Chemically reactive flow of micropolar fluid accounting viscous dissipation and Joule heating

Chemically reactive flow of micropolar fluid accounting viscous dissipation and Joule heating

Accepted Manuscript Chemically reactive flow of micropolar fluid accounting viscous dissipation and Joule heating Muhammad Ijaz Khan, Muhammad Waqas, ...

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Accepted Manuscript Chemically reactive flow of micropolar fluid accounting viscous dissipation and Joule heating Muhammad Ijaz Khan, Muhammad Waqas, Tasawar Hayat, Ahmed Alsaedi PII: DOI: Reference:

S2211-3797(17)31255-X https://doi.org/10.1016/j.rinp.2017.09.016 RINP 934

To appear in:

Results in Physics

Please cite this article as: Khan, M.I., Waqas, M., Hayat, T., Alsaedi, A., Chemically reactive flow of micropolar fluid accounting viscous dissipation and Joule heating, Results in Physics (2017), doi: https://doi.org/10.1016/j.rinp. 2017.09.016

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Chemically reactive flow of micropolar fluid accounting viscous dissipation and Joule heating Muhammad Ijaz Khand>1 , Muhammad Waqasd , Tasawar Hayatd>e and Ahmed Alsaedie d

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

e

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia

Abstract: Inspired by the several applications of non-Newtonian materials, the current investigation manages a theoretical analysis of series solutions in MHD flow of micropolar material towards nonlinear stretchable surface. Mathematical modeling is developed through viscous dissipation, mixed convection, chemical reaction and Joule heating. The phenomenon of heat and mass transfer are investigated simultaneously. The technique of local similarity transformation is utilized in order to transform the governing expressions from PDEs into ODEs. The established non-linear expressions have been tackled analytically by means of homotopic concept. The interference influences and the flow aspects are presented in the form of liquid velocity, temperature and concentration fields. The results described here demonstrate that material parameter boosts the velocity and micro-rotation velocity. It is noticed that thermal and concentration fields are higher when Eckert number and destructive chemical reaction parameter are enhanced. Besides this for the verification of the present findings, the results of presented analysis have been compared with the available works in particular situations.

Keywords: Thermal radiation; Magnetohydrodynamics; Micropolar fluid; Chemical aspect; Convective conditions. 1

Corresponding author:

Email: [email protected], [email protected] (M. Ijaz Khan)

1

1

Introduction

Most substances with less molecular weight like organic/inorganic liquids, inorganic salts solution, salts, molten metals and gases reveal Newtonian flow aspects, i.e. at constant pressure/temperature and in simple shear, both shear stress and shear rate have a direct relation through the constant of proportionality. Such materials are characteristically known as the Newtonian materials. However several substances having industrial importance, particularly of multi-phase type (i.e. emulsions, foams, suspensions and dispersions, slurries etc.) and polymeric solutions and melts (i.e. both man-made and natural) do not adapt the Newtonian theory of direct relationship between shear stress and shear rate in simple shear. Consequently these materials are dierently known as complex, non-linear, rheological complex or non-Newtonian materials [1]. The tendency for investigating non-Newtonian materials has been enhanced with respect to significance of this type of liquid in industry. Several models regarding non-Newtonian materials have been reported. Amongst these models there is micropolar material which was initially addressed by Eringen [2]. In the flow expression of micropolar material, both the conventional velocity vector and micro-rotation vector are connected with each other. Hence micropolar material is useful for describing the flow characteristics at rotation and micro-scale [3, 4]. Distinct flow structures have been implemented for reporting this model. For illustration Rashad et al. [5] reported mixed convection aspect in stratified stretchable flow of micropolar material. Eects of heat sink/source and mixed convection in magnetohydrodynamic (MHD) stretchable flow of micropolar liquid are reported by Sandeep and Sulochana [6]. Animasaun [7] explored temperature dependent conductivity and viscosity impacts in stagnation point flow of micropolar material through melting aspect. Analysis of micropolar liquid via distinct physical aspects are provided by Shehzad et al. [8], Hayat et. 2

