Hydromagnetic flow of Casson nanofluid over a porous stretching cylinder with Newtonian heat and mass conditions

Journal Pre-proof Hydromagnetic flow of Casson nanofluid over a porous stretching cylinder with Newtonian heat and mass conditions Syed Muhammad Raza Shah Naqvi, Taseer Muhammad, Mir Asma

PII: DOI: Reference:

S0378-4371(19)32208-3 https://doi.org/10.1016/j.physa.2019.123988 PHYSA 123988

To appear in:

Physica A

Received date : 5 May 2019 Revised date : 26 September 2019 Please cite this article as: S.M.R.S. Naqvi, T. Muhammad and M. Asma, Hydromagnetic flow of Casson nanofluid over a porous stretching cylinder with Newtonian heat and mass conditions, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123988. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

Journal Pre-proof Hydromagnetic flow of Casson nanofluid is investigated.

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Flow is generated due to a porous stretching cylinder.

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Brownian motion and thermophoresis effects are considered.

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Newtonian heat and mass conditions are implemented.

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*Manuscript Click here to view linked References

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Hydromagnetic flow of Casson nanofluid over a porous stretching cylinder with Newtonian heat and mass conditions Syed Muhammad Raza Shah Naqvi1,2, Taseer Muhammad3,* and Mir Asma4 1

Department of Mathematics, Pusan National University; San 30 Jangjeon-dong Geumjeong-gu,

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Busan 609-735, Korea Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, P. R. China 3 4

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Department of Mathematics, Government College Women University, Sialkot 51310, Pakistan Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia

* Correspondence: [email protected] (Taseer Muhammad)

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Abstract: Blood flow in arteries is an example of Casson fluid flow among other major applications of this fluid. Thus, it would be advantages to discuss the presence of Nano particles in Casson fluid for biomedical applications. In this study, the electrically conducting Casson fluid mixed with Nano size metallic particles, in the existence of magnetic field and thermal radiation, has been addressed owing to a moving boundary cylinder. The convective heat transfer has been augmented with Joule heating and dissipation. Mathematical formulation of the problem involves a

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set of simultaneous nonlinear partial differential equations of high order. These governing equations are distorted to ordinary differential equations via apposite variables. Numerical solution for the subsequent mathematical model of the problem is sought out. Computational results are obtained

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for different ranges of the pertinent parameters and discussed in detail. The results have been elaborated in graphical form for velocity field, heat transfer rate at the surface, concentration function and thermal field. An increase in the intensity of magnetic field and Casson parameter is causing a decrease in the velocity while an enhanced temperature is noted for increasing values of

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Casson and magnetic parameters.

Keywords: Casson nanofluid; Viscous dissipation; Joule heating; Newtonian heat and mass conditions; Stretching cylinder.

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1. Introduction Among other non-Newtonian fluids, Casson fluid is one of its kinds, also known as shear thinning liquid. It behaves like two states of matter. If the applied yield stress is greater than shear stress, it response inherits the properties of solid while in the other case when applied yield stress is smaller or lesser it behaves like liquids. Human blood, jelly, honey and tomato sauce etc. are some of

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its examples [1,2]. Because of its wide range of applications Casson fluids are studied by mathematicians, biomedicals, scientists as well as engineers [3-7]. Addition of Nano particles in Casson fluids enhance its thermal conductivity up to 50 percent [8,9]. Buongiorno [10] presented the

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comprehensive survey of heat convection in nanofluids. Furthermore, magnetic nanoparticles can be guided within the non-magnetic fluids and bodies. In some clinical applications, magnetic fluids are supplied to certain tumor to cure it, through blood flow. It is also well-known hat magnetic nanofluids are much more reliable and effective for tumor treatment as compared to nonmagnetic

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Nano particles [11]. The readers are referred to see some recent studies related to Casson fluid in [12-14]

Casson nanofluids, on different geometries, are discussed by numerous authors in recent years. First time concept of nanofluid flow was used by Choi [9]. In past several studies have examined the additional effects on heat transfer in case of fluids with Nano particles [15,16]. In a similar work, Nadeem et al. [17] analyzed the convective heat transfer and its radiation effects for magnetohydrodynamic (MHD) fluid flow and determined some mathematical solutions by

