Eur. Phys. J. Plus (2017) 132: 338 DOI 10.1140/epjp/i2017-11599-0
THE EUROPEAN PHYSICAL JOURNAL PLUS
Regular Article
Chemical reaction, thermal relaxation time and internal material parameter effects on MHD viscoelastic fluid with internal structure using the Cattaneo-Christov heat flux equation Sabeel M. Khana , M. Hammad, and D.A. Sunnyb Department of Applied Mathematics & Statistic, Institute of Space technology Islamabad, Islamabad 44000, Pakistan Received: 16 May 2017 / Revised: 15 June 2017 c Societ` Published online: 7 August 2017 – a Italiana di Fisica / Springer-Verlag 2017 Abstract. In this article, the influence of thermal relaxation time and chemical reaction is studied on the MHD upper-convected viscoelastic fluid with internal structure using the Cattaneo-Christov heat flux equation for the first time in the literature. The flow-governing equations are formulated and are converted into their respective ordinary differential equations (ODEs) with the application of similarity functions. The resulting system of coupled nonlinear ODEs is solved along with the prescribed conditions at boundary using a finite-difference code in MATLAB. Influence of chemical reaction, thermal relaxation time and internal material parameter on the macroscopic and micropolar velocities as well as on the temperature and concentration profiles is examined along with other physical parameters (e.g., magnetic parameter, Eckert number, Prandtl number and fluid relaxation time). The accuracy of the obtained numerical solution is shown by comparing the physical parameters of interest with particular cases of existing results in the literature.
1 Introduction The study of chemical reaction effects on a MHD viscoelastic fluid flow is of great significance due to its various industrial applications [1,2]. The major applications of such study are of interest in chemical process industry, where the different chemical reactions take place. It also has applications in many other processes, for instance manufacturing of ceramics, production of polymers and food processing. In the literature, considerable effort has been made in order to investigate the influence of chemical reactions on mass and heat transfer using different classes of fluids. For instance, Sandeep and Reddy [3] studied the solutal and thermal effects for the MHD non-Newtonian flow across a melting surface. Improvements were observed in heat transfer rates. Behavioral variations of slip conditions were analyzed by Fetecau et al. [4] on a free MHD flow with convection in an unsteady framework in the presence of radiation and mass diffusion using suitable perturbation technique. Time estimates for steady states were also discussed. Reddy [5] presented an analytical study on the influence of chemical reaction on the MHD flow in a medium with nonzero porosity and with heat source, mixed convection and Ohmic heating. Essawy et al. [6] discussed the uniform normal application of the inflow and external magnetic field to a permeable plate in a porous chemically reactive medium with mass and heat transfer in the magnetic hydrodynamic flow of a nanofluid by considering the Brownian and thermophoresis influences. Srinivasacharya and Reddy [7] studied radiation combined with chemical reaction effects in a power-law fluid through a Darcy porous medium with mixed convection along the plate placed vertically and showed increase in radiation, temperature and flow velocity but a decrease in concentration. Reddy and Gorla [8] used similarity transforms combined with the Runge-Kutta technique to conclude that flow velocity and temperature profile are directly proportional by considering the effect of buoyancy and MHD force on mass and heat transfer with convection in the presence of effects like chemical reaction and thermal radiation. Mulolani et al. [9] obtained solutions using similarities for naturally occurring convection in flows generated by virtue of a vertically moving plate of semiinfinite length containing concentrations along the wall of the plate in a steady laminar flow situation. Chamkha et al. [10] considered diffusive and naturally convective micro-rotational fluid flow along a vertical plate placed in the presence of chemical reaction effects along with inertial and thermal dispersion influence for the linear variation a b
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Eur. Phys. J. Plus (2017) 132: 338
