoelastic fluid with Cattaneo–Christov heat flux model. Appl Math Lett 38:87–93MathSciNetCrossRefMATHGoogle Scholar
- 38.Hayat T, Waqas M, Shehzad SA, Alsaedi A (2016) On 2D stratified flow of an Oldroyd-B fluid with chemical reaction: an application of non-Fourier heat flux theory. J Mol Liq 223:566–571CrossRefGoogle Scholar
- 39.Khan WA, Khan M, Alshomrani AS, Ahmad L (2016) Numerical investigation of generalized Fourier’s and Fick’s laws for Sisko fluid flow. J Mol Liq 224:1016–1021CrossRefGoogle Scholar
- 40.Nadeem S, Muhammad N (2016) Impact of stratification and Cattaneo–Christov heat flux in the flow saturated with porous medium. J Mol Liq 224:423–430CrossRefGoogle Scholar
- 41.Hayat T, Khan MI, Waqas M, Alsaedi A (2017) On Cattaneo–Christov heat flux in the flow of variable thermal conductivity Eyring–Powell fluid. Results Phys 7:446–450CrossRefGoogle Scholar
- 42.Khan M, Ahmad L, Khan WA, Alshomrani AS, Alzahrani AK, Alghamdi MS (2017) A 3D Sisko fluid flow with Cattaneo–Christov heat flux model and heterogeneous-homogeneous reactions: a numerical study. J Mol Liq 238:19–26CrossRefGoogle Scholar
- 43.Zhang Y, Chen B, Li D (2017) Non-Fourier effect of laser-mediated thermal behaviors in bio-tissues: a numerical study by the dual-phase-lag model. Int J Heat Mass Transfer 108:1428–1438CrossRefGoogle Scholar
- 44.Turkyilmazoglu M (2016) An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method. Filomat 30:1633–1650MathSciNetCrossRefMATHGoogle Scholar
- 45.Shehzad SA, Hayat T, Alsaedi A, Chen B (2016) A useful model for solar radiation. Energy Ecol Environ 1:30–38CrossRefGoogle Scholar
- 46.Hayat T, Waqas M, Khan MI, Alsaedi A (2016) Analysis of thixotropic nanomaterial in a doubly stratified medium considering magnetic field effects. Int J Heat Mass Transfer 102:1123–1129CrossRefGoogle Scholar
- 47.Ellahi R, Hassan M, Zeeshan A (2016) Aggregation effects on water base Al
_{2}O_{3}-nanofluid over permeable wedge in mixed convection. Asia-Pac J Chem Eng 11:179–186CrossRefGoogle Scholar - 48.Hayat T, Zubair M, Waqas M, Alsaedi A, Ayub M (2017) Application of non-Fourier heat flux theory in thermally stratified flow of second grade liquid with variable properties. Chin J Phys 55:230–241CrossRefGoogle Scholar
- 49.Khan WA, Alshomrani AS, Khan M (2016) Assessment on characteristics of heterogeneous-homogenous processes in three-dimensional flow of Burgers fluid. Results Phys 6:772–779CrossRefGoogle Scholar
- 50.Muhammad T, Alsaedi A, Shehzad SA, Hayat T (2017) A revised model for Darcy–Forchheimer flow of Maxwell nanofluid subject to convective boundary condition. Chin J Phys. https://doi.org/10.1016/j.cjph.2017.03.006 Google Scholar
- 51.Turkyilmazoglu M (2016) Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method. Mediterr J Math 13:4019–4037MathSciNetCrossRefMATHGoogle Scholar
- 52.Mahapatra TR, Gupta AS (2002) Heat transfer in stagnation-point flow towards a stretching sheet. Heat Mass Transfer 38:517–521CrossRefGoogle Scholar