MHD squeezed Darcy–Forchheimer nanofluid flow between two h–distance apart horizontal plates

Ghulam Rasool; Waqar A. Khan; Sardar Muhammad Bilal; Ilyas Khan

doi:10.1515/phys-2020-0191

Open Access Published by De Gruyter Open Access December 29, 2020

MHD squeezed Darcy–Forchheimer nanofluid flow between two h–distance apart horizontal plates

Ghulam Rasool , Waqar A. Khan , Sardar Muhammad Bilal and Ilyas Khan

From the journal Open Physics

https://doi.org/10.1515/phys-2020-0191

Abstract

This research is mainly concerned with the characteristics of magnetohydrodynamics and Darcy–Forchheimer medium in nanofluid flow between two horizontal plates. A uniformly induced magnetic impact is involved at the direction normal to the lower plate. Darcy–Forchheimer medium is considered between the plates that allow the flow along horizontal axis with additional effects of porosity and friction. The features of Brownian diffusive motion and thermophoresis are disclosed. Governing problems are transformed into nonlinear ordinary problems using appropriate transformations. Numerical Runge–Kutta procedure is applied using MATLAB to solve the problems and acquire the data for velocity field, thermal distribution, and concentration distribution. Results have been plotted graphically. The outcomes indicate that higher viscosity results in decline in fluid flow. Thermal profile receives a decline for larger viscosity parameter; however, Brownian diffusion and thermophoresis appeared as enhancing factors for the said profile. Numerical data indicate that heat flux reduces for viscosity parameter. However, enhancement is observed in skin-friction for elevated values of porosity factor. Data of this paper are practically helpful in industrial and engineering applications of nanofluids.

Keywords: nanofluid; Darcy–Forchheimer medium; magnetohydrodynamics; squeezed flow

1 Introduction

Heat and mass convection is a natural phenomenon in every physical situation involving viscous fluid materials. This phenomenon is quite natural in various structures such as between two parallel plates, over stretching surfaces, and within a cylindrical sphere. Therefore, one can define a fluid flowing between two surfaces as squeezing flow. Because of high significance and demand in industrial and other setups, squeezing flow has received special attention from the researchers working on fluid flow, heat, and mass transport analysis. In particular, the branches of fluid dynamics which belong to mechanical and biochemical engineering, food processing, chemical engineering, industrial processing, and many others are typical examples where the concept of squeezing flow is mostly used. In addition, we have seen this in gears, rolling elements, machine devices, lubrication, automotive engines, grease and oil setups, compression, injection, and shaping. The pioneer approach was reported by Stefan [1], which was highly appreciated by the research community. In this research, squeezing flow between two surfaces was disclosed for the first time with useful comments on the flow profiles. Later on, the squeezing flow has been discussed widely in many articles. For example, Rashidi et al. [2] discussed unsteady and axisymmetric squeezed flow of nanofluids for approximation of analytic solutions to the flow problems. Hayat et al. [3] disclosed the features of three-dimensional and squeezed flow using two parallel sheets because of mixed convection. Hayat et al. [4] reported squeezing flow in rotating frame between two disks. The problems were developed using second grade fluid. In another study, Hayat et al. [5] discussed the findings of flow bounded by porous squeezed enclosure disclosing the features of magnetic field effects. Shahmohamadi and Rashidi [6] reported some good findings on the squeezing flow of nanofluids subject to rotating channel. The lower plate was assumed to be porous. Some recent studies [7,8] are also referred for further understanding the above scenario.

