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S0997-7546(15)30262-4 http://dx.doi.org/10.1016/j.euromechflu.2016.04.003 EJMFLU 3005

To appear in:

European Journal of Mechanics B/Fluids

Received date: 2 October 2015 Revised date: 14 March 2016 Accepted date: 8 April 2016 Please cite this article as: J.C. Umavathi, M.A. Sheremet, S. Mohiuddin, Combined effect of variable viscosity and thermal conductivity on mixed convection flow of a viscous fluid in a vertical channel in the presence of first order chemical reaction, European Journal of Mechanics B/Fluids (2016), http://dx.doi.org/10.1016/j.euromechflu.2016.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Combined effect of variable viscosity and thermal conductivity on mixed convection flow of a viscous fluid in a vertical channel in the presence of first order chemical reaction J.C. Umavathi1,2, M.A. Sheremet3,4* and Syed Mohiuddin1 1

Department of Mathematics, Gulbarga University, Gulbarga, Karnataka, India

2

Department of Engineering, University of Sannio, Piazza Roma 21, 82100 Benevento, Italy

3

Department of Theoretical Mechanics, Tomsk State University, 634050 Tomsk, Russia

4

Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, 634050

Tomsk, Russia *Corresponding author, telephone/fax number: +73822529740, e-mail address: [email protected] (Mikhail A. Sheremet)

Abstract An analysis has been carried out to obtain the flow, heat and mass transfer characteristics of a viscous fluid having temperature dependent viscosity and thermal conductivity in a vertical channel. The energy equation accounts for viscous dissipation, while the first order homogeneous chemical reaction between the fluid and diffusing species is included in the mass diffusion equation. The walls of the channel are maintained at constant but different temperatures. The non-dimensional coupled nonlinear ordinary differential equations are solved analytically using perturbation method and numerically using Runge-Kutta shooting method. The velocity, temperature and concentration distributions are obtained numerically and presented through graphs. Skin friction coefficient and Nusselt number at the walls of the channel are derived and discussed and their numerical values for various values of physical parameters are presented through tables. Keywords: Mixed convection, variable viscosity, variable thermal conductivity, viscous dissipation.

1

Nomenclature a

empirical constant for the viscosity

b

characteristic length of the channel

b%

empirical constant for the thermal conductivity

bv

viscosity variation parameter

bk

conductivity variation parameter

Br

Brinkman number

C1 , C2 concentrations along the walls C0

reference species concentration

C

species concentration of the fluid

D

effective diffusion coefficient

GRT

modified thermal Grashof number

GrT

thermal Grashof number

GRC

modified mass Grashof number

GrC

mass Grashof number

Re

Reynolds number

g

acceleration due to gravity

K

thermal conductivity of the fluid

K0

thermal conductivity at temperature T0

m

wall temperature ratio

n

wall concentration ratio

T0

reference temperature

T

fluid temperature

T1 , T2

wall temperatures

U

velocity

u

dimensionless velocity

u

mean velocity

Y

dimensional coordinate axis 2

y

dimensionless coordinate axis

Greek letters α

dimensionless chemical reaction parameter

βT

coefficient of thermal expansion

βC

concentration expansion coefficient

γ

chemical reaction parameter

θ

dimensionless temperature

μ

dynamic viscosity

μ0

dynamic viscosity at temperature T0

ν

kinematic viscosity

ρ

density of the fluid

ρ0

static density

φ

rescaled species concentration

τ 1 ,τ 2 skin friction ΔC

concentration difference

ΔT

temperature difference

1. Introduction The natural convection process is present in various physical phenomena such as fire engineering, combustion modeling, nuclear energy, heat exchangers, petroleum reservoir etc. Transport phenomena involving the combined influence of thermal and concentration buoyancy are often encountered in many engineering systems and natural environments. There are many applications of such transport processes in industry, notable in chemical distilleries, heat exchangers, solar energy collectors and thermal protection systems. In all such cases of flows, the driving force is provided by a combination of thermal and chemical diffusion effects. In atmosphere flows, thermal convection of the earth by sunlight is affected by differences in water vapour concentration. This buoyancy driven convection due to a coupled heat and mass transfer has also many important applications in energy related engineering. Convection flow driven by temperature and concentration has been studied extensively in the past and various extensions of 3

the problems have been reported in the literature (see Bejan [1], Jang and Yan [2], Umavathi [3] and several references therein). Combined heat and mass transfer problems with chemical reaction are of importance in many processes of industrial applications such as the polymer production and manufacturing of ceramics or glass ware. There are two types of reactions such as (i) homogeneous reaction and (ii) heterogeneous reaction. A homogeneous reaction occurs uniformly throughout the given phase, whereas heterogeneous reaction takes place in a restricted region or within the boundary of a phase. A chemical reaction is said to be first-order, if the rate of reaction is directly proportional to concentration itself. In many industrial processes, the diffusing species can be generated or absorbed into the ambient fluid due to different types of chemical reactions in heat and mass transfer over a moving surface which can greatly affect the properties and quality of the finished products. The combined effects of thermal and mass diffusion in channel flows have been studied by Nelson and Wood [4] and Rajasekhar et al. [5]. Das et al. [6] have studied the effect of mass transfer on the flow started impulsively past an infinite vertical plate in the presence of wall heat flux and chemical reaction. Muthucumaraswamy and Ganeshan [7] have studied the impulsive motion of a vertical plate with heat flux/mass flux/suction and diffusion of chemically reactive species. Seddeek [8] have studied the finite element method for the effect of chemical reaction, variable viscosity, thermophoresis and heat generation/absorption on a boundary layer hydro magnetic flow with heat and mass transfer over a heated surface. Muthuraj and Srinivas [9] investigated the problem of mixed convection of heat and mass transfer through a vertical wavy channel with porous medium. Kumar et al. [10, 11] studied the homogeneous and heterogeneous chemical reactions of immiscible fluids. Patil et al. [12] studied numerically, effects of chemical reaction on mixed convection flow of a polar fluid in a porous medium in the presence of internal heat generation. Pal and Talukdar [13] analytically studied the problem of unsteady hydromagnetic heat and mass transfer for a micropolar fluid bounded by semi-infinite vertical permeable plate in the presence of first-order chemical reaction, thermal radiation and heat absorption. Recently Kumar et al. [14] studied the heat and mass transfer between vertical infinite parallel plates in the presence of first-order chemical reaction. In the above mentioned studies, the researchers have considered constant physical properties. However, it is known that the physical properties of fluid may change significantly 4

