Outcome for chemically reactive aspect in flow of tangent hyperbolic material

    Outcome for chemically reactive aspect in flow of tangent hyperbolic material Muhammad Ijaz Khan, Tasawar Hayat, Muhammad Waqas, Ahmed Alsaedi PII: DOI: Reference:

S0167-7322(16)33875-2 doi:10.1016/j.molliq.2017.01.016 MOLLIQ 6818

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

1 December 2016 27 December 2016 6 January 2017

Please cite this article as: Muhammad Ijaz Khan, Tasawar Hayat, Muhammad Waqas, Ahmed Alsaedi, Outcome for chemically reactive aspect in flow of tangent hyperbolic material, Journal of Molecular Liquids (2017), doi:10.1016/j.molliq.2017.01.016

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Outcome for chemically reactive aspect in flow of tangent hyperbolic material Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b

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Muhammad Ijaz Khana,1 , Tasawar Hayata,b , Muhammad Waqasa and Ahmed Alsaedib

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of

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Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah

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21589, Saudi Arabia

Abstract: Intention in this communication is to predict the influences of chemical reaction

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and heat generation/absorption in nonlinear flow process. Stagnation point flow of tangent hyperbolic fluid towards a stretching sheet with variable thickness is examined. Both stretching and

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free stream velocities are nonlinear. Inclined applied magnetic field is taken. Incoming nonlinear

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modeled problems have been computed for the convergent solutions of velocity, temperature and concentration. Drug force and heat transfer rate are further addressed. Outcome of sundry variables

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are explored.

Keywords: Chemical reaction; tangent hyperbolic fluid; heat absorption/generation; inclined MHD.

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Corresponding author: email address: [email protected] (M Ijaz Khan)

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ACCEPTED MANUSCRIPT Nomenclature fluid velocity components

N ux

Nusselt number

λ

material power law index

τw

shear stress

υ

kinematic viscosity

Cf

skin friction coefficient

Γ

fluid material constant

δ

chemical reaction parameter

σ

electrical conductivity

Sc

Schmidt number

B0

strength of magnetic field

β

heat absorption/generation parameter

ρ

density

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inclination

Pr

Prandtl number

Ha

Hartman number

We

Weissenberg number

ξ

transformed variable

ψ

stream function

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. φ

T

u, v

temperature

cp

specific heat

Q0

heat absorption/generation coefficient

T∞

ambient temperature

A

ratio parameter

k∗

chemical reaction constant

Rex

Reynold number

C

concentration

Ue

free stream velocity

C∞

ambient concentration

n

power law index

qw

wall heat flux

Uw

stretching velocity

1

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Introduction

Flow analysis in presence of heat and mass transfer with chemical reaction has acquired considerable importance. Several processes for instance evaporation, drying, energy transport in a chilling tower and flow in the desert cooler involve mass and heat transport simultaneously. Natural convective procedures comprising the simultaneous impact of heat and mass transfer are further encountered in several industrial applications and natural processes including 2

ACCEPTED MANUSCRIPT plastics, chemical processing and cleaning of materials, insulated cables and production of pulp. Various researchers investigated chemical reaction characteristics considering different

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geometries. For instance, Srinivas [1] addressed chemical reaction characteristics in viscous

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liquid flow in a permeable channel. Magnetohydrodynamic (MHD) chemically reactive flow of Casson material by an exponentially stretched radiative sheet is reported by Reddy [2].

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Khan et al. [3] modeled three-dimensional chemically reactive stretched flow of Burgers liquid

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utilizing non-Fourier flux theory. Chemical reaction and MHD effects in third grade liquid induced by expontential stretched sheet is addressed by Hayat et al. [4]. Lakshmi et al.

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[5] established numerical solutions for thermo-diffusion and diffusion-thermo characteristics in two-phase stretched flow of viscous material in presence of chemical reaction and mag-

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netohydrodynamics. Analysis of magneto viscoelastic material towards stretched sheet with chemical reaction and thermal radiation is developed by Nayak et al. [6].