Technological developments have led the fluid mechanics toward a more productive formulation that is known as nanofluids. This formulation generally contains nano-size metallic particles suspended in the base fluid for shorter period. However, the results of this shorter period suspension are highly effective because the suspension is more efficient and has more thermophysical properties such as electric and thermal conductivity and density. Thermal features and dynamic flexibility in the context of irreversability, entropy, and many other respective features have shown some great improvements for nanofluids. The pioneer study was reported by Choi [9] disclosing the thermo-physical characteristics of base liquid with the addition of nanoparticles. The concept was highly appreciated by research community. Later, Parvin and Chamkha [10] reported free convection and entropy optimization of nanofluids flowing in odd-shaped cavity. Zaraki et al. [11] disclosed the properties of boundary layer convection considering the size, type, and shape of nanoparticles as well as the type of base fluid. Reddy and Chamkha [12] accounted the effect of Soret and Dufour on TiO₂-water and Al₂O₃-water type suspensions passing via stretching sheet. Chamkha et al. [13] disclosed the features of entropy optimization in Cu − O –water nanofluid using magnetic influence. Rasool et al. [14] discussed the effects of porosity and Darcy media in nanofluid flow via stretching surface. Ismael et al. [15] analyzed the entropy optimization in cavity filled with nanofluid via porous medium. Rasool et al. [16] reported flow of nanofluids bounded by a convective and vertically adjusted Riga plate. Many recent articles are typically based on the flow of nanofluids; however, some of them are listed here [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]. Fluid flow analysis via porous medium has many important applications in various mechanical and industrial procedures. For example, the underground water purification process, oil recovery and purification, outlining, pipe developments, and many other procedures are typical examples of fluid flow analysis through porous medium (known as Darcy–Forchheimer flow). The pioneer model was originally defined by Darcy for weak porosity conditions and smaller velocity. Later on, Forchheimer [40] remodeled it using nonlinear factor through velocity and the new name given to this model as Darcy–Forchheimer model. Muskat [41] presented homogeneous fluid flow through Darcy medium. Seddeek [42] disclosed the features of thermophoresis and dissipation in Darcy type fluid flow using the concept of mixed convection. Hayat et al. [43] analyzed the entropy optimization and heat and mass transport mechanism using bidirectional water-based nanofluid flow subject to convective conditions. Sadiq and Hayat [44] reported Darcy–Forchheimer Maxwell type nanofluid flow via convectively heated stretching surface. Umavathi et al. [45] reported numerical investigation on Darcy type nanofluid flow bounded by vertically adjusted rectangular duct. Hayat et al. [46] performed framing of radiation effect and heat generation in Darcy type nanofluid flow using chemical reaction.

In this research, our motivation is based on three novel concepts: first, to involve two parallel plates having filled the gap with a porous medium that has never been reported yet; second, to involve magnetohydrodynamics (MHD) effect in this formulation; and finally, to see the impact of squeezing nature of the model on the fluid flow analysis. This article is organized in the following order. First, a viscous, MHD non-Newtonian nanofluid is considered between the h − distance apart two parallel plates via Darcy medium. The attributes of Brownian diffusive motion and thermophoresis are involved. Second, the so-formulated governing problems are transformed into nonlinear dimensionless problems using suitable transformations. Third, numerical Runge–Kutta method built in MATLAB is applied to solve the problems and acquire the data for velocity field, thermal distribution, concentration distribution, and Nusselt number. All the results have been plotted graphically. Finally, the discussion is provided on the results in detail.

2 Mathematical model

Here we consider a steady squeezed nanofluid flow contained between two horizontally adjusted h − distance apart plates. The location of plates is fixed at x 2 = 0 at one side and x 2 = h at the other side in Cartesian coordinates. The bottom plate is stretched with at the rate of u 1 = z x 1 , where z is a positive constant integer. A uniformly induced magnetic impact is involved at the direction normal to the lower plate. Darcy–Forchheimer medium is considered between the plates, which allows the flow along horizontal axis with additional effects of porosity and friction. The geometry of the problem can be seen in Figure 1. The governing equations (see for reference Sheikholeslami et al. [47]) are:

(1) ∂ u 1 ∂ x 1 + ∂ u 2 ∂ x 2 = 0 ,

(2) u 1 ∂ u 1 ∂ x 1 + u 2 ∂ u 1 ∂ x 2 = − 1 ρ f ∂ p ∂ x 1 + ν ∂ 2 u 1 ∂ x 1 2 + ∂ 2 u 1 ∂ x 2 2 − σ B 0 2 ρ f u 1 − ν K u 1 − F u 1 2 ,

(3) u 1 ∂ u 2 ∂ x 1 + u 2 ∂ u 2 ∂ x 2 = − 1 ρ f ∂ p ∂ x 2 + ν ∂ 2 u 2 ∂ x 1 2 + ∂ 2 u 2 ∂ x 2 2 ,

(4) u 1 ∂ T ∂ x 1 + u 2 ∂ T ∂ x 2 = α ∂ 2 T ∂ x 2 2 + ∂ 2 T ∂ x 1 2 + τ D B ∂ C ∂ x 2 ∂ T ∂ x 2 + ∂ C ∂ x 1 ∂ T ∂ x 1 + D T T h ∂ T ∂ x 1 2 + ∂ T ∂ x 2 2 ,

(5) u 1 ∂ C ∂ x 1 + u 2 ∂ C ∂ x 2 = D B ∂ 2 C ∂ x 1 2 + ∂ 2 T ∂ x 2 2 + D T T 0 ∂ 2 T ∂ x 2 2 + ∂ 2 T ∂ x 1 2 ,

with BCs

(6) u 1 = u w = z x 1 , u 2 = 0 , C = C h , T = T h , at x 2 = 0 ,

(7) u 1 = 0 , C = C 0 , T = T 0 , at x 2 = + h .