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implementing adaptive approach. They analyzed and experimented for stretching sheet. In another study, Rizwan et al. [18] presented some results for exponentially shrinking sheet under convection

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boundary constraints and implemented Casson fluid model. In a similar work [19], MHD Casson fluid was discussed in additional presence of Nano particles. They also considered the exponentially stretching surface. Abolbashri et al. [20] presented the detailed response of entropy generation in case of Casson nanofluids for stretching surfaces. They determined as approximate analytical solution with conventional Homotopy analysis method. Similarly, Hayat et al. [21] has discussed

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convection flow of Casson nanofluid with additional factors. These factors include radiation, caused by particular heat source, with chemical reaction of first order. Pal et al. [22] has further analysed the nonlinear profile of stretching velocity and suction forces for MHD Casson nanofluid flow. They considered the spatial case of vertically stretching surface. Rauf et al. [23] has extended this scenario and discussed MHD boundary layer flow of Casson nanofluids for spatial case of non-unidirectional stretching surface. In another study [24], the author discussed the results regarding porous medium of stretching sheet and they presented a detailed analysis of chemical reaction’s effects on Casson fluid 2

Journal Pre-proof flow. They also analysed the velocity slip effect in this case. Ahmad et al. [25] has investigated Casson fluid flow with particular magnetic field and Newtonian heating model. The author implemented Keller Box technique to obtain a solution to proposed model. Ibrahim et al. [26] has described the effect of radiation presented in form of thermal reaction, generated by heat source their chemical reaction in the case of Casson MHD fluid flow for stretching surface. The effect of Brownian motion and thermophoresis has also been incorporated in their model. They found

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approximated solution using homotopy analysis technique. Another interesting study for non-steady electrically conducting flow was presented by Imran et al. [27]. They analysed the MHD boundary

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layer flow over bodies with specific shape of wedge design. A few attempts have been made where nanofluid behavior has been discussed to [28-32].

Fluid flow over stretching/shrinking cylinder has frequent applications in engineering, industrial process and pharmaceuticals. Najib et al. [33] has discussed the relation of transfer of mass under chemical reaction for stretching cylinder. Qasim et al. [34] presented the MHD boundary layer slip

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flow of ferrofluid along with a stretching cylinder. In a similar study [35], the author discussed stagnation point flow for starching cylinder case and their heat transfer results. Later, Sulochana and Sandeep [36] extended the scope by adding porosity o medium in the case of stretching cylinder. They comprehensively summarized their results values of temperature with additional factors of injection and suction, magnetic field with uniformity and specific shapes of Nano particles Casson nanofluid flow over cylinder is also discussed in many articles. Malik et al. [37] has

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analyzed the case of vertical exponentially stretching cylinder. They presented their findings for heat transfer properties in case of nanofluid. Hayat et. al. [38] has utilized the case of vertically stretching cylinder. They analyzed stagnation point flow of Casson nanofluid with an additional slip constraint.

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Mahdy and Chamkha [39] has utilized built in MATLAB command bvp45 to solve Buongiorno's model for Casson fluid to classify features of non-Newtonian fluid for anti-stretching cylinder case. Maria et al. [11] introduced a generic convective heat and mass constraint in mixed convection flow of Casson fluid with nanoparticles induced by a stretching cylinder.

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It is well established effect the amount of heat transfer at the surface is directly proportional to local surface temperature. This particular effect is regarded as Newtonian heating effect in literature [40]. The Newtonian heating effect have been utilized in many different research and industrial application, like wise mechanical design of heat exchanger and convection flows, mechanical setups in which surface area of the setup absorb heat from the solar radiation [41]. There are several other studies in which these effects has been comprehensively analyzed for different mechanical setups and environmental conditions [42-46]. 3

Journal Pre-proof In a recently published study Hayat at. el. [40] explored the effects of Newtonian heating mass condition at surface. Later, some of other studies further analyzed the same idea on different geometries and explored different effects [47-52]. None of this study in literature has analyzed the combine effect of Newtonian heating at surface with heat source/sink and radiation for Casson fluid flow mixed nanoparticles. In this study, the characteristics of heat transfer over a starching cylinder is comprehensively