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The characteristics of heat transport in magneto non-Newtonian materials have numer-

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ous applications in technology and science for instance establishment of heat exchangers, scheme for chilling of nuclear reactors, insertion of nuclear accelerators, turbo machinery, measurement techniques for blood flow etc. Effectiveness of peristaltic transport in magneto hyperbolic tangent nanomaterial is discussed by Akram and Nadeem [7]. Gaffar et al. [8] explored the characteristics of Biot number in laminar flow of hyperbolic tangent material. Consequences of melting heat and magnetic field in stretched flow of hyperbolic tangent material are reported by Hayat et al. [9]. Mixed convection and heat absorption/generation characteristics in chemically reactive radiative flow of hyperbolic tangent nanomaterial is studied by Hayat et al. [10]. Hayat et al. [11] scrutinized Joule heating and slip effects in MHD peristaltic flow of tangent hyperbolic nanoliquid. Moreover the analysis regarding heat absorption or generation effects has significance in the cooling processes. Such effects 3

ACCEPTED MANUSCRIPT may change the temperature distribution and thus the deposition rate of particle. Also these effects are further significant in metal waste, analysis of reactor safety, spent nuclear fire, fuel

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and combustion investigations and stability of radioactive stuffs. Few recent attempts in this

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direction can be stated through [12-15] and several studies therein.

No doubt stretching surface in presence of variable thickness possess engineering and

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physical backgrounds because the usage of variable thickness is beneficial in order to minimize

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the structural elements weight and enhance the effectiveness of materials [17]. Subhashini et al. [18] established dual solutions for thermal diffusive stretched flow of viscous liquid with

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variable thickness. Characteristics of melting heat and heterogeneous/homogeneous processes in magneto viscous material towards variable thicked surface are analyzed by Hayat et al.

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[19]. Hayat et al. [20] explored melting heat and MHD aspects in nonlinear stretching flow of Williamson nanoliquid over a sheet with variable thickness. Analysis of non-Fourier flux

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et al. [21, 22].

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theory in stagnation point flow of Eyring-Powell and Jeffrey materials is presented by Hayat

In perspective of the aforementioned investigations it is observed that heat absorption/generation characteristics in stagnation point flow towards nonlinear stretching surface with variable thickness is not examined yet. Our motto here is to fill this void by considering such characteristics in nonlinear stretched flow of magneto hyperbolic tangent material. Free stream velocity is also taken nonlinear. Chemical reaction effects are also involved. Governing problems are formulated and solved through homotopic technique [23-35]. Consequence of influential variables occurring into the problems statememt is reported graphically.

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Mathematical model

Let us consider steady two-dimensional stagnation point flow of incompressible tangent hy-

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perbolic fluid in presence of heat absorption/generation and chemical reaction. Stretching

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sheet with variable thickness is considered. Fluid is electrically conducting in the presence of an applied magnetic field. Induced and electric field effects are negligible. Heat transfer

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analysis is discussed in the absence of thermal radiation and viscous dissipation. The sheet is

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stretched with velocity u = Uw (x) = U0 (x + b)n . The temperature and concentration near the surface are considered as T = Tw and C = Cw . We assume that the sheet is not flat and 1−n 2

. Note that the behavior of nonlinear surface depend

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thickness of sheet is y = A∗ (x + b)

upon the different values of n. For n = 1 the nonlinear surface of sheet reduces to flat surface

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whereas for n > 1 and n < 1 the wall thickness parameter increases and decreases respec-

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tively. Furthermore the given flow problem reduces to viscous fluid for λ = 0. A model of chemical reaction in boundary layer flow was first time utilized by Merkin [31]. He presented

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the effectiveness of chemical reaction in flow of viscous fluid in that analysis. The equations comprising the conservation of mass, momentum, energy and concentration are expressed as: ∂u ∂u + = 0, ∂x ∂y

(1)

