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The present study deals with the analysis of heat transfer of the unsteady Maxwell nanofluid flow in a squeezed rotating channel of a porous extensile surface subject to the velocity and thermal slip effects incorporating the theory of heat flow intensity of Cattaneo-Christov model for the expression of the energy distribution in preference to the classical Fourier’s law. A set of transformations is occupied to renovate the current model in a system of nonlinear ordinary differential equations that are numerically decoded with the help of MATLAB integrated function bvp4c. The effects of various flow control parameters are investigated for the momentum, temperature and diffusion profiles, as well as for the wall shearing stress and the heat and mass transfer. The results are finally described from the material point of view. A comparison of heat flux models of Cattaneo-Christov and Fourier is also performed. An important result from the present work is that the squeezing parameter is strong enough in the middle of the channel to retard the fluid flow.

The heat transfer phenomenon is of great concern because of its impact on industrial applications, including cooling of space and nuclear reactors, heat conduction in tissues pasteurization of milk, magnetic targeting of drugs, etc. Fourier [

The practice of adding polymers to mineral oils, known as multi-grade oils, has become recognized since the middle of 1990s [

Nanofluids are the new-generation heat transfer fluids that contain higher thermal conductivity at very low particle concentrations than the conventional fluids. This idea of nanofluid was first developed by Choi [

While Stefan [

The boundary velocity, proportional to the shearing stress at the solid surface, is playing an important role in boundary value problems. For viscoelastic fluids, the slip condition is considerably important [

So far, few attempts have been made to study the transfer of heat and mass through a three-dimensional compression flow in a rotating channel, and therefore, the objective of the current work is to analyze the effect of thermal relaxation factor on the flow flux of time dependent Maxwell viscoelastic nanofluid that is squeezed in rotating parallel plates with porous stretched surface incorporating Cattaneo-Christov heat flux model.

The governing model equations consisting of conservation of mass, momentum, energy and concentration are given by

∂ ρ ∂ t + ∇ ⋅ ( ρ V ) = 0 (1)

ρ d V d t = ∇ ⋅ τ (2)

ρ c p d T d t = − ∇ ⋅ q − p ∇ ⋅ V − c p J s ⋅ ∇ T + Φ + S Q (3)

d C d t = − 1 ρ ∇ ⋅ J s (4)

here, V = ( u , v , w ) is the three-dimensional velocity of the viscous fluid, τ is the Cauchy stress tensor, T is the temperature of the fluid, q is the heat flux, Φ is the viscous dissipation term that describes the conversion of mechanical energy to heat. Also, S Q represents the heat sources, J s is the sum of Brownian and thermophoresis diffusions, ρ and c p are the density and specific heat respectively.

The elastico-viscous behavior of fluid will be realized if elastic stress is applied to the fluid, and the resulting strain will be time dependent characterized by relaxation time. The constitutive equation considering time dependent stress relaxation is [

τ = − p I + S (5)

The extra stress tensor S satisfies the upper convected Maxwell model given by

S + λ S ( d S d t + V ⋅ ∇ S − L S − S L tr ) = μ ( L + L tr ) (6)

here, L tr = ∇ V (i.e., L i j = ∂ u i / ∂ x j ) is the velocity gradient and the superscript tr indicates a transpose, μ is the viscosity, λ S > 0 is the stress relaxation time where λ S = 0 describes the Newtonian fluids.

Cattaneo-Christov model is proposed by adding thermal relaxation time in Fourier’s Law, also called the modified Fourier heat conduction law, presented by [

q + λ T ( ∂ q ∂ t + V ⋅ ∇ q − q ⋅ ∇ V + ( ∇ ⋅ V ) q ) = − κ ∇ T (7)

here, κ is the thermal conductivity and λ T is the thermal relaxation time parameter for the heat flux where λ T = 0 simplifies the expression (7) to classical Fourier’s law.

Buongiorno [

J s = − ρ D B ∇ C − ρ D T ∇ T T a (8)

here, D B is the Brownian diffusion coefficient, D T is the thermal diffusion coefficient and T a is the reference temperature.