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analyzed in a particular scenario. We have considered the viscous dissipation effect in Casson fluid flow mixed with nanoparticles. In addition, the features of heat transfer via heat source/sink and

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radiation effect are discussed. As a whole this study presents a detailed analysis of MHD convective flow with Joule heating for characterizing the heat transfer properties. An efficient numerical scheme is purposed in MATHEMATICA framework to determine the velocity, concentration and temperature profile of the experimental setup. The velocity and temperature distributions are noticed

2. Materials and methods

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improved for various increasing values of magnetic and Casson parameters.

In this section, a mathematical representation of thermal radiation effect with Joule heating on MHD flow of Casson nanofluid over a stretching cylinder is presented. The flow is governed through stretching of cylinder. The convective heat transfer has been incorporated with Joule heating and dissipation. The fluid motion is in axial direction whereas radial direction is perpendicular to it. The

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phenomena are mathematically described as [53].  (ru )  (rv)   0, x r

 u u 1    2u 1 u   e B0 v   1    2  u,  x r r r      r

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T  T v u r  x

(1) 2

k   2T 1 T    2   r r   C p  r

 B   C p

(2)

 1   u  Q T  T   1        r  Cp     2

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2   B2 1  C T DT  T     rqr   0 u 2    DB    , r r C p r r T  r   

u  x, r   u w ( x ) 

u

  2C 1 C  DT   2T 1 T  C C v  DB  2     , x r r r  Tx  r 2 r r   r

(3)

(4)

u0 x T C , v  x , r   v w ,  h1T ,   h2C , as r  0, u  0, T  T , C  C as r  . l r r

(5) 4

Journal Pre-proof where in the afforested expression u and v are velocity components in axial (x) and radial (r) direction respectively,  is the kinematic viscosity of the fluid, Casson Parameter  , strength of the applied magnetic field is B0 whereas induced magnetic field is negligible,  is the fluid density, fluid temperature and particle concentration are T and C respectively and their corresponding uniform flow values are T and C respectively, k is the thermal conductivity, the specific heat is C p ,  B the



 C   C  p

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plastic dynamic viscosity, Q is heat generation/absorption, qr represents additional radiation term, is the ratio of heat capacity of nanoparticles and the heat capacity of base fluid, DB the

p 

cylinder. The stretching velocity is uw 

u0 x l

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Brownian diffusion coefficient, DT the thermophoresis diffusion coefficient, R is the radius of the where u0 the reference velocity, l is the characteristics

given by  

 B 2 c . r

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length and,  is the constant electrical conductivity,  is the dynamic viscosity. Casson Parameter is

Let us define a suitable transformation for considered problem which satisfy equation of continuity represents in equation (1) [53]. 

u0  r 2  Rn 2   l  2 Rn

 u0 x R f    , v   n ,u  l r 

u0 T  T C  C f   ,    ,     l T C

(6)

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where  is the similarity variable, f ( ) the non-dimensional stream function,  ( ) the nondimensional temperature and    is the non-dimensional concentration function. Substitution the

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above appropriate relation in equations (2), (3) and (4) we get,  1 2 2 1    1  2  f   2 f    ff   f '  Ha f   0,   



1  2  (1 

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

  4R 1 )   Pr  Nb    Nt 2   1   Ec Pr f 2   2   Pr f    Ec Pr Haf 2  0, 3   

1  2   '' 2   Sc f   

Nt  1  2    2   0, Nb

f  S , f '  1,    (1   (0)),  '   (1   (0)) as  0, f '  0,   0,   0 as  .