[9] and Waqas et al. [10]. The growing requirement for hydrometallurgical industries and chemical reaction demands the investigation of heat/mass transfers with chemical reaction. No doubt there are several transport mechanisms that are controlled by the joint movement of buoyancy forces subject to thermal/mass diusion in the occurrence of chemical reaction aspect. These processes arise in the combustion and nuclear reactor safety systems, chemical and metallurgical engineering, solar collectors, oxidation and synthesis materials, evaporation at the surface of a water body, energy transfer in a cooling tower, cleaning and chemical processing of materials, manufacturing of ceramics or glassware etc. [11]. In the context of diusion of chemical species, Krishnamurthy et al. [12] examined chemical reaction and porous medium characteristics in thermally radiative stretchable flow of magneto Williamson nanoliquid. Khan and Hashim [13] developed numerical solutions for magneto Carreau material considering temperature dependent conductivity and first order chemical reaction. Combined characteristics of magnetic field and chemical reaction in time dependent stretchable flow of viscous material with heat generation and viscous dissipation is explicated by Daba and Devaraj [14]. Hayat et al. [15] scrutinized chemical reaction and Joule heating aspects in stretchable flow of Carreau liquid through convective heating process. Outcome for inclined MHD aspect in stretching flow of hyperbolic tangent material is elucidated by Khan et al. [16]. The electrically conducting liquid motion across the magnetic field is signified as magnetohydrodynamics (MHD). The consideration of MHD has became a topic of significance due to its widespread demands for improvement of several medical, chemical and industrial devices. Few of these include MHD throttles, magnetic position sensing in DC electric motors, magnetometers, electroslag remelting, automotive fuel level indicator and magnetic devices 3

for separation and purification of alloys [17]. Several attempts regarding MHD aspect have been presented. For instance Bhattacharyya [18] addressed the MHD flow and heat transfer for an incompressible viscous material in the region of a three-dimensional (3D) stagnation point. Characteristics of Joule heating in MHD peristalsis of a nanoliquid with compliant walls are reported by Reddy and Reddy. [19]. Khan and Khan [20] explored electrically conducting power-law nanoliquid flow with convective heating eect. Combined eects of heat and mass transfer aspects in stretchable flow of Newtonian material considering MHD are disclosed by Mabood et al. [21]. Hayat et al. [22] contemplated the impacts of heat source and MHD in permeable stretched flow of Burgers liquid. Modeling and investigation of magneto third-grade material with chemical reacting species is communicated by Hayat et al. [23, 24]. Few recent attempts on MHD flow can be stated through [25-29]. The thermal radiation eect is relevant in regulating the temperature of a system/altering heat transfer rate and controlling thermal boundary layer/altering thermal boundary layer structure eectively. Investigations related with this phenomena have been carried on by several researchers. For illustration Rahman et al. [30] discussed the impact of heat transfer to modified second grade material in the presence of mixed convection and thermal radiation. Thermal radiation characteristics in the peristalsis of MHD Williamson nanoliquid in a tapered asymmetric channel are discussed by Kothandapani and Prakash [31]. Hayat et al. [32] explored thermal radiation impact in slip flow of viscous nanoliquid with melting aspect. Importance of thermal radiation in stretchable flow of viscous liquid considering dierent types of nanoparticles is reported by Aly and Sayed [33]. Hayat et al. [34] reported nonlinear convection and magnetic field characteristics in three-dimensional radiative stretched flow of Maxwell nanoliquid. Thermal radiation and aligned MHD eects in viscous nanoliquid towards stretchable surface are addressed by Rashid et al. [35]. 4

