# Swirling flow of Maxwell nanofluid between two coaxially rotating disks with variable thermal conductivity

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## Abstract

The main concern of this article is to study the Maxwell nanofluid flow between two coaxially parallel stretchable rotating disks subject to axial magnetic field. The heat transfer process is studied with the characteristics of temperature-dependent thermal conductivity. The Brownian motion and thermophoresis features due to nanofluids are captured with the Buongiorno model. The upper and lower disks rotating with different velocities are discussed for the case of same as well as opposite direction of rotations. The von Kármán transformations procedure is implemented to obtain the set of nonlinear ordinary differential equations involving momentum, energy and concentration equations. A built-in numerical scheme bvp4c is executed to obtain the solution of governing nonlinear problem. The graphical and tabular features of velocity, pressure, temperature and concentration fields are demonstrated against the influential parameters including magnetic number, stretching parameters, Deborah number, Reynolds number, Prandtl number, thermal conductivity parameter, thermophoresis and Brownian motion parameters. The significant outcomes reveal that stretching action causes to reverse the flow behavior. It is noted that the effect of Deborah number is to reduce the velocity and pressure fields. Further, the impact of thermophoresis and thermal conductivity parameters is to increase the temperature profile. Moreover, the fluid concentration is reduced with the stronger action of Schmidt number.

## Keywords

Maxwell nanofluid Coaxially rotating disks Variable thermal conductivity Magnetic field Numerical solutions## List of symbols