of wall temperature and concentration. Usman et al. [11] considered the unsteady MHD flow along a plate placed with pores vertically to investigate the behavior of parameters governing the flow characteristics of an incompressible electrically conducted viscous fluid under the presence of chemical reaction, variable suction and thermal radiation. Inverse proportionality of the radiation for changing chemical parameters with heat and velocity was observed. Elkabeir et al. [12] analyzed chemical behavior of an electrically conducting incompressible micropolar fluid of Boussinesq type with viscosity, magnetic and thermal effects on a cone surface and showed inverse relationship between magnetic field and velocity along the boundary of the cone and direct relationship between magnetic effect and concentration and temperature. The two-dimensional chemical effect of mass and heat transfer along a stretchable sheet for an incompressible micropolar fluid, such that the chemical is considered homogenous is studied by Rawat et al. [13]. It was observed that the profile of the boundary layer has inverse proportionality to the thermal flux with the chemical reaction rate being constant. RamReddy et al. [14] applied the Lie group scaling on a free flow with convection over the plate placed in vertical direction with porosity for a micropolar fluid with internal structure to get a similarity transformed system such that the micro-rotation becomes insignificant as microelements are not significantly concentrated at the boundary. Electrically conducting fluid is taken into consideration by Rout et al. [15] with micropolar effects pertinent to the transverse magnetic field in order to investigate the chemical behavior on the boundary layer. Their results show that the chemical reaction is strong enough to reduce the effects of the magnetic field. Darbhashayanam et al. [16] discussed effects of the thermal radiation, free convection, chemical reaction and magnetic force for an incompressible micropolar fluid flow with mass and heat transfer along the vertical plate. They observed that with uniform concentration and temperature the mass transfer and heat transfer rate slows down because of microscopic motion of the fluid particles. El-Fayez [17] investigated the free convective fluid flow in the unsteady framework past a vertical moving plate having variable temperature under the chemical reaction effects. Ibrahim [18] studied thermal radiation and chemical reaction effects with Dufour and Soret effects in the case of MHD convective unsteady mass and heat transfer in a porous medium. Uddin and Kumar [19] investigated the influence of thermal radiation in a transient flow past a heated porous plate with chemical reaction effects under consideration. The effects of chemical reaction in the presence of suction, radiation and heat generation on the MHD unsteady flow passing by a vertical moving plate is also investigated by Ibrahim et al. [20]. In the analysis of boundary layer flows, almost no study is present in the literature, where the chemical reaction effects are studied along with thermal relaxation effects in the context of micropolar theory. Micropolar theories take into account the fluid particle micro-motions at the micro-scale and describe the deformation of the material at the continuum scale after bridging the motions of particle at two different length scales, i.e. the micro-scale and macroscopic continuum scale. These theories greatly emerge after the seminal work of Eringen [1,2] in the mid 1960’s. A detailed review on fluids with internal structure and its applications the reader is referred to the work of Lukaszewicz [9] and Ariman et al. [21,22]. So far, in the available literature, all present investigations on the chemical reaction effects in MHD viscoelastic fluid flow are based on the parabolic heat flux equation [10] in the thermal energy balance. To the best of the authors knowledge, not a single study is available which discusses the effects of chemical reaction in the present context by utilizing the more generic heat flux equation. Here, for the first time, we use the hyperbolic heat flux equation which is named after the seminal work by Cattaneo [23] and Christov [24] as the Cattaneo-Christov heat flux equation, in the literature. Here, we incorporate the thermal relaxation time effects by applying the more generic form of the heat flux equation, known as Cattaneo-Christove heat flux equation, to the thermal balance energy considering the chemical reaction effects within the framework of non-classical continuum theory (refs. [1,2,25] and references therein). This allows us to examine the influence of thermal relaxation time in the analysis of the boundary layer flow in the viscoelastic UCM fluid with chemical reaction. The effects of distinct physical parameters of interest on the development of boundary layer are studied. Moreover, obtained computational results are compared with the existing results in special cases available in the literature. The rest of the article is organized as follows: In sect. 2, the governing flow equations are stated describing the physical problem. In sect. 3, the numerical results obtained by simulations using a finite-difference procedure [26] are provided and discussed for various different physical parameters of interest in both classical and non-classical fluid flow cases. Based on the results obtained and discussion from sect. 3, finally in sect. 4, conclusions are drawn.