Figure 1

Geometry.

Differentiation of equation (2) w.r.t. x 2 and equation (3) w.r.t. x 1 and subtraction yield the following combined momentum equation:

(8) ∂ u 1 ∂ x 1 ∂ u 1 ∂ x 2 + u 1 ∂ 2 u 1 ∂ x 1 ∂ x 2 + ∂ u 1 ∂ x 2 ∂ u 2 ∂ x 2 + u 2 ∂ 2 u 1 ∂ x 2 2 − u 1 ∂ 2 u 2 ∂ x 1 2 − ∂ u 1 ∂ x 1 ∂ u 2 ∂ x 1 − u 2 ∂ 2 u 2 ∂ x 1 ∂ x 2 − ∂ u 2 ∂ x 1 ∂ u 2 ∂ x 2 = ν ∂ 3 u 1 ∂ x 1 2 ∂ x 2 + ∂ 3 u 1 ∂ x 2 3 − ∂ 3 u 2 ∂ x 1 3 − ∂ 3 u 2 ∂ x 1 ∂ x 2 2 − σ B 0 2 ρ f ∂ u 1 ∂ x 2 − ν K ∂ u 1 ∂ x 2 − 2 F ∂ u 1 ∂ x 2 .

Define,

(9) u 1 = z x 1 ∂ f ∂ η , u 2 = − z h f , ( T 0 − T h ) θ ( η ) = ( T − T h ) , ( C 0 − C h ) ϕ ( η ) = ( C − C h ) , η = y h .

Using (9) in (1), (4), (5), and (8), we have

(10) f i v − P ( f ′ f ″ − f f ′ ″ ) − M f ″ − λ f ″ − 2 F r f ′ f ″ = 0 ,

(11) θ ″ + Nb ϕ ′ θ ′ + P Pr f θ ′ + Nt θ ′ 2 = 0 ,

(12) ϕ ″ + P Sc f ϕ ′ + Nt Nb θ ″ = 0 ,

(13) f ' = 1 , f = 0 , θ = 1 = ϕ , at η = 0 , f ' = 0 , f = 0 , θ = 0 = ϕ , at η = 1 ,

where P = h 2 z v is the viscosity parameter, λ = h 2 K is the porosity, F r = F z h x 1 v is the Forchheimer number such that F = C b K is the drag force coefficient and M = σ B 0 2 h 2 ρ f ν is the magnetic parameter. In the energy equation, Pr = v α is the Prandtl factor, Nb = τ D B ν ( C 0 − C h ) is the Brownian motion factor, Nt = τ D T ν T h ( T 0 − T h ) is the thermophoresis factor, and Sc = v D B is the Schmidt factor.

The physical quantities are given as:

(14) P x 1 h C f = f ″ 0 , Nu x = θ ′ ( 0 ) .

3 Numerical solution

MATLAB-based numerical scheme is applied for the solutions of governing problems. The results are plotted graphically for the velocity field and thermal and solute profiles as well as for Nusselt factor and skin-frictional force. The governing equations, i.e., equations (10)–(12), are highly nonlinear problems that are numerically dealt with RK-45 technique accompanied with shooting method. The methods start with conversion of governing equations into first-order differential problems as follows:

(15) f i v = P ( f ′ f ″ + f f ′ ″ ) + M f ″ + λ f ″ + 2 F r f ′ f ″ = 0 ,

(16) f = ξ 1 , f ′ = ξ 2 , f ″ = ξ 3 , f ′ ″ = ξ 4 , f i v = ξ 4 ' , θ = ξ 5 , θ ′ = ξ 6 , θ ″ = ξ 6 ' , ϕ = ξ 7 , ϕ ′ = ξ 8 , ϕ ″ = ξ 8 ' ,