 ∂u ∂u ∂ 2 u σB 2 (x) 2 ∂ 2u √ ∂u dUe u + υ (1 − λ) 2 + 2υΓλ − +v = Ue sin φ (u − ue ) , (2) dx ∂x ∂y ∂y ∂y ∂y 2 ρ  2 ∂T k ∂T Q (x) ∂ T u +v = + (T − T∞ ) , (3) 2 ∂x ∂y ρcp ∂y ρcp u

∂C ∂ 2C ∂C +v = D 2 − k ∗ (C − C∞ ) , ∂x ∂y ∂y

(4)

with u = Uw (x) = U0 (x + b)n , T = Tw , C = Cw at y = A∗ (x + b) 5

1−n 2

,

(5)

ACCEPTED MANUSCRIPT u → Ue (x) = U∞ (x + b)n , T = T∞ , C = C∞ when y → ∞.

(6)

Employing the transformations [19, 20, 22]: ! 2 n + 1 U0 (x + b)n−1 y, υU0 (x + b)n+1 F (ξ), ξ = ψ= n+1 2 υ ! n−1 ′ n+1 u = U0 (x + b)n F ′ (ξ), v = − υU0 (x + b)n−1 (F (ξ) + ξ F (ξ)), 2 n+1 T − T∞ C − C∞ Θ (ξ) = , Φ (ξ) = , Tw − T∞ C w − C∞

(7)

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T

!

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the incompressibility condition (1) is automatically fulfilled while Eqs. (2)-(5) reduced to the forms !

n + 1 ′′ ′′′ 2 2n 2 F F + A − Ha2 Si n2 φ (F ′ − A) = 0, 2 n+1 n+1

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2n ′2 F +F F ′′ +λWe (1 − λ) F − n+1 ′′′

2 β Pr Θ = 0, n+1

(9)

2Sc (ΦF ′ + δΦ) = 0, n+1

(10)

Θ′′ + Pr F Θ′ +

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Φ′′ + ScF Φ′ −

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subject to

F (α) = α

Here α = A1

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n+1 U0 2 υ

1−n , 1+n

(8)

F ′ (α) = 1, F ′ (∞) = A,

(11)

Θ (α) = 0,

Θ (∞) = 1,

(12)

Φ (α) = 0,

Φ (∞) = 1.

(13)

denotes the plate surface and prime represents derivative with respective

to ”ξ”. Letting F (ξ) = f (ξ −α) = f (η), Θ (ξ) = θ(ξ −α) = θ(η) and Φ (ξ) = φ(ξ −α) = φ(η), Eqs. (8)-(10) become 2n ′2 (1 − λ) f ′′′ − f + f f ′′ + λWe n+1

!

2n 2 n + 1 ′′ ′′′ 2 f f + A − Ha2 Si n2 φ (f ′ − A) = 0, 2 n+1 n+1 (14)

θ′′ + Pr f θ ′ +

2 β Pr θ = 0, n+1 6

(15)

ACCEPTED MANUSCRIPT φ′′ + Scf φ′ −

2Sc (φf ′ + δφ) = 0, n+1

(16)

subjected to the boundary conditions 1−n ′ , f (0) = 1, f ′ (∞) = A, 1+n

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(17)

θ (0) = 0, θ (∞) = 1,

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f (0) = α

(18) (19)

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φ (0) = 0, φ (∞) = 1.

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For n = 1 the above problems reduces to viscous fluid.

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The emerging variables in the aforementioned equations are # U∞ σB02 µcp 2U03 (x + b)3n−1 We = Γ , Ha2 = , Pr = , A= , υ U0 ρU0 k υ k Q0 (x + b)1−n . , Sc = , δ = β= U0 ρcp D U0

(20)

The drag force on the surface and local Nusselt number are defined as [10]:

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Cf =

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where τ w and qw are

τw , 1 ρUw2 2

$

N ux =

Γ ∂u + λ√ τ w = µ (1 − λ) ∂y 2



∂u ∂y

xqw k(T − T∞ )

2

%

, qw = −k

(21)



∂T ∂y

.