2 Problem formulations Let us consider a laminar, incompressible, MHD viscoelastic upper-convected Maxwell (UCM) fluid flow with internal structure and heat transfer in two dimensions. The fluid is flowing over a semi-infinite surface, which is stretching on the plane y = 0. We further assume Tw be the constant plate temperature, with T∞ being the ambient fluid temperature. The plate has a constant temperature Cw , with C∞ being the ambient temperature of the fluid. A magnetic field B0 of constant strength is introduced along the vertical y-axis. Now, taking the above assumptions into consideration and
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following [16] one can write the flow-governing equations for the MHD UCM fluid [7,15,16,25] with internal structure in the absence of pressure gradient and viscous dissipation in the incompressible limit case as follows: ∂v ∂u =− , (1) ∂x ∂y 2 k ∂ u σB02 ∂u ∂v ∂ ∂ ∂u ∂u k ∂N · u +v = ν+ − , (2) u +v −λ1 u +v u+gT ∗ (T − T∞ )+gC ∗ (C −C∞ )+ 2 ∂x ∂y ∂x ∂y ∂x ∂y ρ ∂ y ρ ρ ∂y k ∂u γ∗ ∂2N ∂N ∂N +v + 2N + = u , (3) ∂x ∂y ρj ∂y ρj ∂y 2 ∂C ∂C ∂2C +v + γ1 (C − C∞ ) = D 2 u (4) ∂x ∂y ∂y ∂T ∂T +v = −∇ · q + σB02 u2 , (5) ρCp u ∂x ∂y where u, v, N , T , λ1 , ν, ρ, k, σ, γ ∗ , γ1 , C, j, and Cp represents the horizontal velocity field component, vertical velocity field component, micro-rotation component (or angular velocity component), temperature, relaxation time associated with fluid, kinematic viscosity, fluid density, vortex viscosity, electric conductivity, viscosity of spin gradient, solutal parameter, chemical concentration, micro-inertia per unit mass and the specific heat, respectively. Let us consider the definition of heat flux q by Cattaneo [23] as follows: λ2 ∂q q − q · ∇V + ∇ · V q + V · ∇q . ∇T = − − (6) κ κ ∂t Here V = V (u, v), and λ2 represents the velocity vector and the relaxation time parameter associated with heat flux, respectively. After following the work of Christov in [24] and using eq. (5) and (6), the heat flux q can be eliminated from eq. (5); thus the resulting heat transfer equation now becomes hyperbolic in nature and takes the form 2 2 ∂v ∂T ∂2T ∂T ∂T ∂u ∂u ∂T ∂v ∂2T σB02 2 2∂ T 2∂ T +v + λ2 +v + u +v +u = α u u + v + 2u v + u . ∂x ∂y ∂x ∂y ∂x ∂x ∂y ∂y ∂x2 ∂y 2 ∂x∂y ∂y 2 ρcp (7) Thus the set of eqs. (1)–(3) and eq. (7) govern the MHD incompressible UCM-micropolar flow of fluid and heat transfer in connection with the conditions at the boundary u(0, x) = U, u(∞, x) → 0,
v(0, x) = 0,
C(0, x) = Cw ,
C(∞, x) → C∞ ,
T (0, x) = Tw ,
T (∞, x) → T∞ ,
and
N (0, x) = −N0
∂u , ∂y
and N (∞, x) → 0.
The shear stress τw , and the couple stress σw at the surface are, respectively, defined by ∂u ∂u + kN + kN τw = (μ + k) and σw = (μ + k) . ∂y ∂y y=0 y=0 The heat flux qw through the plate wall and the local mass flux mw are given as
∂T ∂C qw = − k and mw = −D . ∂y y=0 ∂y y=0
(8)
(9)
(10)
The skin-friction coefficient Cf x , local Nusselt number N ux , and Sherwood number Shx , which are the parameters of physical importance, are thus calculated by the following relations: Cf x =
τw , ρU 2
N ux =
xqw k(Tw − T∞ )
and
Shx =
xmw . D(Cw − C∞ )
The following von K´ arm´an-type transformations [13] are used to get the similarity functions: T − T∞ 1 Uν U y, (f − ηf ) , u = U f (η), θ(η) = , v=− η= νx Tw − T∞ 2 x C − C∞ U g(η), φ(η) = N =U . νx Cw − C∞
(11)
(12)
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By using eqs. (9)–(12), the following relations are established for local Nusselt number N ux , skin-friction coefficient Cf x and mass transfer rate Sh N ux = −
θ (0) Re−1/2 x
f (0) Cf x = (1 + Δ) √ Rex
,
and Sh = −
φ (0) Re−1/2 x
,
(13)
0 where Δ = k( 1−N μ ) is the material parameter representing the ratio of vortex to dynamic viscosity. Using transformations from eq. (12) into eqs. (1)–(4) and eq. (7) we arrive at the following set of nonlinear coupled ODEs: 1 β 2 β 2 − βf f f − ηf f − M f + Grθ + Gmφ + Kg = 0 (14) 1+K − f f + 2 2 2 1 1 γ θ + fθ − 3f f θ + f 2 θ + M Ecf 2 = 0 (15) Pr 2 2 (2 + K)g + (gf + f g ) − β ∗ (2g + f ) = 0 (16) Sc f φ = 0. (17) φ − Scγ1 φ + 2
Also the conditions in eq. (8) take the form f (η) = 1, and
f (η) = 0,
f (η) = 0,
θ(η) = 1,
φ(η) = 1,
g(η) = −N0 f (η) at
g(η) = 0,
θ(η) = 0,
φ(η) = 0,
at
η = 0,
η → ∞.