with temperature (Herwig and Gersten [15], Lai and Kulacki [16] and Pop et al. [17], Chiam [18], Abel et al. [19]). The variations of properties with temperature has several practical applications in the field of metallurgy and chemical engineering; in the extrusion process, the heat-treated materials traveling between a feed roll and wind-up roll or on conveyor belt possess the feature of a moving continuous surface. The increase of temperature leads to a local increase in the transport phenomena by reducing the viscosity across the momentum boundary layer and so rate of heat transfer at the wall is also affected. Therefore, to predict the flow behaviour accurately it is necessary to take into account the viscosity variation for incompressible fluids. Gary et al. [20] and Mehta and Sood [21] showed that, when this effect is included the flow characteristics may change substantially compared to constant viscosity assumption. For lubricating fluids heat generated by internal friction and the corresponding rise in the temperature affects the viscosity of the fluid and so that the fluid viscosity no longer be assumed constant. Mukhopadhyay et al. [22] investigated the MHD boundary layer flow with variable fluid viscosity over a heated stretching sheet. Li et al. [23] investigated the variable properties effects on three-dimensional liquid flow in a rectangular channel. The authors investigated the hydrodynamically fully developed, thermally developing liquid flow by neglecting viscous dissipation effects. The obtained results showed that temperature dependent properties significantly affect the heat transfer characteristics of the flow. Another study that investigated the temperature variable properties in liquid microscale flow was performed by Gulhane and Mahulikar [24]. The authors mainly focused on incompressible, laminar, and axisymmetric flow. Both thermally developing and simultaneously developing flow conditions were examined for constant wall heat flux and constant wall temperature boundary conditions. The obtained results showed that thermal conductivity significantly affects the flow and temperature fields. Zamora and Hernández [25] investigated the influence of variable properties on the laminar air flow induced by natural convection in a vertical asymmetrically-heated channel. They presented the detailed analysis of how variable properties effect on the induced mass flow rate and the size of the recirculation region in a channel formed by isothermal and adiabatic plates. Hernández and Zamora [26] extended their analysis to observe the effects of variable properties considering non-uniform heating of the plates and also for non-Boussinesq conditions. 5

The viscous dissipation effect, which is a local production of thermal energy through the mechanism of viscous stresses, is a ubiquitous phenomenon and it is encountered in viscous fluid flow. When compared with other thermal influences on fluid motion the effect of heat released by viscous dissipation covers a wide range of magnitudes from being negligible to being significant. Gebhart [27] discussed this range at length and stated that “a significant viscous dissipation may occur in natural convection in various devices which are subject to large decelerations or which operate at high rotational speeds. In addition, important viscous dissipation effects may also be present in stronger gravitational fields and in processes wherein the scale of the process is very large, e.g., on larger planets, in large masses of gas in space, and in geological processes in fluids internal to various bodies.” Nield et al. [28] studied the effect of fully developed forced convection in a parallel plate channel filled by a saturated porous medium, with walls held either at uniform temperature or at uniform heat flux, with the effects of viscous dissipation and flow work. Barletta and Nield [29] studied mixed convection with viscous dissipation and pressure work in a lid-driven square enclosure. Recently, Pinarbasi et al. [30] investigated the effect of variable thermal conductivity and variable viscosity on the flow characteristics and flow rates of an inelastic fluid, including viscous heating. Therefore keeping in view the effects of variable viscosity on the flow, in this problem it is assumed that the viscosity and thermal conductivity of the fluid are exponential functions of the temperature. The analytical solutions are obtained using perturbation method which has restrictions of the perturbation parameter and this restriction is relaxed by solving the governing equations numerically using Runge-Kutta method with shooting technique.

2. Mathematical formulation Consider a steady laminar, fully developed flow of an incompressible viscous fluid between two parallel plates. The distance between the plates is 2b and the origin of coordinate axis is located in the mid-plane of the channel as shown in Fig 1. The two plates are kept at two constant temperatures T1 for the left plate and T2 for the right plate. The channel is assumed to occupy the region of space −b ≤ Y ≤ b . A fluid rises in the channel driven by buoyancy forces. The no-slip boundary condition is imposed on the parallel plates for the velocity, and since the

6

plates are infinite in the X-direction, the physical variables are invariant in these directions and the problem is essentially one-dimensional with velocity component U (Y ) along the X-axis.