The analysis regarding mixed convection is significant in interpreting the heat transport processes intricate in industry. Utilizations of mixed convective liquid flows may be found in the cooling of electronic devices and the operation of heat exchangers [36]. Hayat et al. [37] investigated nonlinear radiative mixed convective stretchable flow of Oldroyd-B liquid considering convective heating phenomena. Slip flow of three dimensional magneto viscous nanoliquid with nonlinear thermal radiation and mixed convection is modeled and examined by Mahanthesh et al. [38]. Mamourian et al. [39] reported the implementation of Taguchi approach in Cu—water nanoliquid with entropy generation and mixed convection. Impacts of chemical reaction and heat generation in nonlinear slip flow of Casson liquid submerged in a permeable medium with convective heating and mixed convection are addressed by Ullah et al. [40]. In light of the aforestated review, the emphasis of present attempt is to address the influences of magnetic field and mixed convection in nonlinear stretchable flow of micropolar liquid. The novelty of the present study includes the following aspects: (i) It is essential to analyze thermal radiation eect in the current investigation as it has meaningful part in space technology and industrial processes comprising high operating temperatures in order to attain high thermal e!ciency. (ii) Viscous dissipation characteristics plays an imperative part like an energy source which alters the temperature distribution and therefore altering the heat transport rate. Due to this reason that viscous dissipation is accounted in the current analysis. (iii) Instead of utilizing constant surface temperature, the idea of convective heat and mass conditions is implemented. These condition characterizes the heat/mass transport rates through the surface which are proportional to the local dierence in temperature/concentration with ambient conditions. Specifically, consideration of convective boundary condition boosts the temperature/concentration. (iv) Meaningful series solutions 5

are constructed via homotopic approach [41-47]. Besides this the structure of current analysis is as per follows. Flow formulation is arranged in Section 2. Section 3 describes the detailed analysis of the analytical solutions for the non-dimensional velocity, temperature and concentration. Characteristics of several physical variables on dimensionless profiles for the velocity, temperature, concentration, skin friction, Nusselt and Sherwood numbers are addressed through graphs and Tables in section 4. Section 5 reports the conclusion of presented analysis.

2

Formulation

Here two-dimensional (2D) mixed convective flow of incompressible micropolar material bounded by impermeable stretchable surface is investigated. Consideration of Lorentz force give rise to magnetohydrodynamics. There is no electric field in magnetohydrodynamics. Heat transfer process is performed in the presence of viscous dissipation, thermal radiation, convective heating and Joule heating. Eects of convective mass transfer and chemical reaction are also retained. The thermophysical properties of liquid are assumed to be constant. Detailed flow assumptions can be understood through Fig. 1.

Fig. 1. Schematic diagram. The governing expressions under the usual boundary layer approximation are [10, 13, 6

47]:

Cx Cy + = 0> C{ C| µ ¶ n C 2 x n CQ + +  C| 2  C| E 2 ({) +j ( W (W  W" ) +  F (F  F" ))  x>  µ ¶ CQ  C2Q n Cx CQ +y = 2Q + > x  C{ C| m C| 2 m C|

(1)

Cx Cx +y = x C{ C|

µ ¶ µ ¶2 CW n1 C 2 W +n Cx CW +y = x + 2 C{ C| fs C| fs C| µ ¶ 2 W E 16 C CW W3 > + 0 x2 + + W fs 3n fs C| C| x

CF CF C2F +y = G 2  n2 (F  F" ) = C{ C| C|

(2) (3)

(4) (5)

Here (x> y) are the velocity components in ({> |) directions respectively,  the kinematic viscosity,  the density of fluid, n the vortex viscosity, j the gravitational acceleration, ( W >  F ) the (thermal, concentration) expansion coe!cients,  the electrical conductivity, E ({) = E0 {

q31 2

the non-uniform magnetic field,  the spin gradient viscosity, m the microin-

ertia, Q the micro-rotation velocity, n1 the thermal conductivity of liquid,  the dynamic viscosity,  W the Stefan-Boltzmann constant, nW the mean absorption coe!cient, (W> F) the (temperature, concentration), (W" > F" ) the ambient (temperature, concentration), G the mass diusion and n2 the reaction rate, n2 A 0 for destructive chemical reaction and n2 ? 0 for the constructive chemical reaction.