(18)
In eqs. (14)–(17) the dimensionless parameters β is fluid Deborah number, M denotes the magnetic parameter, Pr represents the Prandtl number, Rd stands for the radiation parameter, γ is the Deborah number associated with temperature, K is the micropolar constant, Gr and Gm represents thermal and solutal buoyancy parameter, respectively, Sc denotes Schmidt number and β ∗ is the micro-inertia coupling parameter and are defined as β=
λ1 U , 2x
M=
σB02 x , ρU
Pr =
ν μcp = , α k
γ=
λ2 U , 2x
K=
k , μ
β∗ =
kλ1 ν , jβ
Sc =
ν , D
γ1 =
kl x . U
(19)
3 Numerical results and discussion Numerical results are presented and discussed in this section, which are obtained by solving the coupled nonlinear flow problem described by nonlinear differential equations (14)–(17), pertinent to boundary conditions in eq. (18). In this respect, we denote, here, f by x1 , θ with x4 , g by x6 , and φ with x8 . This allows rewriting the set of ODEs in eqs. (14)–(17) equivalently in the following system of first-order nonlinear coupled differential equations: ⎧ ⎪ x1 = x2 , ⎪ ⎪ ⎪ ⎪ ⎪ x2 = x3 , ⎪ ⎪ ⎪ ⎪ 2 ⎪ 1 ⎪x = ⎪ β ηx2 x3 + 2x1 x2 x3 − x1 x3 + 2M x2 − 2Kx7 − Grx4 − Gmx8 , ⎪ 3 2 ⎪ (2 + 2K − βx1 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x4 = x5 , ⎪ ⎪ ⎪ ⎪ Pr ⎨ 3γx1 x2 x5 − x1 x5 − 2M Ecx22 , x5 = 2 (20) 2 − Pr γx1 ⎪ ⎪ ⎪ ⎪ ⎪ x6 = x7 , ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ x7 = {(β ∗ (2x6 + x3 ) − (x6 x2 + x1 x7 ))} , ⎪ ⎪ ⎪ 2 + K ⎪ ⎪ ⎪ ⎪ ⎪ x8 = x9 , ⎪ ⎪ ⎪ ⎪ ⎪ Sc ⎪ ⎩x9 = Scγ1 x8 − x1 x9 , 2 subjected to the boundary condition x1 (0) = 0, x2 (0) = 1, x3 (0) = p, x4 (0) = 1, x5 (0) = q, x6 (0) = 1, y7 (0) = r, x8 (0) = 1, and x9 (0) = s. Now, starting with some suitable initial choices for p, q, r, and s, the finite-difference–based collocation method implemented in MATLAB with function bvp4c [27] is utilized to solve the IVP in eq. (20). Instead
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Table 1. A comparison of the Nusselt number N ux Rex2 by the present method with those in the literature when K = 0. −θ (0) Salleh et al. [28] Pr
Qasim et al. [29]
Present
Keller-box method
RK-4
RK-5
bvp4c [26]
–
0.46317
0.46360
0.47101994484306
1
0.58202
0.58198
0.58202
0.58386578477887
3
1.16525
1.16522
1.16525
1.16501916594711
5
1.56805
1.56806
1.56805
1.56787143176487
7
1.89540
1.89548
1.89542
1.89523965855700
10
2.30800
2.30821
2.30800
2.30785450232703
100
7.76565
7.76249
7.75826
7.76553982755546
0.72
Table 2. For Pr = 100, γ1 = 0.5, β = 0.1, Gr = 0.1, Gm = 0.1, K = 1, and β ∗ = 0.5. using bvp4c [27]. −θ (0)
−f (0)
−φ (0)
5.5440101346727
0.314659521675166
0.385958959400054
0.05
5.42478915082161
0.345955264608128
0.384626912597721
0.1
5.31055617289054
0.375474106567454
0.383395522470885
1
8.78177656362278
0.315259573094346
0.385955750779464
2
10.9403900690356
0.383614128099016
0.383102633328462
3
12.9254217904409
0.383871886943333
0.383102132051892
0.2
12.7426468798878
0.383550832577042