Figure 1. Physical configuration The physical properties characterizing the fluid except density, viscosity and thermal conductivity are assumed to be constant. As customary, the Boussinesq approximation and the equation of state

ρ = ρ0 ⎡⎣1 − βT (T − T0 ) − βC ( C − C0 ) ⎤⎦

(1)

will be adopted. The flow and heat transfer of viscous fluid is examined considering the viscosity and thermal conductivity dependent on the temperature. The momentum equation, the energy balance equation and mass transfer equations governing the motion of an incompressible fluid in the presence of viscous dissipation with the variable viscosity and variable thermal conductivity are given by ∂p ⎞ =0 ⎟ + ρ0 g βT (T − T0 ) + ρ0 g βC ( C − C0 ) − ∂X ⎠

d dY

⎛ dU ⎜μ ⎝ dY

d dY

⎛ dT ⎞ ⎛ dU ⎞ ⎜K ⎟+μ⎜ ⎟ =0 ⎝ dY ⎠ ⎝ dY ⎠

(2)

2

(3)

7

D

d 2C − γ ( C − C0 ) = 0 dY 2

(4)

where T0 is the reference temperature (as defined by Barletta and Zanchini [31]). The boundary conditions for the velocity, temperature and concentration fields are given as U = 0, T = T1 , C = C1 at Y = − b

(5)

U = 0, T = T2 , C = C2 at Y = b

(6)

The fluid viscosity μ is assumed to vary with temperature as [32, 33]

μ = μ0 e− a (T −T ) = μ0 (1 − a (T − T0 ) 0

)

(7)

where second and higher order terms of the exponent are neglected as they are small and the subscript 0 denotes the reference state and a is an empirical constant for the viscosity. In Eq. (7) the viscosity μ is assumed to be depended on temperature exponentially. The parameter a may take positive values for liquids such as water, benzene or crude oil. In some gases like air, helium or methane a may be negative, i.e., the coefficient of viscosity increases with temperature [34, 35]. This type of model can find applications in many processes where preheating of the fuel is used as a means to enhance heat transfer effects. In addition, for many fluids such as lubricants, polymers, and coal slurries where viscous dissipation is substantial, an appropriate constitutive relation where viscosity is a function of temperature should be used. The thermal conductivity of the fluid is assumed as [33]

K = K0e

− b% (T −T0 )

(

= K 0 1 − b% (T − T0 )

)

(8)

The thermal conductivity of the fluid is assumed to vary linearly with temperature as can be seen in Eq. (8) (the second and higher order terms of b% (T − T0 ) are neglected as they are very small) where the parameter b% may be positive for some fluids such as air or water vapor or negative for others like liquid water or benzene [35, 36]. The thermal conductivity changes approximately linearly with temperature in the range from 0 °F to 400 °F [37]. Equations (2)–(4) determine the velocity, temperature and concentration distribution and can be written in a dimensionless form using the following dimensionless parameters

8

u=

C − C0 T − T0 T −T Gr U Y γ b2 b 2 ∂p ,y= ,m= 1 2,φ = ,θ = , GRT = T , α = , P= , ΔT ΔC ΔT u b Re D μ0u ∂x

Gr g β C b3 ΔC μ 0u 2 g β b3 ΔT ub GRC = C , GrT = T 2 = B r , GrC = , , Re = 2 Re K 0 ΔT υ υ υ

(9)

In terms of the non-dimensional variables as in Eq. (9), Eqs. (2)–(4) take the form d 2u dθ du − bv + (1 + bvθ )( GRT θ + GRC φ ) − (1 + bvθ ) P = 0 2 dy dy dy 2

2

2

(10) 2

⎛ dθ ⎞ ⎛ du ⎞ ⎛ du ⎞ ⎛ du ⎞ d 2θ 2 − bk ⎜ ⎟ + Br ⎜ ⎟ + ( bk − bv ) θ Br ⎜ ⎟ − bk bvθ Br ⎜ ⎟ = 0 2 dy ⎝ dy ⎠ ⎝ dy ⎠ ⎝ dy ⎠ ⎝ dy ⎠

(11)

d 2φ −α φ = 0 dy 2

(12)

where ΔT = T2 − T0 , ΔC = C2 − C0 , bv = aΔT is the variable viscosity parameter, bk = b% ΔT is the variable conductivity parameter, m =

T1 − T2 C − C2 is the wall temperature ratio, n = 1 is the wall ΔT ΔC

concentration ratio. The non-dimensional form of the boundary conditions can be written as follows

u = 0, θ = 1 + m, φ = 1 + n at y = −1

(13)

u = 0, θ = 1, φ = 1 at y = 1

(14)

3. Solutions

The solution for φ can be obtained directly by integrating Eq. (12), and the solution is given by

(

φ = c1e −

αy

+ c2 e

αy

)

(15)

where c1 and c2 are integrating constants evaluated using boundary conditions as defined in Eqs. (13), (14) and are given as below c1 =

(1 + n ) e α − e− e2

α

− e −2

α

α

, c2 =

1 − c1e − e

α

α

.