To demonstrate the physical model of present analysis, it is considered that the flow is laminar, unsteady and three dimensional. An incompressible ( ∇ ⋅ V = 0 ), electrically conducting elastico-viscous Maxwell nanofluid is being squeezed between two infinite parallel plates rotating about y-axis. To explain the physical configuration, the Cartesian coordinate system is introduced such a way that x-axis is measured along the plate surface and y-axis is perpendicular to the plates, shown in

will not deviate of in the z-direction i.e., ∂ ∂ z = 0 . The gap width between those

plates in the minimal separation region is taken as time dependent given by h ( t ) = l 1 − α t , where l is the steady gap width and α is a constant having

dimension time_{−1}. For α > 0 the two plates are squeezed until they touch t = 1 / α and for α < 0 the two plates are always separated. The upper plate placed at y = h ( t ) is squeezing towards the lower plate with a vertical velocity

v h = d h ( t ) d t . This plate is stretched with a velocity u h = α h x / ( 1 − α t ) in the

positive x-direction with velocity slip parameter r 1 and thermal slip parameter r 2 . The lower plate is fixed at y = 0 assumed to be porous in which the fluid flows with suction velocity v 0 = − V 0 / ( 1 − α t ) . A uniform magnetic field of density B 0 is applied to along y-direction and the external electric field is assumed zero.

The boundary conditions of the present physical models are

u − r 1 ∂ u ∂ y = u h , v = v h , w = 0 , T − r 2 ∂ T ∂ y = T H , C = C H at y = h ( t ) u = 0 , v = v 0 , w = 0 , T = 0 , C = 0 , at y = 0 } (9)

Now in order to find the approximate solutions of the model it is essential to make the model equations dimensionless using the following non-dimensional variables [

η = y l ( 1 − α t ) 1 2 , u = α x 2 ( 1 − α t ) f ′ , v = − α l 2 ( 1 − α t ) 1 2 f , w = α x 2 ( 1 − α t ) g , T = T H θ , C = C H F (10)

Using the above transformations, the Equation (1) is satisfied and the Equations (2)-(4) are reduced to

( 1 − S q β S ( η 2 − 2 η f + f 2 ) ) f i v − S q β S ( 9 η f ‴ + 2 η f ″ 2 − 2 f f ″ 2 − 2 f ′ 2 f ″ − 8 f f ‴ + 4 f ′ f ″ + 15 f ″ ) − M ( f ′ ′ + β S ( η f ‴ + 3 f ′ ′ − f f ′ ′ ′ − f ′ f ′ ′ ) ) − S q ( η f ‴ + 3 f ″ + f ′ f ″ − f f ‴ + 2 ω g ′ ) = 0 (11)

( 1 − S q β S ( η 2 − 2 η f + f 2 ) ) g ″ − S q β S ( 7 η g ′ + 2 η f ′ g ′ + 4 f ′ g − 6 f g ′ − 2 f f ′ g ′ + 8 g ) − S q ( η g ′ + 2 g + f ′ g − f g ′ − 2 ω f ′ ) − M ( g + β S ( η g ′ + 2 g − f g ′ ) ) = 0 (12)

( 1 − P r S q β T ( η 2 − 2 η f + f 2 ) ) θ ″ − S q P r ( η θ ′ − f θ ′ ) − S q P r β T ( 3 η θ ′ − η f ′ θ ′ − 3 f θ ′ + f f ′ θ ′ ) + P r ( N B θ ′ F ′ + N T θ ′ 2 ) + E c P r ( 4 δ 2 f ′ 2 + f ″ 2 + g ′ 2 + δ 2 g 2 ) = 0 (13)

F ″ − S q L e ( η F ′ − f F ′ ) + N t N b θ ″ = 0 (14)

The dimensionless boundary conditions are

f = 1 , f ′ = γ u + ε u f ″ , g = 0 , θ = 1 + ε u θ ′ , F = 1 , a t η = 1 f = f w , f ′ = 0 , g = 0 , θ = 0 , F = 0 at η = 0 } (15)

here, ω = 2 ( 1 − α t ) α Ω is the rotation parameter; β S = λ S α 2 ( 1 − α t ) is the Maxwell parameter; S q = α l 2 2 ν is the squeeze number; M = σ B 0 2 h 2 μ is the Magnetic field parameter; P r = ν ρ C p κ f is the Prandtl number; E c = ρ u a 2 ρ C p T H is the Eckert number; γ u = α h α is the stretching parameter; ε u = r 1 h is the velocity slip parameter; ε T = r 2 h is the thermal slip parameter; δ = h x is the characteristic length ratio; f w = 2 V 0 α h is the suction parameter.