0.383117717088527
0.4
12.3770004428913
0.382908478104254
0.383148893826986
0.8
11.6453142347209
0.381622985461411
0.383211261913557
7.01879094501818
0.385085141750695
0.451147277684931
0.5
7.02381528366018
0.39184797873891
0.590107548516742
0.7
7.31032297891396
0.394084983070013
0.649273374274791
Sc
γ
M
Ec
0.22
0
0.01
0.1
0.3
1
of infinite domain [0, ∞) of the problem, a finite domain [0, 12] is chosen for consideration after a certain number of computational experiments with varying values of non-dimensional physical parameters. Macroscopic velocity, concentration, micro-rotation and the temperature profiles computed are shown graphically in both Cauchy and non-classical continuum cases. Influence of different dimensionless physical parameters, for instance, thermal relaxation time parameter, solutal parameter, micro-structural parameter, magnetic number, Schmidt number, Prandtl number, Eckert number, fluid and thermal Deborah numbers are calculated for both the Cauchy and micropolar cases of the fluid flow. In all these numerical simulations, the numerical values for M , β, Pr, Ec, γ, K, β ∗ , γ1 , Sc, Gr and Gm are taken to be 0.01, 0.1, 100, 0.1, 0.0, 0, 0.5, 0.5, 0.25, 0.5, 0.1 and 0.1 unless otherwise specified. To show the accuracy of our calculated numerical results, a comparison for the numerical values of Nusselt number is tabulated in table 1 with varying values of the Prandtl number with the results available in the literature for a special case of our study. It is shown that our results are in strong agreement with the results obtained by Salleh et al. [28] using the Keller-box method and those by Qasim et al. [29] using the Runge-Kutta methods (RK-4 and RK-5), in the available special case where the flow is assumed to be a classical fluid flow. In table 2, the effects of different physical parameters, for instance, Schmidt number, Deborah number associated with temperature, magnetic parameter and Eckert numbers on the heat transfer rate, skin-friction coefficient and mass transfer rates, are investigated. It is observed that an increase in the Schmidt number causes a rise in the heat transfer rate. The skin-friction coefficient also increases with an increment in the Schmidt number. Moreover, the rate of mass transfer also rises with a rise in the Schmidt number. This enhancement of the Nusselt number, skin-friction coefficient and couple stress in the case of a non-classical fluid flow is in accordance with the observation of Rout et al. [15], where they studied the
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Eur. Phys. J. Plus (2017) 132: 338 1.2 For Sc = 0.22, and 1 = 0.00
1
For Sc = 0.22, and 1 = 0.25 For Sc = 0.22, and 1 = 0.50 For Sc = 0.50, and 1 = 0.00
0.8
For Sc = 0.50, and = 0.25 1
For Sc = 0.50, and = 0.50 1
()
0.6 Sc = 0.22 0.4
0.2
0 Sc = 0.5 −0.2 0
2
4
6
8
10
12
Fig. 1. Profiles of concentration for fixed Sc values and varying γ1 .