5

(7)

(8)

(9) (10)

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vl 2 u  0 Rn

where   

  B0 2 l is the Hartman number,  is the curvature parameter, Ha   u0 

the radiation parameter, Pr 

cp k

is the Prandtl number, Ec 

is

D is the Eckert number, Nb   B 

uw 2 c pT

 v 

 D 

v

is the Brownian motion parameter, N t   T  is the thermophoretic parameter, Sc  is the Lewis DB  T v  vl u0

heat and mass parameters respectively.

and    h2

vl u0

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is the suction/injection parameter,   h1

number,

are the conjugate

Cf 

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Skin friction coefficient  C f  Nusselt number  Nux  and Sherwood number  Shx  are 2 w xqw xqm , Nu x  , Shx  , 2 k Tw  T  DB  Cw  C   uw

(11)

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where  w is the shear stress, the local heat flux qw and local mass flux qm are given by  u 

 T 

 C 

 w     , qw  k   , qm   DB   .  r r  R  r r  R  r r  R

In dimensional form

1 1 1     1 1  1  C f Re x 2  f   0  , Nux Re x 2   1   , Shx Re x 2    1   , 2    0     0 

v



u0  x 

(13)

2

vl

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where Re x 

uw  x 

(12)

is the Reynolds number.

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The resulting set of governing equations (7)-(9) with boundary conditions, presented in equation (10), is nonlinear and partially coupled. It is hard to find any analytical solution of this problem. This set of equations can be transformed into first order coupled ordinary differential equations using

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reduction of order. We apply the following substitutions: f   z1 , f   z2 ,   z3 ,    z4

(14)

The resulting system of first order equations is as follows: 

2

1  2  z2   z1

 Haz1  fz2







   2 z2 , 1 1  

  Ec Haz12  fz3   2 z3  4R   2 2 , 1  1   z3   Pr  N b z3 z4  N t z3  1   Ec z2    (1  2  ) P (1  2  ) r   3     

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(15)

(16)

Journal Pre-proof  N  (1  2 ) z   2 z  , (1  2 ) z4    2 z4  Sc fz4   t  3 3  N b   

(17)

f (0)  S , z1 (0)  1, z3 (0)   (1   (0)), z4 (0)   '(1   (0)), z1 ()  0, ()  0,  ()  0

(18)

In equation (10) we replace infinity with some finite value. The numerical results of the above set of equations have been obtained and presented in graphical form of non-dimensional velocity

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f '   , temperature    and concentration    .

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3. Results and discussion

This section presents the results regarding behavior of different parameters on velocity, temperature and particle concentration. The decrease in radius of cylinder causes the increase in the values of α, which results in decrease of contact area of the fluid with fluid boundary and hence increase in the velocity as demonstrated in Figure 1a. The effect of suction parameter S (S>0) on

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velocity f '   is displayed in Figure 1b. The curve of axial velocity maps down with increase in S because some of fluid is sucked into the cylinder. Similarly, the higher values of Casson parameter β slow down the flow velocity f '   as presented in Figure 1c. The magnetic parameter Ha which appears due to Lorentz force that retards the flow and in addition to reduction of thickness of boundary layer and hence consequently decrease in velocity f '   as seen in Figure 1d. 1.0

1.0

S 0

0 0.8

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0.1

f'

0.6

0.2

0

2

4

6

0.6

S 1 S 1.5

0.4 0.2 0.0

8

(a)

0

2

4

(b)

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0.0

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0.4

S 0.5

f'

0.3

0.6

0.8

7

6

8

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1.0

Ha 0

0.7

0.8

1.4

0.6 0.4

0.6

Ha 0.8 Ha 1.2

0.4

2

0.2

0.2

0

2

4

6

0.0

8

0

of

0.0

Ha 0.4

f'

f'

2

0.8

2

4

6

8

(d)

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(c)

Figure 1. (a) The plot of velocity under the effect of curvature parameter; (b) The plot for curves of velocity under the effect of Suction parameter; (c) The plot for curves of velocity

Hartman number.

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under the effect of Casson parameter; (d) The plot for curves of velocity under the effect of

Figure 2a is mapped to elaborate the influence of curvature parameter  on    . It seems that fluid cools down near the boundary of cylinder and it gains higher temperature away from the surface with increase in  . Physically increase of    is because of decay in heat transport rate while this decay is caused by increase in boundary layer thickness because of increase in  . Figure 2b discloses the effect of Casson parameter β on temperature function    . It is seen that temperature

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curve rises up for higher values of β. Similarly, the increase in Eckert number demonstrate a rapid rise in thermal distribution as depicted in Figure 2c. Physically, storage of energy caused the increase

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in the motion of liquid particles for larger Eckert number. Due to this increase in temperature is also observed. The increase in conjugate heat parameter exibites an escalation in thermal distribution and its associated thermal layer as depicted in Figure 2d. For larger conjugate heat parameter, there is enhancement in heat transfer coefficient which describes more heat is shifted from heated cylinder

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surface to cooled liquid surface.