7

2.1

Relevant boundary conditions

The proper boundary conditions proposed by the dynamics of the problem are given by: x = xz ({) = f{q > y = 0> Q = p0 n1 CW = k1 (Wi  W )>  G C|

Cx > C|

CF = k2 (Fi  F) at | = 0> C|

x $ 0> Q $ 0> W $ W" > F $ F" as | $ 4=

(6) (7)

Here xz ({) = f{q denotes the stretching velocity, y = 0 means there is no suction/injection, p0 the boundary parameter having range 0  p0  1 and (k1 > k2 ) the convective (heat, mass) transfer coe!cients.

2.2

Local similarity renovation

Consider the following dimensionless variables in order to obtain the non-dimensional form of governing expressions: r r f (q + 1) q31 f (q + 1) q31 W  W" q = { 2 |> Q = f{ { 2 j () >  () = >   Wi  W" r  ¸ f (q + 1) q31 F  F" q1 0 q 0 { 2 i () + i () = (8) ! () = > x = f{ i () > y = Fi  F" 2 q+1 In the aforestated expression (8),  the similarity variable, (i () >  () > ! ()) are the dimensionless (velocity, temperature, concentration). Also the utilization of x = f{q i 0 () q ¤ q31 £ q31 0 2 and y = f(q+1) { i () in Eq. (1) satisfied it identically. i () + 2 q+1

2.3

Transformed profiles of governing expressions

Now invoking the expression (8) in the expressions given in (2)-(7), one can obtains the following nonlinear dierential systems:

(1 + N) i 000 + ii 00 

2q 02 2 2 i + Nj0 + ( + Q!)  Kd2 i 0 = 0> q+1 q+1 q+1 8

(9)

µ ¶ 3q  1 0 2N N i j (2j + i 00 ) = 0> 1+ j 00 + ij 0  2 q+1 q+1 µ ¶ 4 2 1 + U 00 + Pr i0 + Pr Hf (1 + N) i 002 + Pr HfKd2 (i 0 ) = 0> 3 !00 + Vfi !0  Vf! = 0>

(10) (11) (12)

³ ´ here primes signify dierentiation with respect to > N = n the material parame³ ´ ³ ´ Ju{W { ter,  = Ju the thermal buoyancy parameter, Q = the concentration buoyJu{ Re2{ µ ¶ j (Wi 3W" ){q31 ancy parameter, Ju{ = W  2 the Grashof numbers due to temperature, Ju{W µ ¶ ¡ ¢ j F (Fi 3F" ){q31 = the Grashof numbers due to concentration, Re{ = xzy { the local 2 ³ ´ ³ ´ W 3 E 2 Reynolds number, Kd2 = f0 the Hartman number, U = 4nnWW" the thermal radiaµ ¶ ´ ³ x2z s tion parameter, Hf = f W 3W the Prandtl number, Vf the Eckert number, Pr = f n1 s( i ") ¡ ¢ ¡ ¢ = G the Schmidt number and  = nf2 the chemical reaction parameter.

2.4

Transformed boundary conditions

Making use of Eq. (8) in Eqs. (6) and (7) yields: i (0) = 0> i 0 (0) = 1> j (0) = p0 i 00 (0) > 0 (0) =  1 (1   (0)) > !0 (0) =  2 (1  ! (0)) i 0 (4) $ 0> j (4) $ 0>  (4) $ 0> ! (4) $ 0. ³ In the above expressions  1 =

k1 n1

p ´ f

¡ and  2 =

Biot numbers respectively.

9

k2 G

p ¢ f

(13) (14)

the thermal and concentration

2.5

Physical quantities

Here surface drag force (Fi ), Nusselt number (Q x{ ) and Sherwood number (Vk{ ) are Fi =

z > x2z

{tz > n1 (Wi  W" ) {tp Vk{ = > G(Fi  F" ) Qx{ =

(15) (16) (17)

here  z the wall shear stress and (tz > tp ) the (heat, mass) fluxes are given by ¶¯ µ ¯ Cx + nQ ¯¯ >  z = ( + n) C| |=0 µ ¶ µ ¶ 3 CW 16 W W" CW tz = n1  > C| |=0 3nW C| |=0 µ ¶ CF tp = G = C| |=0