phenomenon by using the parabolic (Fourier’s [30]) heat flux model, which is different in our case. We observe these effects by utilizing a more generic and effective model of heat conduction called the hyperbolic (Cattaneo-Christov) heat flux model. The Deborah number associated with temperature increases by increasing the thermal relaxation time parameter and effect of rising thermal relaxation time is shown. It is found that with an increase in the thermal relaxation time, the skin-friction coefficient and Nusselt number increase. However, a decrement is found in the mass transfer rate with increasing thermal relaxation time. A variation in the magnetic and Eckert number effect the mass transfer rate, skin-friction coefficient and heat transfer rate. With an increase in the magnetic parameter a decrease in heat and mass transfer rate is found, whereas the skin-friction coefficient increases with incrementing the magnetic number. The heat transfer rate and coefficient of skin friction are decreasing by increasing the Eckert number, while the mass transfer rate, on the other hand, increases. In fig. 1 the profiles of concentration are shown for varying values of the solutal parameter and Schmidt number in the case of a non-classical fluid flow. It is seen that in order to control the growth of the solutal boundary layer, a higher solutal stratification is required. This result is in accordance with the observation of [15]. Moreover, it is observed from fig. 1 that the solutal boundary layer thickness is higher for smaller values of Schmidt number in the case of micropolar fluid flow. In fig. 2 it is seen that the concentration boundary layer distribution is affected by variation of the Schmidt number. Also, it is observed that in the case of heavier species, the concentration boundary layer is lower. Thus by increasing the Schmidt number the solutal boundary layer decreases in thickness for both cases of Cauchy/classical continuum flow and in non-classical/micropolar fluid flow. This observation is in agreement with the observation of Mishra et al. [31]. In addition to this, the solutal boundary layer is significantly lower in the case of micropolar fluid flow than it is in the case of Cauchy fluid flow. In fig. 3 the development of velocity boundary layer is depicted with varying values of the Schmidt number in both cases: the classical continuum flow and the micropolar continuum fluid flow. It is found that the thickness of the velocity boundary layer is decreasing in both classical and micorpolar fluid flow cases with increasing values of the Schmidt number. Moreover, the velocity boundary layer thickness is lower in the case of classical continuum fluid flow as compared to the case of non-classical continuum flow. Figure 4 exhibits the micro-rotational velocity profiles for varying values of the Schmidt number in the classical as well as in the non-classical fluid flow case. It is observed that the micro-rotations of the fluid particles are increasing by increasing the Schmidt number near the plate surface, while away from the plate surface, the fluid particles show opposite behavior. The micro-rotational velocity of the fluid particles away from the surface decreases by increasing the Schmidt number. Thus the variation in the Schmidt number introduces a micro-rotational shear band in the fluid medium, where on its two sides, the fluid particles experience counter-rotational effect.
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1.2 Classical continuum case, K=0, Sc = 0.25
1
Micropolar continuum case, K = 1, Sc = 0.25 Classical continuum case, K=0, Sc = 0.5 Micropolar continuum case, K = 1, Sc = 0.5
0.8
Classical continuum case, K=0, Sc = 1.0 Micropolar continuum case, K = 1, Sc = 1.0 ()
0.6
0.4
Sc = 0.25, 0.5, 1.0
0.2
0
−0.2 0
2
4
6
8
10
12
Fig. 2. Concentration profiles with varying of Sc in case of classical and micropolar flow. 1
For Sc= 0.25, Classical continuum case
0.8
For Sc= 0.25,Micropolar continuum case For Sc= 0.5, Classical continuum case 0.6
For Sc= 0.5,Micropolar continuum case
f ’ ()
For Sc= 1.0, Classical continuum case For Sc= 1.0,Micropolar continuum case 0.4 Sc = 0.25, 0.5, 1.0
0.2
0 0
2
4
6
8
10
12
Fig. 3. Velocity profiles with varying of Sc in the case of classical and micropolar flow.
In fig. 5 thermal relaxation time effects are depicted on the development of temperature boundary layer. The temperature boundary layers distribution is shown for varying Schmidt number and the Deborah number associated with thermal relaxation in the case of non-classical continuum flow. It is seen that for lower values of the Schmidt number the thickness of temperature boundary layer decreases by increasing the thermal relaxation time parameter near the surface. However for higher values of the Schmidt number the temperature boundary layer thickness increases near the surface with increasing thermal relaxation time parameter. Contrary to this, away from the plate surface, the temperature boundary layer thickness increases for lower values of the Schmidt number and by increasing thermal relaxation time. However, for larger values of Schmidt number the behavior of the thermal boundary layer is opposite. In this case, a decrement in the thickness of the thermal boundary layer is observed with an increasing value of thermal time relaxation.