Equations (10) and (11) are coupled nonlinear equations, and it is difficult, in general, to solve analytically. However approximate analytical solutions can be found using regular

9

perturbation method. Small values of Br ( < 1) facilitate finding analytical solutions of Eqs. (10)– (12) in the form u = u0 + Br ⋅ u1 + KK

(16)

θ = θ 0 + Br ⋅θ1 + KK

(17)

where the second and higher order terms on the right-hand side give a correction to u0 , θ 0 accounting for the dissipative effects. Substituting Eqs. (16) and (17) into Eqs. (10) and (11) and equating the coefficients like power of Br to zero, we obtain Zeroth order equations: d 2 u0 dθ du − bv 0 0 + GRT (1 + bvθ 0 ) θ 0 + GRC (1 + bvθ 0 ) φ − (1 + bvθ 0 ) P = 0 2 dy dy dy

(18)

2

⎛ dθ ⎞ d 2θ 0 − bk ⎜ 0 ⎟ = 0 2 dy ⎝ dy ⎠

(19)

The corresponding boundary conditions are u0 = 0, θ 0 = 1 + m at y = −1

(20)

u0 = 0, θ 0 = 1 at y = 1

(21)

First order equations: du dθ d 2u1 du dθ − bv 0 1 − bv 1 0 + GRT (1 + 2bvθ 0 ) θ1 + GRc bvθ1φ − bvθ1 P = 0 2 dy dy dy dy dy 2

2

(22)

2

⎛ du ⎞ ⎛ du ⎞ dθ dθ ⎛ du ⎞ d 2θ1 − 2 bk 0 1 + ⎜ 0 ⎟ + ( bk − bv ) θ 0 ⎜ 0 ⎟ − bv bkθ 0 2 ⎜ 0 ⎟ = 0 2 dy dy dy ⎝ dy ⎠ ⎝ dy ⎠ ⎝ dy ⎠

(23)

The corresponding boundary conditions are u1 = 0, θ1 = 0 at y = −1

(24)

u1 = 0, θ1 = 0 at y = 1

(25)

Eqs. (18), (19) and (22), (23) are still coupled and nonlinear. Hence we shall further perform a perturbation analysis considering variable conductivity parameter bk as a perturbation parameter. The solutions of Eqs. (18) and (19) are assumed as follows u0 = u00 + bk u01

(26)

θ 0 = θ 00 + bkθ 01

(27) 10

Substituting Eqs. (26) and (27) into Eqs. (18) and (19) and equating the coefficients like power of bk , we obtain the following boundary value problem Zeroth order equations: d 2u00 dθ du − bv 00 00 + (1 + bvθ 00 )( GRT θ 00 + GRCφ ) − (1 + bvθ 00 ) P = 0 2 dy dy dy

(28)

d 2θ 00 =0 dy 2

(29)

First order equations: d 2u01 dθ du dθ du − bv 00 01 − bv 01 00 + (1 + bvθ 00 ) GRT θ 01 + bvθ 01 ( GRT θ 00 + GRC φ ) − bvθ 01 P = 0 2 dy dy dy dy dy

(30)

2

d 2θ 01 ⎛ dθ 00 ⎞ −⎜ ⎟ =0 dy 2 ⎝ dy ⎠

(31)

The corresponding boundary conditions are ⎧u00 = 0, θ 00 = 1 + m at y = −1, ⎨ ⎩u00 = 0, θ 00 = 1 at y = 1;

⎧u01 = 0, θ 01 = 0 at y = −1, ⎨ ⎩u01 = 0, θ 01 = 0 at y = 1;

(32)

Integrating Eqs. (28)–(31) we will obtain u00 = c7 + c8e

m1 y

+ k1 ye −

αy

+ k2 ye

αy

+ k3 e −

αy

+ k4 e

αy

+ k5 y 3 + k6 y 2 + k7 y

(33)

θ 00 = c3 y + c4

(34)

θ 01 = l1 y 2 + c5 y + c6

(35)

u01 = c9 + c10 e m 1 y + g1 y 2 e − + g 4 ye −

αy

+ g5 ye

4

αy

3

αy

αy

+ g 2 y 2e

+ g3 y 2 em 1 y

+ g 6 ye m 1 y + g 7 e −

αy

+ g8 e

αy

(36)

2

+ g9 y + g10 y + g11 y + g12 y

where c4 =

l1 =

2+m , c3 = 1 − c4 , c6 = −l1 , c5 = 0 , 2

(

c32 1 , c8 = m 1 − m 1 − ( k1 + k2 ) e 2 e −e

c7 = −c8e

− m1

+ k1e

α

+ k2 e −

(

α

− k3 e

α

α

+ e−

− k4 e−

α

α

) + (k − k ) (e 3

4

+ k5 − k6 + k7 ,

11

α

− e−

α

) − 2k − 2k ) , 5

7

(

)

(

⎛ ( g − g + g − g ) e α − e− α − ( g + g ) e α + e− 1 2 7 8 4 5 1 c10 = m 1 − m 1 ⎜ ⎜⎜ + g e − m 1 − e m 1 − g e − m 1 + e m 1 − 2 g − 2 g e −e 6 10 12 ⎝ 3

(

c9 = −c10 e

− m1

− g8 e −

− g1e α

α

− g 2e−

)

α

− g 3e

− m1

(

+ g 4e

α

)

+ g5e−

α

+ g6e

− m1

α

− g7 e

) ⎞⎟ , ⎟⎟ ⎠

α

− g9 + g10 − g11 + g12

The solution of Eq. (30) is not found analytically as it is very lengthy. The solutions of the first order Eqs. (22) and (23) are also not found analytically as it is very tedious.

3.2. Numerical solutions

The analytical solutions obtained in the above section are valid for small values of perturbation parameters. Further it is seen in the above section that it is not possible to find solutions of even the first order. Hence we resort to solve the governing equations by numerical methods using Runge-Kutta shooting method (RKSM). The validity of RKSM is justified by comparing the solutions with the results obtained by the perturbation method and the values are displayed in tables. The perturbation method and RKSM solutions agree very well in the absence of perturbation parameter.