(18)

(19) (20)

Substituting Eqs. (18)-(20) into Eqs. (15)-(17), we obtains the surface drag force, Nusselt number and Sherwood number in dimensionless forms as: [email protected]

Fi Re{

Qx{ [email protected] {

r

q+1 (1 + (1  p0 )N) i 00 (0)> 2 µ ¶ 4 =  1 + U 0 (0)> 3 =

Vk{ [email protected] = !0 (0)= {

3

(21) (22) (23)

Computational procedure and convergence analysis

This section intends to compute the series solutions of established nonlinear systems (9)-(12) subject to boundary conditions (13) and (14) via homotopy concept. We adopted homotopy in view of the following aspects. (i) The homotopy method does not comprise any large/small parameters. (ii) The convergence of established expressions can be justified easily. 10

(iii) HAM delivers great liberty to choose the linear operators and base functions. In the present problem we are dealing with semi infinite domain situation. Therefore in order to achieve fast convergence we select the exponential type base functions as: i0 () =

¡ ¢ 1  h3 >

j0 () = p0 h3 > 1 exp()> 1 + 1 2 !0 () = exp()> 1 + 2 0 () =

(24) (25) (26) (27)

with Li = i 000  i 0 >

(28)

Lj = j 00  j 0 >

(29)

L = 00  >

(30)

L! = !00  !=

(31)

The aforestated linear operators fulfills the following characteristics: Li (F1 + F2 h + F3 h3 ) = 0>

(32)

Lj (F4 h + F5 h3 ) = 0>

(33)

L (F6 h + F7 h3 ) = 0>

(34)

L! (F8 h + F9 h3 ) = 0>

(35)

where Fl (l = 1  9) specify the arbitrary constants. It is well-known that the homotopy concept has superb flexibility to interpret the auxiliary variables (}i > }j > } > }! ) about control and adjustment of the convergence of series solutions. We have sketched the ~-curves in Fig. 1 in order to interpret the appropriate values of 11

(}i > }j > } > }! ). Clearly acceptable values for the established solutions are in the ranges 1=55  ~i  0=25> 1=50  ~j  0=20, 1=45  ~  0=45 and 1=75  ~!  0=10= Moreover the convergence of developed series solutions is also justified numerically in Table 1. Here 15th order approximation is su!cient for the convergence of Eq. (10) however 20th order approximations are enough for convergence of Eqs. (9), (11) and (12). . O

 1 5 (F  +D  J  Q  3U  m

J

 6F  J 

I

+ J + T + /  I +  /



 

I

+/ J +/ T +/ I +/

 

 

 

 

 

² f ² g²T²I

 



Fig. 2. ~curves for i> j>  and != Table 1: Series solutions convergence for distinct order of approximations when N =  = 0=2> Kd = 0=1>  =  1 = 0=6> q = 1=5> Pr = 1=2> p0 =  2 = 0=5> Q = U = Hf = 0=3 and Vf = 1=0= Order of approximations i 00 (0) j0 (0) 0 (0) !0 (0) 1

0.9788

0.5144

0.3001

0.3311

5

0.9286

0.5059

0.2314

0.3322

10

0.9222

0.5038

0.2257

0.3328

15

0.9224

0.5038

0.2260

0.3328

20

0.9224

0.5038

0.2260

0.3328

25

0.9224

0.5038

0.2260

0.3328

30

0.9224

0.5038

0.2260

0.3328

12

4

Analysis

This section aims to highlight the features of several variables like material parameter (N) > power law index (q) > thermal buoyancy parameter () > concentration buoyancy parameter (Q) > Hartman number (Kd) > boundary parameter (p0 ) > thermal radiation parameter (U) > Prandtl number (Pr) > Eckert number (Hf) > thermal Biot number ( 1 ) > Schmidt number (Vf) > chemical reaction parameter () and concentration Biot number ( 2 ) on velocity (i 0 ) > temperature () and concentration (!). This objective is achieved via illustration and interpretation of Figs. 3-18.