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Eur. Phys. J. Plus (2017) 132: 338 0.3 0.028
0.25
Classical continuum case, for Sc = 0.25
Sc = 0.25, 0.5, 1.0
0.026
Micropolar continuum case, for Sc = 0.25
0.024
Sc = 0.25, 0.5, 1.0
0.2
Classical continuum case, for Sc = 0.50
0.022
Micropolar continuum case, for Sc = 0.50
0.02 0.018
0.15
Classical continuum case, for Sc = 1.0
Sc = 0.25, 0.5, 1.0
g()
0.016
Micropolar continuum case, for Sc = 1.0
0.014 0.012
0.1
0.01
4.4
4.5
4.6
4.7
4.8
4.9
5
5.1
0.05
0
−0.05 0
2
4
6
8
10
12
Fig. 4. Micro-rotation profiles with varying of Sc in the case of classical and micropolar flow.
0.9
= 0, Sc = 0.22, K=1
for Sc = 0.22
0.1
= 1, Sc = 0.22, K=1
0.8
()
= 2, Sc = 0.22, K=1
0.08
0.7 0.6
0.06
0.5
0.04
0.4
= 0, Sc = 0.50, K=1 = 1, Sc = 0.50, K=1 = 2, Sc = 0.50, K=1 for Sc = 0.50
0.02
for Sc = 0.50
0.3 0 0.2 for Sc = 0.22
−0.02 0
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 −0.1 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig. 5. Temperature profiles for fixed Sc and varying temperature relaxation time for the case of micropolar flow.
In fig. 6 the influence of thermal relaxation time is illustrated on the micro-rotational velocity profiles for the case of micropolar fluid particles. It is observed that the micro-rotations of the fluid particles increase with an increase in the thermal relaxation time effect. In fig. 7 the effect of solutal variation is depicted on the micro-rotations of the continuum fluid particles for two distinct values of the Schmidt number. It is observed that the micro-rotations of the micropolar fluid particles near the surface increase with an increase in the chemical reaction parameter. While, on the other hand, away from the surface the fluid, particles behave differently. The micro-rotations of the fluid particles away for the surface exhibit decreasing velocities by incrementing the chemical reaction parameter. Thus as a consequence the micro-rotational boundary layer thickness increases near the plate surface and decreases away from the plate surface for an increasing value of γ1 . Moreover, it is observed that for smaller values of the Schmidt number the micro-rotational boundary layer is lower near the plate surface and higher away from the plate surface in comparison to large values of the Schmidt number. In fig. 8 the velocity boundary layers are depicted for varying values of solutal parameter and for two distinct values of the Schmidt number in the case of non-classical continuum flow.
Eur. Phys. J. Plus (2017) 132: 338
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0.159
0.14
= 0, Sc = 0.22, K=1
0.1585
0.12
= 1, Sc = 0.22, K=1 0.158
0.1
= 2, Sc = 0.22, K=1
= 0, 1, 2 0.1575
g ()
0.08
0.157 0.06 0.1565 0.05
0.04
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.02 0 −0.02 0
2
4
6
8
10
12
Fig. 6. Concentration profiles varying temperature relaxation time for the case of micropolar flow.
0.16
For = 0.15, Sc = 0.22, Micropolar continuum case 1
For 1 = 0.25, Sc = 0.22, Micropolar continuum case
0.14 0.12
g ()
0.1
For 1 = 1.0, Sc = 0.22, Micropolar continuum case For = 2.0, Sc = 0.22, Micropolar continuum case 1
0.17
For 1 = 0.15, Sc = 0.5, Micropolar continuum case
1 = 0.15, 0.25, 0.5, 1.0
For 1 = 0.25, Sc = 0.5, Micropolar continuum case For 1 = 0.5, Sc = 0.5, Micropolar continuum case
0.165
0.08
For 1 = 1.0, Sc = 0.5, Micropolar continuum case
−3
4.6 0.16
x 10
1 = 0.15, 0.25, 0.5, 1.0, 2.0
4.4
0.06
4.2
0.04
4
0.155
3.8
0.02
3.6 0.15
= 0.15, 0.25, 1.0, 2.0
3.4
1
0 0
0
0.2
0.4
0.6
2
0.8
1
7.5
4
7.51
7.52
6
7.53
7.54
7.55
7.56
7.57
8
7.58
7.59
7.6
10
12
Fig. 7. Micro-rotation profiles for fixed Sc and varying values of values of γ1 in case of micropolar flow.