But the temperature function decreases with increments in the value of suction parameter S as we can observe in Figure 2e. While the effect of Brownian motion parameter N b shown in Figure 2f. It is examined that for larger N b the associated boundary layer thickness is heightened as well as    . Physically a rise in N b , an enhancement of    occurs due to increase in the haphazard

motion of molecules. An increase in Prandtl number Pr caused the reduction in thermal diffusivity which can be seen in Figure 2g. The Figure 2h, Figure 2j and Figure 2i reveal the effects of Hartman 8

Journal Pre-proof number Ha , radiation and thermophoresis parameters on temperature respectively. It is noticed that temperature curve rises up with increase in the values of these parameters. High Lorentz force resists the movement of fluid while Hartman number causes this increase in the Lorentz force consequently a part of energy is transported into heat. Increase in    occurred because of this phenomenon. Now in cause of radiation parameter, temperature field enhances because of decay in absorption

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coefficient which caused by increase in radiation parameter. In case of increase in thermophoresis parameter cause the increase in the difference between reference temperature and wall temperature. 0.6

0.6 0

0.5

0.4

0.05

0.3

0.15

0.2

0.3

p ro

0.5

2

0.3

3

0.1

2

4

6

2.5

8

0.0

Pr e-

0

(a)

0

2

4

6

8

(b)

0.5

Ec 0.1

2.0

Ec 0.3 Ec 0.6

1.5

Ec 0.9

0.4

0.1 0.2

0.3

0.3

0.2

0.5

2

4

6

0.4

0.1 0.0

8

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0

al

1.0

0.0

1.4

0.4

0.2

0.1 0.0

0.7

0

2

(c)

0.6

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0.3 0.2 0.1 0.0

0

2

4

8

0.6 Nb 0.01

0.5

Nb 0.03

S 0.2

0.4

6

(d)

S 0

0.5

4

S 0.4

0.4

Nb 0.06

S 1

0.3

Nb 0.1

0.2 0.1 6

0.0

8

(e)

0

2

4

(f)

9

6

8

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1.2

Pr 0.9

1.0

Pr 1.3

1.0

0.8

Pr 1.5

0.8

Ha 0.35

Pr 2

0.6

Ha 0.45

0.4

0.4

0.2

0.2

0.0

0

2

4

6

0.0

8

Ha 0.25

0

(g)

4

6

8

(h)

0.6

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1.0 R 0.1

0.8

Nt 0

0.5

R 0.2

0.6

Nt 0.05

R 0.3

0.4

Nt 0.1

R 0.4

0.3

Nt 0.2

0.4

Pr e-

0.2

0.2 0.0

2

of

0.6

Ha 0

0.1

0

2

4

(i)

6

0.0 0

8

2

4

6

8

(j)

Figure 2. (a) The plot for curves of temperature under the effect of Curvature parameter; (b) The plot for curves of temperature under the effect of Casson parameter; (c) The plot for

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curves of temperature under the effect of Eckert number; (d) The plot for curves of temperature under the effect of Conjugate heat parameter; (e) The plot for curves of

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temperature under the effect of Suction parameter; (f) The plot for curves of temperature under the effect of Brownian motion parameter; (g) The plot for curves of temperature under the effect of Prandtl number; (h) The plot for curves of temperature under the effect of Hartman number; (i) The plot for curves of temperature under the effect of radiation parameter; (j) The plot for curves of temperature under the effect of thermophoretic

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parameter.