3.3. Skin friction and Nusselt number

In addition to the velocity and temperature fields, the following physical quantities can be defined. The shear stress is defined by ⎛

dU ⎞

τ = ⎜μ ⎟ ⎝ dY ⎠Y = ±b

(37)

By introducing the non-dimensional quantities as defined in Eq. (9) we get the dimensionless skin friction at each boundary as ⎛ du ⎞

⎛ du ⎞

τ 1 = e − b (1+ m ) ⎜ ⎟ and τ 2 = e − b ⎜ ⎟ ⎝ dy ⎠ y = −1 ⎝ dy ⎠ y =1 v

v

(38)

The Nusselt number is given by (as defined in [38])

12

⎛ dT ⎞ b⎜ K ⎟ ⎝ dy ⎠ y = ± b Nu = − K 0 ΔT

(39)

which in the dimensionless form become ⎛ dθ ⎞ ⎛ dθ ⎞ Nu1 = − (1 + bk (1 + m ) ) ⎜ and Nu2 = − (1 + bk ) ⎜ ⎟ ⎟ ⎝ dy ⎠ y = −1 ⎝ dy ⎠ y = +1

(40)

The values for the skin friction and Nusselt number are evaluated for different governing parameters and shown in Table 1.

4. Results and discussion

The problem of heat and mass transfer in a vertical channel with the combined effect of temperature dependent viscosity and thermal conductivity in the presence of first order chemical reaction is analyzed. The governing equations which are highly non-linear and coupled are solved analytically using perturbation method and numerically using Runge-Kutta shooting method. The solutions for concentration distribution are obtained directly by integrating the Eq. (12) using the boundary conditions (13) and (14). The solutions for the governing equations are found using regular perturbation method valid for small values of the perturbation parameters. The Brinkman number Br is used as the first perturbation parameter and the thermal conductivity parameter bk is used as the second perturbation parameter. For large values of the perturbation parameter the solutions cannot be used and hence we relax these restrictions by finding the solutions numerically. Runge-Kutta shooting method is used to find the numerical solutions. The results are drawn and presented graphically for governing parameters such as viscosity variation parameter bv , conductivity variation parameter bk , thermal Grashof number GRT , mass Grashof number GRC , Brinkman number Br, wall temperature ratio m and first order chemical reaction parameter α. The effect of viscosity variation parameter bv on the velocity and temperature is displayed in Figs. 2 and 3 respectively. For negative values of bv the velocity profiles is shifted to left wall (cold wall), for positive values of bv the velocity profiles tend towards the hot wall (right wall). It is seen from Figs. 2 and 3 that as the viscosity variation parameter bv increases 13

both the velocity and temperature increases. One can also observe that the velocity and temperature profiles for constant viscosity ( bv = 0 ) lies above bv < 0 and below bv > 0 . The effect of viscosity variation parameter bv on the flow was the similar result observed by Attia [39] on the MHD channel flow of dusty fluid. The effect of variable thermal conductivity bk for constant viscosity shows that as bk increases both the velocity and temperature profiles are suppressed as seen in Figs. 4 and 5. Here also the profiles for constant thermal conductivity ( bk = 0 ) lies below for bk < 0 and above for bk > 0 (which contradicts for variable viscosity parameter Figs. 2 and 3). The effect of bk is the similar result observed in [38, 40]. The effect of thermal Grashof number GRT and mass Grashof number GRC on the velocity and temperature fields are displayed in Figs. 6 to 9. As the thermal Grashof number GRT and mass Grashof number GRC increases the velocity and temperature fields increases. Physically an increase in the thermal Grashof number GRT and mass Grashof number GRC implies an increase of buoyancy force which supports the motion. Since the thermal Grashof number GRT acts as the driving mechanism of the driving force in the momentum equation, the velocity and/or velocity gradient increases and therefore the effect of dissipation increases, which results in the enhancement of temperature field. The effects of GRT and GRC on the flow were the similar results observed by Shivaiah and Anand Rao [41] for the flow past a porous vertical plate for constant properties. The effect of Brinkman number Br on the velocity and temperature field is shown in Figs. 10 and 11 respectively. It is evident from these figures that as the Brinkman number Br increases the flow is promoted. An increase in the Brinkman number Br results in the increase in dissipation effects which results in the increase of temperature and as a consequence velocity increases for the increase in buoyancy force in the momentum equation. The profiles for the effects of wall temperature ratio m on the flow field are displayed in Figs. 12 and 13. As the wall temperature ratio m increases both the velocity and temperature fields are enhanced. It is quite natural that physically an increase in m for values of m > 1 means that the wall temperature at the left wall is higher than the wall temperature at the right wall which will enhance the temperature gradient. The effects of first order chemical reaction parameter α on the 14

velocity, temperature and concentration fields are depicted in Figs. 14 to 16. It is evident from these figures that velocity, temperature and concentration are reduced. Physically an increase in