(17) ξ 4 ' = P ( ξ 2 ξ 3 − ξ 1 ξ 4 ) + M ( ξ 3 ) + λ ξ 3 + 2 F r ξ 2 ξ 3 ,

(18) ξ 6 ' = − P Pr ξ 1 ξ 6 − Nb ξ 8 ξ 6 − Nt ξ 6 2 ,

(19) ξ 8 ' = − P Sc ξ 1 ξ 8 − Nt Nb ξ 6 ' ,

subject to:

(20) ξ 1 ( 0 ) = 0 , ξ 2 ( 0 ) = 1 , ξ 1 ( 1 ) = 0 and ξ 2 ( 1 ) = 1 , ξ 3 ( 0 ) = α 1 , ξ 5 ( 0 ) = 1 , ξ 5 ( 1 ) = 0 , ξ 6 ( 0 ) = α 2 , ξ 7 ( 0 ) = 1 , ξ 7 ( 1 ) = 0 , ξ 8 ( 0 ) = α 3 .

The fifth-order RK45-numerical techniques efficiently solved this problem for various values of χ ∞ , and subsequently the results are found accurate in region 0 ≤ χ ≤ 1 subject to suitable numerical values assigned to each fluid parameter.

4 Discussion

This section explores the graphical results and their physical justifications for the three main profiles, i.e., velocity, temperature, and concentration of nanoparticles. Figures 2–18 are plotted to disclose the impact of various fluid parameters involved in the present flow model. In particular, Figures 2 and 3 are graphical display of the influence of Forchheimer parameter on the velocity field (f and fʹ), respectively. A closer look explores that velocity field shows reduction for larger Forchheimer number. Physically, the relation of Forchheimer number with drag force coefficient is responsible for this trend in velocity field. For larger Forchheimer number, an intensive drag force coefficient results in higher amount of friction offered to the fluid flow. Therefore, a decline is noticed in velocity profile. Figures 4 and 5 show the impact of magnetic field on fluid flow through Darcy medium. The impact of magnetic field is inversely related to the fluid flow. Physically, a surface normal implementation of magnetic field creates certain bumps in the direction of fluid flow. Therefore, a decline is noticed in both f and f ′ . Impact of viscosity parameter P on both f and f ′ is shown in Figures 6 and 7, respectively. Physically, for larger values of P, the inverse relation of kinematic viscosity confirms the enhancement in dynamic viscosity, and therefore, a decline in velocity field is justified for larger values of viscosity parameter. Figure 8 shows the impact of viscosity parameter on thermal distribution. The relevant boundary layer shows declining trend for augmented values of viscosity parameter. Impact of Brownian diffusive motion and thermophoresis on thermal distribution is shown in Figures 9 and 10. The non-predictive movement of nanoparticles because of the Brownian motion rises for stronger thermophoretic force, resulting in a more rapid transport from hot region to the colder region. Therefore, a rise in thermal distribution is noticed for both the parameters. For elevated values of viscosity parameter, one can see an enhancement in the concentration distribution shown in Figure 11, which equally confirms the mathematical expression of viscosity parameter and its physical significance in fluid flow. For elevated values of Brownian diffusive motion parameter, concentration profile shows reduction. Physically, the random motion reduces for larger Brownian; however, the case is opposite in case of thermophoresis. At Nb = 0.01 , the impact of Brownian diffusion is quite obvious; however, the impact becomes slighter for further improvement in the Brownian diffusion, whereas a linear enhancement is seen in concentration distribution for Nt = 0.01 , 0.1 , 0.2 , 0.3 , and so on. The variation in Nusselt number is shown in Figures 14 and 15. In Figure 14, the cross effect of viscosity parameter and thermophoresis is involved, whereas in Figure 15, thermophoresis is replaced with Brownian diffusion. The rate of heat flux reduces in both cases. Figures 16 and 17 are plotted to see the fluctuation in skin-friction for porosity factor and magnetic parameter, respectively. The larger friction produced by Forchheimer medium and retardation offered by magnetic impact result in enhancement of skin-friction. The impact of Prandtl number on thermal profile is shown in Figure 18. A decline is noticed for larger values of Pr.

$Figure 2 Consequences of F r {F}_{\text{r}} on f ( η ) f(\eta ) .$

Figure 2

Consequences of F r on f ( η ) .

$Figure 3 Consequences of F r {F}_{\text{r}} on f ′ ( η ) f^{\prime} (\eta ) .$

Figure 3

Consequences of F r on f ′ ( η ) .