4.1

Characteristics of velocities

This subsection communicates the influences of distinct variables on velocity (i 0 ) through Figs. 3-10. Impact of N on i 0 is discussed in Fig. 3. Here i 0 enhances via larger N= In fact larger N corresponds to less viscosity liquid which results in the enhancement of i 0 = Fig. 4 highlights the behavior of q on i 0 = As expected i 0 and associated thickness layer are reduced when q is incremented. Variation in i 0 via  is exhibited in Fig. 5. Clearly i 0 and related thickness of boundary layer rise when  is incremented. Enhancement in  illustrates that buoyancy force due to gravity rises which boosts the velocity (i 0 ) and corresponding layer thickness. Fig. 6 interprets the influence of Q on i 0 = Both i 0 and associated thickness of boundary layer boosts via larger Q= Physically concentration buoyancy force increments which corresponds to higher velocity. Analysis of the characteristics of Kd is depicted in Fig. 7. Here i 0 decays when Kd is taken larger. The liquid velocity for the hydrodynamic situation is greater when compared with hydromagnetic situation. Physically Lorentz force which arises in magnetic parameter (Kd) becomes greater via increase in Kd. In other words greater Lorentz force reduce the velocity. Fig. 8 portrays the role of N on micro-rotation 13

velocity (j) = It is noted that initially j decays for higher estimation of N= However it is concluded that micro-rotation velocity (j) is higher when N is incremented. It is due to the fact that viscosity of the material decays which results in the improvement of micro-rotation velocity (j) = Moreover it is also observed that micro-rotation velocity (j) is higher when N A 0 in comparison with N = 0= Influence of power index (q) on micro-rotation velocity (j) is explained in Fig. 9. Clearly micro-rotation velocity (j) decays when q is incremented. Fig. 10 interprets the characteristics of boundary parameter (p0 ) on micro-rotation velocity (j). Larger estimation of p0 results in the enhancement of micro-rotation velocity (j) = It is also noted there is no micro-rotation velocity when p0 = 0=

4.2

Characteristics of temperature and concentration

Salient characteristics of distinct variables on temperature () and concentration (!) are argued through Figs. 11-18. Analysis of the radiation parameter (U) is addressed in Fig. 11. No doubt thermal radiation has a foremost impact on the temperature of the fluid. Here it is noticed that temperature distribution augments when radiation parameter is incremented. It is due to the fact that mean absorption coe!cient (nW ) decays via increase in U which is accountable for the augmentation of thermal field. Fig. 12 signifies the impact of Pr on temperature () = Actually larger Pr correspond to less thermal diusivity which decays the temperature and related thickness layer. Variation of Hf on temperature () is disclosed in Fig. 13. Clearly temperature () rises via larger Hf. In fact heat energy in the liquid is saved via larger Hf. It is due to friction forces that ultimate boosts the temperature () = Fig. 14 describes the influence of Biot number ( 1 ) on temperature () = Larger  1 is accountable for an increment in temperature () and corresponding thickness layer. Physically the involvement of larger heat transfer coe!cient correspond to high temperature. The vari14

ation of temperature () with respect to Hartman number (Kd) is discussed in Fig. 15. As expected larger Kd yields higher temperature. Occurrence of Lorentz force in Kd provides more resistance which corresponds to higher temperature. Fig. 16 illustrates the features of Schmidt number (Vf) on concentration (!) = As Schmidt number (Vf) involves Brownian diusion coe!cient which in fact decays for higher estimation of Vf= Therefore concentration (!) and related thickness layer are reduced when Vf is incremented. Salient characteristics of chemical reaction parameter () are portrayed in Fig. 17. Here larger  yields lower concentration distribution. Larger values of  correspond to higher rate of destructive chemical reaction which dissolves or terminates the liquid specie more eectively. Thus concentration distribution (!) decays. Fig. 18 explores the behavior of  2 on concentration distribution (!) = Here ! and corresponding thickness layer increments via larger  2 = Physically mass transfer coe!cient (k2 ) rises which yields higher concentration distribution.