α leads to the increase in the number of solute molecules undergoing chemical reaction results in decrease in fluid field. This is the similar result observed by Damesh and Shannak [42] for viscoelastic fluid and Bhattacharyya [43] for viscous fluid with constant properties. The effect of variable viscosity parameter bv , thermal Grashof number GRT , mass Grashof number GRC , Brinkman number Br and the wall temperature ratio m on the concentration fields show the invariant effect, because the concentration equation does not include these parameters. The effect of variable viscosity parameter bv , variable thermal conductivity parameter bk , thermal Grashof number GRT , mass Grashof number GRC , Brinkman number Br, the wall temperature ratio m and first order chemical reaction parameter α on skin friction and Nusselt number is shown in Table 1. Increasing the values of bv increases the skin friction and Nusselt number at the left wall, and decreases the skin friction and Nusselt number at the right wall in magnitude. Fixing bv = 0.2 and increasing values of bk decreases the skin friction at both the left and right wall, whereas the Nusselt number decreases at the left wall and increases at the right wall in magnitude. The skin friction increases at both the plates in magnitude as the thermal Grashof number GRT , mass Grashof number GRC , Brinkman number Br and wall temperature ratio m increases whereas it decreases with the first order chemical reaction parameter α. The Nusselt number increases at both the plates as GRT , GRC and Br increases. The Nusslet number increases in magnitude at both the plates as the wall temperature ratio m increases whereas it decreases at the left wall and increases at the right wall in magnitude as the first order chemical reaction parameter α increases. The analytical solutions obtained by regular perturbation methods are valid only for small values of perturbation parameter. To overcome this restriction the governing equations are solved using Runge-Kutta shooting method. The validity of Runge-Kutta shooting method is justified by comparing the results obtained by perturbation method and Runge-Kutta shooting method and displayed in Tables 2. It is viewed from Table 2 that the analytical and numerical solutions agree very well for small values of Brinkman number and the error increases as Brinkman number Br increases for velocity and temperature fields. The justification of the 15

solutions obtained by Runge-Kutta shooting method is further justified by comparing with the theoretical data of Hernández and Zamora [26] and experimental results of Guo and Wu [44] for constant and variable properties. It is seen from Fig. 17 that the value of temperature obtained in the absence of first order chemical reaction is 0.652 ( m = − 1, P = 0, bv = 0, Br = 2 ) and nearly the same value obtained by Guo and Wu [44]. Also the temperature profiles for the variable properties and for constant properties agree with the data of Guo and Wu [44]. Further analysis is made about the effects of pressure work and viscous dissipation on the flow. The values of Nusselt number are evaluated for different cases and shown in Tables 3 and 4 for gas and for highly viscous fluids. Tables 3 and 4 indicate that the values of Nusselt number are not varied in a significant way in the presence or in the absence of pressure work for the convection flow.

5

4

u

3

m = - 1, bk = 0 . 2, Br = 0 . 01, GRT =5, GRC =5, α = 0 . 5, n = 0 . 3, P = 0. 5

0. 9 0. 5 0 - 0. 5 bv = - 0 . 9

2

1

0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 2. Velocity profiles for different values of viscosity variation parameter bv

16

1.0

0.8

0.6

θ

m = - 1, bv = - 0 . 9, - 0 . 5, 0, 0 . 5, 0 . 9 bk = 0 . 2, B r = 0 . 01, G RT = 5, G RC = 5, α = 0 . 5, n = 0 . 3, P = 0.5

0.4

0.2

0.0 -1.0

-0.5

0.0

0.5

y

1.0

Figure 3. Temperature profiles for different values of viscosity variation parameter bv

m = - 1, bv = 0. 2, Br = 0. 01, GRT = 5, GRC = 5, 3 α = 0.5, n = 0. 3, P =0.5 4

u

2

bk = - 0. 4, - 0. 2, 0, 0. 2, 0. 4 1

0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 4. Velocity profiles for different values of conductivity variation parameter bk 17

1.0

m = - 1, bv = 0 . 2, B r = 0 . 01, GRT = 5, GRC = 5, α = 0 . 5, n = 0 . 3, P = 0.5

0.8

0.6

θ

bk = - 0.4, - 0.2, 0, 0.2, 0.4

0.4

0.2

0.0 -1.0

-0.5

0.0

0.5

y

1.0

Figure 5. Temperature profiles for different values of conductivity variation parameter bk

10

8

6

u

m = - 1, bv = 0. 2, bk = 0. 5, Br = 0. 01, GRC = 5, α = 0. 5 n = 0. 3 P = 0.5

18 15 10 5

4

GRT = 1 2

0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 6. Velocity profiles for different values of thermal Grashof number GRT 18

1.0

θ 0.5

m = - 1, bv = 0. 2, GRT =1, 5, 10, 15, 18 bk = 0. 5, Br = 0. 01, GRC = 5, α = 0. 5 n = 0. 3 P = 0. 5

0.0 -1.0

-0.5

0.0

0.5

y

1.0

Figure 7. Temperature profiles for different values of thermal Grashof number GRT

7.5

20 15

6.0

10

4.5

u

m = - 1, bv = 0.2, bk = 0.2, Br = 0.01, GRT = 5, α = 0.5, n = 0.3, P = 0.5

5

3.0

GRC = 1 1.5

0.0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 8. Velocity profiles for different values of mass Grashof number GRC 19

1.0

θ 0.5

m = - 1, bv = 0. 2, GRC=1,5, 10, 15, 20 bk = 0. 2, Br = 0.01, GRT = 5, α = 0. 5, n = 0. 3, P = 0. 5

0.0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 9. Temperature profiles for different values of mass Grashof number GRC

m = - 1, bv = 0. 2, bk = 0. 2, GRT = 1, 0.6 GRC = 1, 0.5 α = 0.5, n = 0.3, 0.4 P = 0.5

Br = 0.1,0.5,1,1.5,2

0.7

u

0.3 0.2 0.1 0.0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 10. Velocity profiles for different values of Brinkman number Br 20

1.0

0.8

θ 0.6

m = - 1, Br = 0.1, 0.5, 1, 1.5, 2 bv = 0.2, bk = 0.2, GRT =1, GRC =1, α = 0.5, n = 0.3, P = 0.5