Finally, the physical attentions in the existing study are the skin friction coefficient C f , the local Nusselt number Nu and the Sherwood number (Sh) defined as

C f ∝ f ″ , N u ∝ − θ ′ and S h ∝ − F ′ (16)

Equations (11)-(14) combined with the boundary conditions (15) are solved numerically using finite difference code developed by a MATLAB boundary value problem solver, known as bvp4c. The analysis is made for various values of the pertinent parameters such as Brownian motion parameter N B , squeezing parameter S q , Maxwell parameter β S , thermal relaxation parameter β T , rotation parameter ω , stretching parameter γ u , velocity slip parameter ε u and thermal slip parameter ε T . The step size is taken as η = 0.01 and the tolerance criteria are set to 10^{−6}. On the basis of the present model, [ 0,1 ] is measured as the domain of a channel. First of all, comparison of the current model is arranged with [

Lightaqua | Lightaqua Mustafa et al. [ | Lightaqua Present work | ||||
---|---|---|---|---|---|---|

Lightaqua Pr | − f ′ ( 1 ) | − θ ′ ( 1 ) | F ′ | − f ′ ( 1 ) | − θ ′ ( 1 ) | F ′ |

−1.0 | 3.026324 | 3.02632355855 | 3.026323 | 2.170091 | 3.319899 | 0.804559 |

−0.5 | 5.98053 | 5.98053039715 | 5.98053 | 2.617404 | 3.129491 | 0.781402 |

0.01 | 14.43941 | 14.4394132325 | 14.439411 | 3.007133 | 3.047091 | 0.761225 |

0.5 | 1.513162 | 1.51316180648 | 1.513161 | 3.336448 | 3.026327 | 0.744224 |

2.0 | 3.631588 | 3.63158826816 | 3.631587 | 4.167387 | 3.118553 | 0.701813 |

When the elastic stress is applied to the non-Newtonian fluid, the time during which the fluid achieves its stability is the relaxation time, which is greater for highly viscous fluids. The Maxwell parameter β S deals with the fluid relaxation time to its characteristic time scale. Here β S = 0 gives the result for Newtonian viscous incompressible fluid. The fluid with a small Maxwell parameter exhibits liquid-like activities but large Maxwell parameter communicates with solid-like materials able to conduct and retain heat better. Therefore, it is observed physically that gradually increasing the Maxwell parameter can increase the fluid viscosity, which enhances resistance to flow and, as a result, the hydrodynamic boundary layer thickness reduces for Maxwell fluid, as shown in

The result found from

Finally, from the point of view of physical interest, the skin friction coefficient is useful to estimate the total frictional drag exerted on the surface. The Nusselt Number is used to characterize the heat flux from a heated solid surface to a fluid. Additionally,

β T = 0 | β T = 0.3 | ||||
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Sq | β S | C f η = 1 | C f η = 0 | C f η = 1 | C f η = 0 |

0.1 | 0.5 | −4.659543 | 4.056332 | −4.659543 | 4.056332 |

0.2 | −4.757606 | 4.214218 | −4.757606 | 4.214218 | |

0.3 | −4.861431 | 4.377962 | −4.861431 | 4.377962 | |

0.6 | −4.921623 | 4.486911 | −4.921623 | 4.486911 | |

0.7 | −4.983358 | 4.598035 | −4.983358 | 4.598035 |

β T = 0 | β T = 0.3 | ||||
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Sq | β S | N u η = 1 | N u η = 0 | N u η = 1 | N u η = 0 |

0.1 | 0.5 | 0.253098 | −4.942629 | 0.261092 | −5.325179 |

0.2 | 0.269455 | −5.387022 | 0.283571 | −6.271893 | |

0.3 | 0.286328 | −5.861222 | 0.304701 | −7.396247 | |

0.6 | 0.290002 | −5.867768 | 0.308438 | −7.405840 | |

0.7 | 0.293886 | −5.874447 | 0.312386 | −7.415598 |

The present paper is to study the effect of thermal relaxation factor on the flow of Maxwell nanofluid squeezing in the parallel rotating plates with porous stretched surface incorporating Cattaneo-Christov heat flux model. The major outcomes drawn from the study of the present model can be summarized as follows:

1) The thermal boundary layer thickness rises for the Brownian motion parameter, squeezing parameter, stretching parameter and velocity slip parameter.

2) The thermal boundary layer thickness decreases for the thermal slip parameter.

3) The hydrodynamic boundary layer thickness is reduced for the squeezing parameter, Maxwell parameter and stretching parameter.

4) The velocity distributions are higher for the velocity slip parameter.

5) The concentration is elevated for Brownian motion parameter and squeezing parameter.

In conclusion of the current study, it can be argued that the squeezing parameter and the stretching parameter that have the velocity control phenomena, can improve the heat transfer in the nanofluid. This study will provide a great opportunity to develop the cooling performance of mechanical system like automotive radiators and nuclear reactors.

We thank the Editor and the referee for their comments. Their support is greatly appreciated.

The authors declare no conflicts of interest regarding the publication of this paper.

Karim, M.E. and Samad, M.A. (2020) Effect of Brownian Diffusion on Squeezing Elastico-Viscous Nanofluid Flow with Cattaneo-Christov Heat Flux Model in a Channel with Double Slip Effect. Applied Mathematics, 11, 277-291. https://doi.org/10.4236/am.2020.114021