(22)

y=0

Dimensionless form of drag force and Nusselt number are   " n+1 ′′ − 12 f (η)  "  2 (1 − λ) 2   ReCf =  = −θ′ (0) .  , N ux Re x x +λWe n+1 (f ′′ (η))2 2

(23)

η=0

1

with Rex2 =

3

"

U0 (x+b)n−1 υ

represents the local Reynold number.

Homotopic procedure and convergence solutions

The initial approximations (f0 , θ0 , φ0 ) and auxiliary linear operators (£f , £θ , £φ ) for homotopy analysis solutions are as follows: 7

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n−1 f0 (η) = Aη + (1 − A)(1 − exp(−η)) + α n+1 ,

(24)

Lf (f ) =

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φ0 (η) = 1 − exp (−η) ,

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θ0 (η) = 1 − exp (−η) ,

d3 f df d2 θ d2 φ − = − θ, L ( ) = − φ. (θ) , L φ φ θ dη 3 dη dη 2 dη 2

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The properties satisfied by operators are

(25)

(26)

Lθ [C4∗ exp(η) + C5∗ exp(−η)] = 0,

(27)

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Lf [C1∗ + C2∗ exp(η) + C3∗ exp(−η)] = 0,

Lφ [C6∗ exp(η) + C7∗ exp(−η)] = 0.

(28)

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with C1∗ to C7∗ as the arbitrary constants.

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The general solutions can be obtained as:

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⋆ (η) + C1∗ + C2∗ eη + C3∗ e−η , fm (η) = fm

(29)

θm (η) = θ⋆m (η) + C4∗ eη + C5∗ e−η ,

(30)

φm (η) = φ⋆m (η) + C6∗ eη + C7∗ e−η .

(31)

In this case the values of the constants C1∗ to C7∗ are

C2∗

=

C4∗

=

C6∗

= 0,

C1∗

, ⋆ ∂fm (η) ,, ⋆ − fm (0) , =− ∂η ,η=0

, ⋆ (η) ,, ∂fm = , C5∗ = −θ∗m (η) , , ∂η ,η=0 ⋆ , ∂φ (η) m , . C7∗ = ∂η ,

C3∗

(32)

η=0

Keeping in mind the goal about meaningful series solutions of boundary layer problems, we will to determine the convergence region. For this purpose we have plotted the - curves 8

ACCEPTED MANUSCRIPT of the functions f ′′ (0) , θ ′ (0) and φ′ (0) for the admissible values of f , θ and φ . Admissible values of f, θ and φ are noted −2.2 ≤ f ≤ −0.2, −1.9 ≤ θ ≤ −0.9 and −1.9 ≤ φ ≤ −0.6

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respectively.

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Fig. 1: h−curves for f, θ and φ.

Table I. Series solutions convergence when λ = Ha = W e = n = α = A = β = δ = 0.1,

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Pr = 0.3, φ = π4 , Sc = 0.5.

′′

′

′

−f (0)

θ (0)

g (0)

1

0.8998

0.9997

0.9996

0.8993

0.9986

0.9978

0.8986

0.9971

0.9957

0.8980

0.9950

0.9939

0.8980

0.9950

0.9939

0.8980

0.9950

0.9939

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order of approximations

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10

20

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25

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From Table 1 we can see that 20 th order HAM iteration is sufficient to get the convergence of series solutions.

Fig. 2: Effect of λ on f ′ . Fig. 3: Effect of We on f ′ . Fig. 4: Effect of A on f ′ . Fig. 5: Effect of Ha on f ′ . Fig. 6: Effect of Pr on θ. Fig. 7: Effect of β on θ. Fig. 8: Effect of Sc on φ. Fig. 9: Effect of δ on φ.

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ACCEPTED MANUSCRIPT Fig. 10: Effects of A and Ha on skin friction. Fig. 11: Effects of α and n on skin friction.