4.3

Characteristics of surface drag force

Influences of distinct variables (i.e. Kd> > N and Q) on surface drag force are discussed in this subsection through Table 2. Clearly surface drag force rises for higher estimation of Kd and N whereas it diminishes via larger  and Q= Table 5 interprets the comparative analysis of obtained solutions with [47]. It is concluded from this Table that our employed technique has reasonable agreement with the technique used in [47].

4.4

Characteristics of heat and mass transfer rates

Tables 3 and 4 are displayed to scrutinize the features of emerging variables on heat mass transfer rates. It is found that heat transfer rate is higher when Pr> U>  and  1 are augmented. However there is decay in heat transfer rate when Hf and Kd are enhanced (for 15

detail see Table 3). Analysis of mass transfer rate is made through Table 4. Clearly mass transfer rate is increasing function of  2 > Vf and = Table 6 communicates the comparison of heat transfer rate with the analysis reported by [47]. This Table reveals that both HAM and numeric approaches have good agreement. Q

 O

 +D

 J

J

 m

J

 5

(F

1

 6F

 3U

.





O

 +D

 J 

 D

J

 5

J

(F

1

 6F

 3U









Q



.





Q



.



Q



.





Q



I +K/

I +K/



.











 



 K









 K

Fig. 3. i 0 via N= Q

 .

 +D

 J

J

 m

J

 5

(F





Fig. 4. i 0 via q= 1

 6F

 3U



Q



 .

 +D

 J 

J

 m

J

 5

(F

 O

6F

 3U









1





1





1





O



O O





I +K/

I +K/



1

O











 



 K











 .

O

 J

J

 m

J









K

Fig. 5. i 0 via = Q



 5

(F

1

Fig. 6. i 0 via Q=  6F

 3U











+D



+D



+D



 O

 +D

 J

J

 m

J

 5

(F

1

 6F





.



.



.



.



 3U



I +K/

J + K/



+D

Q





























K

 K

Fig. 7. i 0 via Kd=

Fig. 8. j via N= 16









O

 +D

 J

J

 m

J

 5

(F

1

 6F





Q



Q



Q



Q







Q

 .

 +D

O

 J 

 J

J

 5

(F

1

 6F





J + K/



 3U

 3U

m



m



m



m





J +K /

.

 







 

Q



 .

O



 +D













K

K

Fig. 9. j via q=

Fig. 10. j via p0 =

 J 

J

 m

J

 (F

1

 6F

 

 3U



Q

 .

O

 +D

 J

J

 m

J

 5



(F

1

 6F

5



3U



5



3U



5



3U



5



3U





T +K/



T +K/





 











 



 K









Fig. 11.  via U= Q

 .

O

 +D

 J

J

 m 

J

 5





Fig. 12.  via Pr =

1

 6F

 3U







(F



(F



(F (F

Q

 .

 +D

O

 J

 m 

J

 5

(F

1











T + K/

T +K/

 K



 6F

J



J



J



J



 3U





 

 



 K





Fig. 13.  via Hf=

 





 K



Fig. 14.  via  1 =

17







Q

 .

O

 J 

 m 

J

 5

J

(F

1

 3U

+D



+D



+D



+D



Q



O

 +D

 J 

J

 m

J

 5

(F







1

 3U

6F



6F



6F



6F















Q



 .



O

 +D











K

 K

Fig. 15.  via Kd=

Fig. 16. ! via Vf=

 J

 m

J

 5

(F

1



 6F

J



J



J



J



 3U





Q



 .

O



 +D

 J 

J

 m



 5

(F

 

I + K/



T + K/

 .