0.4

0.2

0.0 -1.0

-0.5

0.0

0.5

y

1.0

Figure 11. Temperature profiles for different values of Brinkman number Br

12

bv = 0.2, bk = 0.2, GRT = 5, GRC = 5, Br = 0.01, α = 0.5, n = 0.3, P = 0.5

2

1 8

0

u

-1

4

m = -2

0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 12. Velocity profiles for different values of wall temperature ratio m 21

2 3

1

2

bv = 0.2, bk = 0.2, Br = 0.01, GRT = 5,GRC= 5, α = 0.5, n = 0.3, P = 0.5

0

θ

1

-1 0

m = -2 -1 -1.0

-0.5

0.0

y

0.5

1.0

Figure 13. Temperature profiles for different values of wall temperature ratio m

5

4

3

u

m = - 1, bv = 0.5, bk = 0.5, Br = 0.01, GRT = 5, GRC = 5, n = 0.3, P=0.5

α = 0, 0.5, 1, 1.5

2

1

0 -1.0

-0.5

0.0

y

0.5

1.0

Figure 14. Velocity profiles for different values of chemical reaction parameter α 22

1.0

0.8

0.6

θ

m = - 1, bv = 0.5, bk = 0.5, Br = 0.01, GRT = 5, GRC = 5, n = 0.3, P = 0.5

α = 0, 0.5, 1, 1.5 0.58

0.4 0.56

0.54

0.2

0.52

0.50

0.48

0.0 -1.0

0.00

-0.5

0.0

y

0.05

0.5

0.10

1.0

Figure 15. Temperature profiles for different values of chemical reaction parameter α

α=0

1.2

1.0

α

0.8

m = - 1, bv = 0.5, bk = 0.5, B r = 0.01, G RT = 5, G RC = 5, n = 0.3, P = 0.5

0.6 -1.0

-0.5

0.5

1 1.5 0.0

y

0.5

1.0

Figure 16. Concentration profiles for different values of chemical reaction parameter α 23

Figure 17. Comparison of temperature profiles with theoretical and experimental data of other authors

Table 1: Values of skin friction and Nusselt number

τ1 bv

τ2

Nu1

Nu2

bk = 0.2, m = −1, Br = 0.01, GRT = 5, GRC = 5, α = 0.5, n = 0.3, P = 0.5

-0.9 -0.5 0 0.5 0.9 bk

4.83700221 -4.34555288 -0.47806663 -0.65088990 5.62306456 -6.70876137 -0.50936745 -0.60961594 6.41346025 -7.59907147 -0.57845637 -0.47441882 7.00061319 -5.58342860 -0.70391984 -0.18855914 7.28364951 -1.42016981 -0.88965210 0.17607247 bv = 0.2, m = −1, Br = 0.01, GRT = 5, GRC = 5, α = 0.5, n = 0.3, P = 0.5

-0.4 -0.2

7.02235504 6.90230100

-7.47574087 -7.37937305 24

-0.92382785 -0.80253599

-0.08802383 -0.15599026

0 0.2 0.4 GRT

6.78589252 -7.28143788 -0.70256590 -0.25167063 6.67364970 -7.18246657 -0.61965512 -0.38145368 6.56597320 -7.08298272 -0.55045492 -0.55290339 bv = 0.2, bk = 0.5, m = −1, Br = 0.01, GRC = 5, α = 0.5, n = 0.3, P = 0.5

1 5 10 15 18 GRC

5.14582557 -4.54334559 -0.46290170 -0.83030650 6.51394197 -7.03320799 -0.52014901 -0.65700325 8.53498612 -10.4600385 -0.63270931 -0.27730513 11.3586546 -14.7191354 -0.83471666 0.50335749 14.2710654 -18.5823144 -1.08212141 1.66507887 bv = 0.2, bk = 0.2, m = −1, Br = 0.01, GRT = 5, α = 0.5, n = 0.3, P = 0.5

1 5 10 15 20 Br

2.24514272 -3.51094488 -0.47934963 -0.60535312 6.67364970 -7.18246657 -0.61965512 -0.38145368 12.42200499 -11.95930439 -0.99630224 0.17968159 18.47909716 -16.99995626 -1.65878711 1.14808555 25.02687941 -22.44598539 -2.76104816 2.76224920 bv = 0.2, bk = 0.2, m = −1, GRT = 1, GRC = 1, α = 0.5, n = 0.3, P = 0.5

0.1 0.5 1 1.5 2 m

0.89108285 -1.03258351 -0.48344381 -0.60940365 0.92319393 -1.06296768 -0.61765240 -0.36822070 0.97374154 -1.11068117 -0.82603992 0.00101903 1.04485181 -1.17759695 -1.11341493 0.50140899 1.17003404 -1.29491063 -1.60350948 1.33674302 bv = 0.2, bk = 0.2, Br = 0.01, GRT = 5, GRC = 5, α = 0.5, n = 0.3, P = 0.5

-2 -1 0 1 2

α

2.96016734 -5.16567920 -0.68848128 -1.36456578 6.67364970 -7.18246657 -0.61965512 -0.38145368 9.74907183 -9.28297934 -0.60108956 0.57802713 11.51975937 -11.52914444 -1.03315372 1.60360269 11.30812541 -14.21835711 -3.30679529 2.93312314 bv = 0.5, bk = 0.5, m = −1, Br = 0.01, GRT = 5, GRC = 5, n = 0.3, P = 0.5