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Fig. 12: Effects of Pr and β on Nusselt number.

Discussion

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Fig. 13: Effects of Pr and n on Nusselt number.

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The group of transformed equations (11-13) subject to boundary conditions (14-16) are computed with the help of homotopy analysis method (HAM). To examine the effects of various

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parameters on different fields, i.e., velocity, temperature and concentration, we assign values to the parameters as λ = 0.5, Ha = 0.5, We = 0.2, n = 0.5, α = 0.2, A = 0.3, φ = π4 , β = 0.01,

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Sc = 1.1 and δ = 0.9. We use graphs to discuss the behavior of diverse parameters such as

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material power law index, magnetic parameter, Weissenberg number, ratio parameter, heat absorption/generation parameter, Schmidt number and chemical reaction parameter. Fig. 2

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explores variations of λ (material power law index) on velocity field. Both velocity field and associated layer thickness decrease for higher λ. It is due to the fact that for rising values of material power law index the viscosity of fluid increases which provides the resistance to the fluid motion. Therefore velocity field decreases. Behavior of Weissenberg number ( We ) on the velocity field is shown in Fig. 3. It is noted that velocity profile shows decreasing behavior for larger We . Since Weissenberg number We is the ratio of relaxation time of the fluid and a specific time therefore for larger relaxation time the thickness of the fluid increases and consequently velocity of fluid particle decreases. Fig. 4 portrays ratio parameter A effect on velocity profile. It is clear here that velocity field and corresponding boundary layer are enhanced through an increase of A. For A = 1 the fluid particle and sheet always move at the same velocity. There is no formation of boundary layer for A = 1. Velocity profile for differ10

ACCEPTED MANUSCRIPT ent values of magnetic parameter Ha is illustrated in Fig. 5. It is revealed from this Fig. that increasing values of M decelerate the fluid velocity leading to thinner momentum boundary

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layer in the flow domain. This behavior implies that the transverse magnetic field develops

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a resistive force called Lorentz force which decelerates the fluid motion. Unless the magnetic field is applied to the conducting fluid it would have never been opposed by Lorentz force.

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The behavior of Prandtl number on temperature field in shown in Fig. 6. Since Prandtl

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number is a function of viscosity, thermal conductivity and specific heat. Therefore it is the ratio of kinematic viscosity to thermal diffusivity. Also Pr controls the relative thickness

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of velocity and thermal boundary layers. Keeping these views in mind and understanding the implications of Fig. 6 one can remark that lower Prandtl fluids having higher thermal

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conductivities contribut faster diffusion of heat in thicker thermal boundary structure when compared with higher Prandtl fluids in thinner boundary region. Hence it can be ensured that

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Prandtl number can be used to enhance the cooling rate in the flow of conducting fluids. Fig.

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7 is focused to describe the behavior of heat absorption/generation parameter on temperature field. Temperature and thermal boundary layer significantly enhance with increasing values of β. In fact larger values of heat absorption/generation parameter (β) transfer more heat to the fluid particle. Therefore temperature of fluid particle increases. Fig. 8 demonstrates of Sc effect on concentration. The ratio of momentum diffusivity to mass diffusivity gives a Schmidt number. Rising values of Sc decrease the transfer rate of mass from one place to another place and as a result concentration distribution decreases. Influence of chemical reaction on concentration field is shown in Fig. 9. Chemical reaction parameter δ regulates the temperature of stretched surface. It is disclosed that φ increases for higher estimation of chemical reaction parameter δ < 0. However φ has reverse behavier for δ > 0. It is due to the fact that higher estimation of δ correspond to larger rate of δ which generates the fluid 11

ACCEPTED MANUSCRIPT specie more efficiently and therefore φ enhances. Concentration on chemically reactive flow decays for larger δ. Behavior of Hartman number (Ha), ratio parameter (A), wall thickness 1

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parameter (α) and power law index (n) on surface drag force coefficient Cf Rex2 is shown in

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Figs. 10 and 11. One can see that surface drag force enhances with an increase of ratio parameter (A) while it decreases for larger values of power law index n. Figs. 12 and 13 depict

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the effects of Prandtl number (Pr), heat absorption/generation parameter ( β) and power law −1

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index (n) on Nusselt number N ux Rex 2 . No doubt the presence of Prandtl number and heat absorption/generation parameter provide a reduction of the local Nusselt number. It can be

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further noted that Nusselt number decreases for rising values of Prandtl number and power

Conclusions

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5

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law index n.