$Figure 4 Consequences of M on f ( η ) f(\eta ) .$

Figure 4

Consequences of M on f ( η ) .

$Figure 5 Consequences of M on f ′ ( η ) f^{\prime} (\eta ) .$

Figure 5

Consequences of M on f ′ ( η ) .

$Figure 6 Consequences of P on f ( η ) f(\eta ) .$

Figure 6

Consequences of P on f ( η ) .

$Figure 7 Consequences of P on f ′ ( η ) f^{\prime} (\eta ) .$

Figure 7

Consequences of P on f ′ ( η ) .

$Figure 8 Consequences of P on θ ( η ) \theta (\eta ) .$

Figure 8

Consequences of P on θ ( η ) .

$Figure 9 Consequences of Nb on θ ( η ) \theta (\eta ) .$

Figure 9

Consequences of Nb on θ ( η ) .

$Figure 10 Consequences of Nt on θ ( η ) \theta (\eta ) .$

Figure 10

Consequences of Nt on θ ( η ) .

$Figure 11 Consequences of P on ϕ ( η ) \phi (\eta ) .$

Figure 11

Consequences of P on ϕ ( η ) .

$Figure 12 Consequences of Nb on ϕ ( η ) \phi (\eta ) .$

Figure 12

Consequences of Nb on ϕ ( η ) .

$Figure 13 Consequences of Nt on ϕ ( η ) \phi (\eta ) .$

Figure 13

Consequences of Nt on ϕ ( η ) .

Figure 14

Variation in Nu for viscosity parameter P and thermophoresis Nt.

Figure 15

Variation in Nu for viscosity parameter P and Brownian diffusion Nb.

$Figure 16 Variation in C f {C}_{\text{f}} for porosity factor λ \lambda .$

Figure 16

Variation in C f for porosity factor λ .

$Figure 17 Variation in C f {C}_{\text{f}} for magnetic parameter M.$

Figure 17

Variation in C f for magnetic parameter M.

$Figure 18 Consequences of Pr on θ ( η ) \theta (\eta ) .$

Figure 18

Consequences of Pr on θ ( η ) .

5 Conclusions

Here the impact of MHD and Darcy medium on nanofluid flow contained between two h − distance apart plates is taken into account. A viscous nanofluid saturates the given porous medium. The characteristics of Brownian diffusion and thermophoresis are disclosed. Numerical scheme using MATLAB is applied to solve the problems and acquire the data for velocity field, temperature, and concentration distributions. Salient features of the study are listed below:

Velocity field shows reduction for larger Forchheimer number. The drag force coefficient is responsible for this trend.
The impact of magnetic field is inversely related to the fluid flow. A decline is noticed in velocity profile.
For larger values of P, the inverse relation of kinematic viscosity confirms a decline in velocity field.
The non-predictive movement of nanoparticles because of the Brownian diffusion rises for stronger thermophoretic force, resulting in a more rapid transport from hot region to the colder region.
For elevated values of viscosity parameter, an enhancement in the concentration distribution is noticed.
The rate of heat flux reduces for Brownian diffusion and thermophoresis.
Skin-friction receives enhancement for elevated porosity factor and magnetic parameter.

Nomenclature

B 0: magnetic number
C f: skin-friction
D B: Brownian diffusion in m 2 s − 1
D T: thermophoretic diffusion in m 2 s − 1
h: distance between plates in m
M: non-dimensional magnetic number
MHD: magnetohydrodynamics
Nb: Brownian diffusion parameter
Nt: thermophoresis parameter
Nu: Nusselt number
P: viscosity parameter
Pr: Prandtl number
RK45: Runge–Kutta method
Sc: Schmidt number
T: temperature K⁻¹
u 1 = z x 1: velocity in ms⁻¹
u 1 , u 2: velocity components in ms⁻¹
x 1 , x 2: Cartesian coordinates in m
z: stretching rate in s − 1
α: thermal diffusivity in m 2 s − 1
ν: viscosity (kinematic) in m 2 s − 1
ρ: density in kg m − 3

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Received: 2020-07-11

Revised: 2020-09-29

Accepted: 2020-09-30

Published Online: 2020-12-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

MHD squeezed Darcy–Forchheimer nanofluid flow between two h–distance apart horizontal plates

Abstract

1 Introduction

2 Mathematical model

3 Numerical solution

4 Discussion

5 Conclusions

Nomenclature

References

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