0 0.5 1 1.5

7.79755382 6.83943080 6.16180899 5.65565555

-6.09676080 -5.47710012 -5.03963658 -4.71387680

-0.64924533 -0.58190669 -0.54120285 -0.51440632

25

-0.19786260 -0.38785343 -0.50324224 -0.57964414

Table 2 Comparison of analytical and numerical solutions on the velocity and temperature for bv = 0.5, bk = 0.5, m = −1, GRT = 0.5, GRC = 0.5, α = 0.5, n = 0.3, P = 0.5 Velocity Br = 0 Br = 1 Br = 2 y Analytical Numerical Analytical Numerical Analytical Numerical 1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.4 0.12566 0.12556 0.12566 0.12733 0.12566 0.12923 0.0 0.12340 0.12368 0.12340 0.12561 0.12340 0.12767 -0.4 0.08818 0.08859 0.08818 0.09001 0.08818 0.09153 -1 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 Temperature Br = 0 Br = 1 Br = 2 y Analytical Numerical Analytical Numerical Analytical Numerical 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1 0.65757 0.64750 0.65074 0.64750 0.64434 0.64750 0.4 0.45082 0.43750 0.44427 0.43750 0.43814 0.43750 0.0 0.26224 0.24750 0.25655 0.24750 0.25121 -0.4 0.24750 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 -1

To understand the flow nature for different types of fluids, the governing parameters are redefined and the rate of heat transfer is evaluated and shown in Tables 3 and 4. The Rayleigh number is the product of thermal Grashof number and Prandtl number, and Brinkman number is the product of Eckert number and Prandtl number, hence the thermal Grashof number and Brinkman number are redefined in terms of Rayleigh number and Brinkman number. The values of the other parameters are taken as bv = 0.2, bk = 0.2, P = 200, m = −1, Ec = 10−10 , GRC = 5,

α = 0.5, n = 0.3 . In Tables 3 and 4 case-1 is related with the values of Nusselt number in the presence of pressure work and viscous dissipation effects, case-2 is for the values in the absence of pressure work but in the presence of viscous dissipation and case-3 is for the values in the absence of both pressure work and viscous dissipation.

26

Table 3: Prandtl number is 0.71 for gas Case-1 Ra

Nu _

Case-2 Nu+

Nu _

Case-3 Nu+

Nu _

Nu+

102 -0.45317381 -0.66420752 -0.45317313 -0.66420824 -0.45317312 -0.66420828 103 -0.45317977 -0.66418844 -0.45317451 -0.66420481 -0.45317312 -0.66420828 104 -0.45446763 -0.66093680 -0.45331090 -0.66386491 -0.45317312 -0.66420828 105 -0.63055165 -0.24321471 -0.46723589 -0.62930426 -0.45317312 -0.66420828 106 -0.63167205 -0.24108653 -0.63179989 -0.24084394 -0.45317312 -0.66420828

Table 4: Prandtl number is 1000 for highly viscous liquid Case-1 Ra

Nu _

Case-2 Nu+

Nu _

Case-3 Nu+

Nu _

Nu+

102 -0.45407585 -0.66287710 -0.45317320 -0.66420815 -0.45317311 -0.66420827 103 -0.45407447 -0.66287978 -0.45317322 -0.66420812 -0.45317311 -0.66420827 104 -0.45406075 -0.66290639 -0.45317342 -0.66420770 -0.45317312 -0.66420828 105 -0.45392970 -0.66315709 -0.45318163 -0.66418806 -0.45317312 -0.66420828 106 -0.45323762 -0.66412296 -0.45388224 -0.66244992 -0.45317312 -0.66420828 107 -0.51435110 -0.50702938 -0.53144870 -0.47371423 -0.45317312 -0.66420828

27

5. Conclusion

The problem of mixed convective flow of a viscous fluid in a vertical channel was analyzed for the variation of combined effect of viscosity and thermal conductivity on the temperature. The analytical solutions were found by perturbation parameter method valid for small values of perturbation parameter and numerical solutions were found by Runge-Kutta shooting method valid for any values of governing parameters. The Runge-Kutta shooting method and perturbation method show good agreement in the absence of Brinkman number. The following results were drawn. 1. Increase in the variable viscosity enhances the flow and heat transfer whereas increase in the variable thermal conductivity suppresses the flow and heat transfer. 2. The thermal Grashof number, mass Grashof number, wall temperature ratio and Brinkman number enhance the flow. The increase in chemical reaction parameter suppresses the flow. 3. The variable viscosity parameter, variable thermal conductivity parameter, thermal Grashof number, mass Grashof number, wall temperature ratio, Brinkman number and first order chemical reaction parameter were invariant on concentration field. 4. The values of Nusselt number are not varied in a significant way in the presence or absence of pressure work for the convection flow. 5. The solutions obtained by Runge-Kutta shooting method and perturbation method are exact in the absence of Brinkman number and the error increases as the Brinkman number increases. 6. The obtained results were agreed very well with Attia [33] for variable properties, with theoretical and experimental data of Guo and Wu [44] and with Umavathi [45] for constant properties.

Acknowledgment

J.C. Umavathi is thankful for the financial support under the UGC-MRP F.43-66/2014 (SR) Project and also to Prof. Maurizio Sasso, supervisor and Prof. Matteo Savino cocoordinator of ERUSMUS MUNDUS “Featured eUrope and South/south-east Asia mobility Network FUSION” for their support to do Post-Doctoral Research. 28

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32

•

Effects of variable viscosity and thermal conductivity on heat transfer and fluid flow are analyzed.

•

The governing equations are solved analytically and numerically.

•

An increase in chemical reaction parameter suppresses the flow.

•

The thermal and mass Grashof numbers, wall temperature ratio and Brinkman number enhance the flow.

33