It is found that the thickness of the velocity boundary layer decreases with an increasing value of the solutal parameter. Moreover, the velocity boundary layer profiles in the case of lower Schmidt number are higher than in the case of large Schmidt number for the fixed values of solutal parameters.
4 Conclusions Here, we study the influence of thermal relaxation time and chemical reaction on MHD upper-convected viscoelastic fluid with internal structure using Cattaneo-Christov heat flux equation for the first time in the literature. The resulting system of PDEs is transformed into their respective system of ODEs by suitable similarity transformations. The obtained boundary value problem is implemented in MATLAB using the finite-difference approach [27]. The effects of thermal relaxation time along with many other dimensionless physical parameters are shown on the development of velocity, angular velocity, solutal bounday and thermal boundary layer. The accuracy of the calculated numerical values is shown by comparing the mass transfer rate with the results available in the literature for a special case of our study. Some interesting findings of this investigation are summarized.
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Eur. Phys. J. Plus (2017) 132: 338 1.2 = 0.15, Sc = 0.22, K=1
= 0.15, 1.0, 2.0
0.16 1
1
1
= 1.0, Sc = 0.22, K = 1
0.15
1
= 2.0, Sc = 0.22, K = 1
0.14 0.8
1
= 0.15, Sc = 0.5, K = 1 1
0.13
1 = 1.0, Sc = 0.5, K = 1
0.12 0.11
0.4
0.09
f ’ ()
0.6
1 = 2.0, Sc = 0.5, K = 1
0.1
0.08 3.7
0.2
3.8
3.9
4
4.1
4.2
4.3
0
−0.2 0
2
4
6
8
10
12
Fig. 8. Velocity profiles for fixed Sc and varying values of values of γ1 in case of micropolar flow.
1) An interesting and new feature of the flow is observed with the variation of the Schmidt number. It introduces a micro-rotational shear band (a finite width of large shearing deformation) in the fluid medium where on the two sides of which the fluid particles experiences counter-rotational effect. 2) Thermal relaxation time increases the skin-friction coefficient and the Nusselt number for its larger values. However, the mass transfer rate decreases by incrementing thermal relaxation time. 3) The micro-rotations of the fluid particles near the plate surface are increasing with incrementing the chemical reaction parameter while, away from the plate surface fluid, particles experiences less micro-rotation with an increasing value of the chemical reaction parameter. 4) It is observed that, in order to control the growth of the solutal boundary layer, a higher solutal stratification is required. This result is in accordance with the observation of [15]. 5) An increment in the Schmidt number causes a rise in the heat transfer rate, skin-friction coefficient and couple stress in the case of non-clasical fluid flow. This observation, in the case of hyperbolic (Cattaneo-Christov) heat flux model, is in accordance with the observations of Rout et al. [15] where they utilized parabolic (Fourier’s) heat flux model. 6) The mass and heat transfer rates are found to decrease, whereas the coefficient of skin friction increases by increasing the magnetic number. The coefficient of skin friction and heat transfer rate decrease with an increment in the Eckert number while the mass transfer rate increases. 7) For lower values of the Schmidt number the thickness of the thermal boundary layer decreases by increasing value of the thermal relaxation time near the plate surface, whereas, for larger Schmidt number, the thickness of temperature boundary layer increases by increasing the thermal relaxation time. 8) The fluid particle experiences large micro-rotations with larger values of thermal relaxation time. This is because with the larger value of thermal relaxation time the temperature of the particle increases and they become lighter and therefore experiences large micro-rotations. This research work is partially supported under the startup research grant scheme with grant 21-557/SRGP/R&D/HEC/2014 by HEC Pakistan. The authors are also thankful to the anonymous reviewers for their constructive comments and suggestions in order to improve the quality of this paper.
Eur. Phys. J. Plus (2017) 132: 338
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