The effects of chemical reaction and heat absorption/generation parameter in MHD flow

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of tangent hyperbolic fluid towards a variable thicked surface is investigated. The major conclusions drawn from this analysis can be summarized as follows: • Behaviors of A, W e and λ on f ′ (η) are qualitatively similar. • Larger Pr and β magnify the temperature θ (η) . • Concentration distribution φ (η) decays via larger δ and Sc. 1

• Skin friction coefficient Cf Rex2 has opposite behaviors for higher A and n. −1

• Nusselt number N ux Rex 2 augments when Pr is enhanced however reverse situation is examined for larger Pr .

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sheet with convective condition, Int. J. Heat Mass Transfer 102 (2016) 766-702. [26] R. Ellahi, M. Hassan and A. Zeeshan, Aggregation effects on water base Al2O3-nanofluid over permeable wedge in mixed convection, Asia-Pacific J. Chem. Eng. 11 (2016) 179186.

[27] T. Hayat, M. I. Khan, M. Waqas, A. Alsaedi and T. Yasmeen, Diffusion of chemically reactive species in third grade flow over an exponentially stretching sheet considering magnetic field effects, Chin. J. Chem. Eng. (2016) DOI: 10.1016/j.cjche.2016.06.008. [28] L. Zheng, C. Jiao, Y. Lin and L. Ma, Marangoni convection heat and mass transport of power-law fluid in porous medium with heat generation and chemical reaction, Heat Transfer Eng. (2016) DOI: 10.1080/01457632.2016.1200384. 16

ACCEPTED MANUSCRIPT [29] S. A. Shehzad, T. Hayat, A. Alsaedi and B. Chen, A useful model for solar radiation, Energy, Ecology and Environment 1 (2016) 30-38.

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[30] W. A. Khan, A. S. Alshomrani and M. Khan, Assessment on characteristics of

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heterogeneous-homogenous processes in three-dimensional flow of Burgers fluid, Res.

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Phys. 6 (2016) 772-779.

[31] J. H. Merkin, A model for isothermal homogeneous-heterogeneous reactions in boundary

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layer flow, Math. Comput. Model 24 (1996) 125-136.

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[32] T. Hayat, M. Waqas, M. I. Khan and A. Alsaedi, Impacts of constructive and destructive chemical reactions in magnetohydrodynamic (MHD) flow of Jeffrey liquid due to

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nonlinear radially stretched surface, J. Mol. Liq. 225 (2017) 302-310.

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[33] M. Turkyilmazoglu, An effective approach for evaluation of the optimal convergence

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control parameter in the homotopy analysis method, Filomat 30 (2016) 1633-1650. [34] T. Hayat, M. I. Khan, A. Alsaedi and M. I. Khan, Homogeneous-heterogeneous reactions and melting heat transfer effects in the MHD flow by a stretching surface with variable thickness, J. Mol. Liq. 223 (2016) 960-968. [35] T. Hayat, Z. Hussain, T. Muhammad and A. Alsaedi, Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness, J. Mol. Liq. 221 (2016) 1121-1127.

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ACCEPTED MANUSCRIPT Highlights • Stagnation point flow of tangent hyperbolic fluid in the presence of inclined MHD is

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modeled. • Characteristics of heat transfer are explored in the presence of heat genera-

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tion/absorption effects.

seen for concentration distribution.

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• Heat transfer rate increases with chemical reaction parameter while opposite trend is

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• Skin friction and Nusselt number have been numerically analyzed

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