I + K/

T + K/



 6F





1

 6F

J



J



J



J





 3U



 







 





 K















K

Fig. 17. ! via =

Fig. 18. ! via  2 =

18





³ ´ [email protected] Table 2: Surface drag force Fi{ Re{ via distinct values of Kd> > N and Q when  =  1 = 0=6> q = 1=5> Pr = 1=2> p0 =  2 = 0=5> U = Hf = 0=3 and Vf = 1=0= Kd  0.0

N

Q

0.2 0.1

[email protected]

Re{ Fi

0.3 -1.1290

0.3

-1.1747

0.6

-1.3031

0.1

0.0

-1.2503

0.4

-1.0329

0.8

-0.8514

0.2 0.0

-1.0779

0.25

-1.1482

0.5

-1.2136

0.2

0.0 -1.1458 0.5 -1.1273 1.0 -1.1083

19

³ ´ Table 3: Heat transfer rate Qx{ [email protected] via distinct values of  1 > > Kd> U> Pr and { Hf when  = 0=6> q = 1=5> Pr = 1=2> p0 =  2 = 0=5> U = 0=3 and Vf = 1=0= 1



Kd U

0.3 0.2 0.1

Pr

Hf

Qx{ [email protected] {

0.3 0.7 0.1 0.2121

0.5

0.2877

0.7

0.3404

0.6 0.0

0.2942

0.4

0.3324

0.8

0.3545

0.2 0.0

0.3176

0.3

0.3028

0.6

0.2609

0.1

0.0

0.2430

0.2

0.2925

0.4

0.3390

0.3 1.0

0.3022

1.3

0.3224

1.6

0.3371

1.2 0.0 0.3987 0.4 0.2900 0.8 0.1893

20

Table 4: Mass transfer rate

³ ´ Vk{ [email protected] via distinct values of  2 > Vf and  when {

N =  = 0=2> Kd = 0=1> q = 1=5> Pr = 1=2> p0 = 0=5>  1 = 0=5 and Q = U = Hf = 0=3= 2

Vf

Vk{ [email protected] {



0.6 1.0 0.6 0.3743 0.7

0.4109

0.8

0.4435

0.5 1.1

0.3384

1.2

0.3435

1.3

0.3480

1.0 0.0 0.2739 0.5 0.3266 1.0 0.3514

21

Table 5: Comparative results of surface drag force with [48] via dierent q when N = 0 = Kd =  = Q = p0 =

q

Cortell [48] Present results

0.0

0.627547

0.6276

0.2

0.766758

0.7669

0.5

0.889477

0.8895

0.75 0.953786

0.9539

1

1.0000

1.0000

1.5

1.061587

1.0616

3

1.148588

1.1486

7

1.216847

1.2169

10

1.234875

1.2349

20

1.257418

1.2575

100

1.276768

1.2768

22

Table 6: Comparative results of heat transfer rate with [48] via dierent Hf and q when Pr = 1=0> Kd = U = 0 and  1 $ 4= Hf

5

q

Cortell [48] Present results

0.0 0.2

0.610262

0.6102

0.5

0.595277

0.5952

1.5

0.574537

0.5748

3.0

0.564472

0.5648

10.0 0.554960

0.5550

0.1 0.2

0.574985

0.5752

0.5

0.556623

0.5568

1.5

0.530966

0.5310

3.0

0.517977

0.5181

10.0 0.505121

0.5055

Conclusions

Prime theme of current investigation is report the simultaneous characteristics of mixed convection and magnetohydrodynamics in nonlinear stretchable flow of micropolar fluid. Consideration of Joule heating, thermal radiation, convective heating and viscous dissipation reports the heat transfer aspect. Mass transfer analysis is made through first order chemical reaction and convective mass conditions. Based on the whole analysis, some key findings are developed as: • Impacts of material parameter (N) on both velocities (i.e. i 0 and j) are similar. • Larger Hartman number (Kd) corresponds to lower velocity and higher surface drag 23

force. • Boundary parameter (p0 ) yields higher micro-rotation distribution. It is noted that there is no micro-rotation velocity when p0 = 0= • Higher estimation of Prandtl number corresponds to lower temperature. Heat transfer rate improves when Prandtl number is incremented. • The influences of thermal radiation parameter and Biot number are noted similar. • Chemical reaction parameter and Schmidt number have similar impacts on concentration distribution. • Both heat and mass transfer rates are higher via larger Prandtl and Schmidt numbers.

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