It is evident from the Figure 3a that flow near stretched surface causes decrease in mass concentration function    and rises up as it is observed away from the surface. In addition to this positive increase in α results in enhancement of thickness of boundary layer. Figure 3b reveal the effects of parameters S. The mass concentration    reduces in magnitude with increment in suction parameter. Figure 3c disclose the ramifications of parameters N b on    . Increase in collisions and 10

Journal Pre-proof the random movement of liquid’s macroscopic particles reduces the mass concentration    Since weaker Brownian diffusion coefficient rises for higher Sc that retards boundary layer thickness and    retarded due to increment in coefficient of Brownian diffusion. While Brownian diffusion

coefficient enhances as Sc goes up. Increase in Sc results decrease in    can be seen in Figure 3d. The effect of parameters β and  on the function    are mapped in Figure 3e and Figure 3f

of

respectively. The curve of    shows a sharp rise with increase of these parameters in addition to the enhancement of thickness of boundary layer corresponding to concentration also enhances. N t is

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sketched in Figure 3g to observe the effect on    . It is noted that N t has the same effect as the previous two parameters. If we describe it physically increase in the fraction distributions volume occurs as more nanoparticles moves away from the hot surface. 0.8

0.7 0

0.6

0.03 0.06

0.4

0.12

0.3

S 0.2

0.4

S 0.3

0.2

0.2 0.1 0.0 0

0.0

2

4

6

8

10

12

0

2

4

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2.0

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Nb 0.07 Nb 0.08

1.5

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Nb 0.09 Nb 0.1

1.0 0.5

2

4

6

8

10

12

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0

11

8

Sc 0.7 Sc 0.9 Sc 1.1 Sc 1.3

0

2

4

6

(d)

(c)

6

(b)

(a)

0.0

S 0.1

0.6

Pr e-

0.5

S 0

8

10

12

Journal Pre-proof 2.0

1.2 0.7 1.4

1.5

1.0

0.1

1.0

0.2

2

0.8

0.3

3

0.6

0.4

0.4 0.5

0.2 0

2

4

6

8

10

0.0

12

0

2

4

6

8

10

12

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0.0

(f)

(e)

Nt 0.06

1.5

Nt 0.08

p ro

2.0

Nt 0.1

1.0

Nt 0.12

0.0

0

Pr e-

0.5 2

4

6

8

10

12

(g)

Figure 3. (a) The plot for curves of concentration under the effect of Curvature parameter; (b) The plot for curves of concentration under the effect of Suction parameter; (c) The plot

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for curves of concentration under the effect of Brownian motion parameter; (d) The plot for curves of concentration under the effect of Schmidt number; (e) The plot for curves of concentration under the effect of Casson parameter; (f) The plot for curves of concentration

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under the effect of Conjugate mass parameter; (g) The plot for curves of concentration under the effect of Thermophoretic parameter. Table 1 displays the values of different parameters which are used in this research work. For some limiting cases, comparison with previously published results in the literature is made and an

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excellent agreement is obtained (for detail see [53] and Table 2). Table 1: Different parameters values used in this work. 

2



Ha

R

Pr

Ec

Nb

Nt

Sc





S

0.1

0.1

0.3

1

0.1

0.1

0.1

0.6

0.4

0.35

0.1

12

Journal Pre-proof Table 2: Comparative values of

with [53] for various values of Ha when

and Ha

1.0000

1.00000

0.2

1.0198

1.01980

0.5

1.1180

1.11803

0.8

1.2806

1.0

1.4142

of

0.0

1.28063

p ro

1.41421

4. Conclusions

In this study, MHD flow of Casson nanofluids over a stretching cylinder in the presence of thermal radiation with suction/injection has been addressed. It has enormous applications and among

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them clinical applications have much focus. Key points of our study are: 

Increase in curvature parameter results rise in velocity, temperate and concentration.



Increase in suction causes decrement in all three profiles. And Brownian motion parameter have same effect on temperature and concentration.



Hartman number, radiation and thermophoresis increase results as direct increment of temperature and Casson parameter increase originate decline in velocity while incline in concentration.

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Conjugate mass parameter has same inclining effect on both velocity and concentration. Acknowledgment

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This work was supported by Basic Science Research program through the National Research Foundation of Korea (NRF) funded by the Ministry of education, science and technology. (2017R1A51015722, 2017R1D1A3B04033516). Conflict of interest

The authors declare no conflict of interest.

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

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