*u*,*v*,*w*Velocity components

- \(r,\varphi ,z\)
Cylindrical coordinate system

**V**Velocity vector

- \(\nabla\)
Nabla

- \(\nu ,\mu\)
Kinematic and dynamic viscosities

*p*,*T*Fluid pressure and temperature

**S**Extra stress tensor

*λ*_{1}Relaxation time parameter

*c*_{p}Specific heat at constant pressure

**L**Gradient of velocity vector

**A**_{1}First Rivlin–Ericksen tensor

*B*_{0}Magnetic field strength

*s*_{1}Lower disk stretching rate

*s*_{2}Upper disk stretching rate

- Ω
_{1} Lower disk rotation rate

- Ω
_{2} Upper disk rotation rate

*T*_{1}Lower disk temperature

*T*_{2}Upper disk temperature

- \(\rho ,C\)
Fluid density and concentration

**B**Magnetic field

*k*(*T*)Variable thermal conductivity

*η*Dimensionless variable

*d*Vertical distance between disks

- \(\left( {\rho c_{p} } \right)_{f}\)
Specific heat of the base fluid

- \(\left( {\rho c_{p} } \right)_{p}\)
Heat capacity of the nanofluid

*D*_{B}Brownian diffusion coefficient

*D*_{T}Thermophoretic diffusion coefficient

*k*_{∞}Fluid thermal conductivity

**J**Current density

*σ*Electrical conductivity

**q**Heat flux

*f′*Dimensionless radial velocity

*g*Dimensionless azimuthal velocity

*f*Dimensionless axial velocity

*P*Dimensionless pressure

*θ*Dimensionless temperature

*ϕ*Dimensionless concentration

- Ω
Rotation parameter

*M*Magnetic parameter

*β*_{1}Deborah number

*Pr*Prandtl number

*Re*Local Reynolds number

*S*_{1}Lower disk stretching parameter

*S*_{2}Upper disk stretching parameter

*ε*Thermal conductivity parameter

- \(\varLambda\)
Pressure gradient parameter

*Nt*Thermophoresis parameter

*Nb*Brownian motion parameter

*Sc*Schmidt number

*Nu*_{r1}Local Nusselt number at lower disk

*Nu*_{r2}Local Nusselt number at upper disk

*Sh*_{r1}Local Sherwood number at lower disk

*Sh*_{r2}Local Sherwood number at upper disk

## Notes

## References

- 1.Von Kármán T (1921) Uber laminare and turbulente Reibung. Z Angew Math Mech 1:233–252CrossRefGoogle Scholar
- 2.Cochran WG (1934) The flow due to a rotating disk. Proc Camb Philos Soc 30:365–375CrossRefGoogle Scholar
- 3.Millsaps K, Pohlhausen K (1952) Heat tranasfer by laminar flow from a rotating-plate. J Aeronaut Sci 19:120–126MathSciNetCrossRefGoogle Scholar
- 4.Turkyilmazoglu M (2009) Exact solutions corresponding to the viscous incompressible and conducting fluid flow due to a porous rotating disk. J Heat Transf 131:091701CrossRefGoogle Scholar
- 5.Turkyilmazoglu M (2012) Effects of uniform radial electric field on the MHD heat and fluid flow due to a rotating disk. Int J Eng Sci 51:233–240MathSciNetCrossRefGoogle Scholar
- 6.Sheikholeslami M, Hatami M, Ganji DD (2015) Numerical investigation of nanofluid spraying on an inclined rotating disk for cooling process. J Mol Liq 211:577–583CrossRefGoogle Scholar
- 7.Khan M, Ahmed J, Ahmad L (2018) Chemically reactive and radiative von Kármán swirling flow due to a rotating disk. Appl Math Mech Engl Ed 39:1295–1310CrossRefGoogle Scholar
- 8.Turkyilmazoglu M (2018) Fluid flow and heat transfer over a rotating and vertically moving disk. Phys Fluids 30:063605. https://doi.org/10.1063/1.5037460 CrossRefGoogle Scholar
- 9.Khan M, Ahmed J, Ahmad L (2018) Application of modified Fourier law in von Kármán swirling flow of Maxwell fluid with chemically reactive species. J Braz Soc Mech Sci Eng 40:573CrossRefGoogle Scholar
- 10.Batchelor GK (1951) Note on a class of solutions of the Navier–Stokes equations representing steady rotationally symmetric flow. Q J Mech Appl Math 4:29–41MathSciNetCrossRefGoogle Scholar
- 11.Stewartson K (1953) On the flow between two ratating coaxial disks. Proc Camb Philos Soc 49:333–341CrossRefGoogle Scholar
- 12.Lance GN, Rogers MH (1962) The axially symmetric flow of a viscous fluid between two infinite rotating disks. Proc R Soc A 266:109–121MathSciNetzbMATHGoogle Scholar
- 13.Yan WM, Soong CY (1997) Mixed convection flow and heat transfer between co-rotating porous disks with wall transpiration. Int J Heat Mass Transf 40:773–784CrossRefGoogle Scholar
- 14.Soong CY, Wu CC, Liu TP, Liu TP (2003) Flow structure between two co-axial disks rotating independently. Exp Therm Fluid Sci 27:295–311CrossRefGoogle Scholar
- 15.Jiji LM, Ganatos P (2010) Microscale flow and heat transfer between rotating disks. Int J Heat Fluid Flow 31:702–710CrossRefGoogle Scholar
- 16.Turkyilmazoglu M (2016) Flow and heat simultaneously induced by two stretchable rotating disks. Phys Fluid 28:043601CrossRefGoogle Scholar
- 17.Das A, Sahoo B (2018) Flow and heat transfer of a second grade fluid between two stretchable rotating disks. Bull Braz Math Soc New Ser 49:531–547MathSciNetCrossRefGoogle Scholar
- 18.Choi SUS (1995) Enhancing thermal conductivity of fluids with nanoparticle. ASME Int Mech Eng Cong Exp 66:99–105Google Scholar
- 19.Buongiorno J (2006) Convective transport in nanofluids. J Heat Transf 128:240–250CrossRefGoogle Scholar
- 20.Kuznetsov AV, Nield DA (2011) Double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate. Int J Therm Sci 50:712–717CrossRefGoogle Scholar
- 21.Turkyilmazoglu M (2015) Anomalous heat transfer enhancement by slip due to nanofluids in circular concentric pipes. Int J Heat Mass Transf 85:609–614CrossRefGoogle Scholar
- 22.Turkyilmazoglu M (2015) Analytical solutions of single and multi-phase models for the condensation of nanofluid film flow and heat transfer. Euro J Mech B Fluids 53:272–277MathSciNetCrossRefGoogle Scholar
- 23.Pourmehran O, Rahimi-Gorji M, Hatami M, Sahebi SAR, Domairry G (2015) Numerical optimization of microchannel heat sink (MCHS) performance cooled by KKL based nanofluids in saturated porous medium. J Taiwan Inst Chem Eng 55:49–68CrossRefGoogle Scholar
- 24.Hatami M, Song D, Jing D (2016) Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition. Int J Heat Mass Transf 98:758–767CrossRefGoogle Scholar
- 25.Hatami M (2017) Nanoparticles migration around the heated cylinder during the RSM optimization of a wavy-wall enclosure. Adv Powder Technol 28:890–899CrossRefGoogle Scholar
- 26.Tang W, Hatami M, Zhou J, Jing D (2017) Natural convection heat transfer in a nanofluid-filled cavity with double sinusoidal wavy walls of various phase deviations. Int J Heat Mass Transf 115:430–440CrossRefGoogle Scholar
- 27.Hatami M, Jing D (2017) Optimization of wavy direct absorber solar collector (WDASC) using Al
_{2}O_{3}–water nanofluid and RSM analysis. Appl Therm Eng 121:1040–1050CrossRefGoogle Scholar - 28.Hatami M, Zhou J, Geng J, Song D, Jing D (2017) Optimization of a lid-driven T-shaped porous cavity to improve the nanofluids mixed convection heat transfer. J Mol Liq 231:620–631CrossRefGoogle Scholar
- 29.Hatami M, Ganji DD (2014) Motion of a spherical particle on a rotating parabola using Lagrangian and high accuracy multi-step differential transformation method. Powder Technol 258:94–98CrossRefGoogle Scholar
- 30.Hatami M, Ganji DD (2014) Motion of a spherical particle in a fluid forced vortex by DQM and DTM. Particuology 16:206–212CrossRefGoogle Scholar
- 31.Dogonchi AS, Hatami M, Domairry G (2015) Motion analysis of a spherical solid particle in plane Couette Newtonian fluid flow. Powder Technol 274:186–192CrossRefGoogle Scholar