Enhanced heat transfer in the flow of dissipative non-Newtonian Casson fluid flow over a convectively heated upper surface of a paraboloid of revolution

    Enhanced heat transfer in the flow of dissipative non-Newtonian Casson fluid flow over a convectively heated upper surface of a paraboloid of revolution J.V. Ramana Reddy, V. Sugunamma, N. Sandeep PII: DOI: Reference:

S0167-7322(16)33714-X doi:10.1016/j.molliq.2016.12.100 MOLLIQ 6783

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

21 November 2016 28 December 2016 29 December 2016

Please cite this article as: J.V. Ramana Reddy, V. Sugunamma, N. Sandeep, Enhanced heat transfer in the flow of dissipative non-Newtonian Casson fluid flow over a convectively heated upper surface of a paraboloid of revolution, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.12.100

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ACCEPTED MANUSCRIPT Enhanced heat transfer in the flow of dissipative non-Newtonian Casson fluid flow over a convectively heated upper surface of a paraboloid of revolution V.Sugunamma2*

1,2

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Department of Mathematics, S.V.University, Tirupati-517502, India. Department of Mathematics, Vellore Institute of Technology, Vellore-632014, India.

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3

N. Sandeep3*

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J.V.Ramana Reddy1

Email: [email protected] 2Email: [email protected] 3Email: [email protected]

Abstract

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An analysis is carried out on Casson fluid flow embedded with magnetic nanoparticles. The

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flow is assumed to be over an upper surface of paraboloid of revolution. The effects of viscous dissipation and nonlinear thermal radiation are considered. Convective boundary conditions are supposed. Fe3O4 nano particles are mixed with Casson fluid. In most of the

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research articles related to MHD nanofluid/ferrofluid flows authors are not taking into consideration the electrical conductivity of base fluid and solid particles. But in the present

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study the electrical conductivity of Fe3O4 and H2o is also accounted. The transformed governing equations with adequate similarity variables are solved by means of an effective R-

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K-F integration scheme. The impacts of a few selected parameters on the usual profiles (thermal and velocity) are inspected comprehensively with the aid of plots. Numerical treatment for reduced Nusselt number and wall friction is depicted in tabular form. Results enable us to state that mounting values of temperature ratio or Eckert number accelerates the fluid temperature. A better heat transfer performance on the Casson ferrofluid is perceived when compared to that of Casson fluid. Keywords: Magnetohydrodynamic (MHD), Casson fluid, ferrofluid, convection, dissipation.

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ACCEPTED MANUSCRIPT 1. Introduction Natural convective heat transfer has innumerable importance in geophysical and industrial

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processes. The MHD flows also have tremendous applications in power generators and

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especially in the areas of geophysics and astrophysics. In addition to this, the

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magnetohydrodynamic flows of non-Newtonian type have widespread importance in purification of molten metals and metallurgy industries. It is worth to mention that the applications with these fluids are found to be more than that of the fluids with Newtonian

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fluids. The technological industrial applications of these fluids can be found in heat

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exchangers, design of diffusers, oil recovery etc. Many investigations have been accomplished by the previous researchers to outline the non-Newtonian fluid flows in various

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cases. For instance, a simple mathematical model was given by Chen and Chen [1] to outline

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the heat transfer effects on convective non-Newtonian flows past a flat plate. Liao [2] deliberated on non-Newtonian fluid flows past a stretching sheet with the assistance of HAM.

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Soares and Ferreira [3] studied the cross flow in a power law fluid past a cylinder by employing the finite difference scheme. Hassanien et al. [4] interrogated the boundary layer

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motion of power law fluid across a stretchable surface and deduced that power law index helps to control the heat in the flow. Abbas et al. [5] and Krishna et al. [6] presented dual solutions to inspect the heat transfer in dissimilar non-Newtonian fluids along a stretching surface. Saleem and Nadeem [7] studied the heat and mass transport effects in a nonNewtonian fluid flow due to rotating cone. Casson fluid is a familiar non-Newtonian fluid which is a shear thinning fluid and display yield stress. Blood, sauces and juices comes under this fluid. In the present study, the Fe3O4 nanoparticles are combined with Casson fluid to acquire Casson-ferro fluid. The investigation of Casson fluid flow caused by a moving surface with dissipation was carried out by Mustafa et al. [8], where time dependence is also considered. Further, the same type of

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ACCEPTED MANUSCRIPT study on the flow generated by stretching of a surface was inspected by Mustafa et al. [9]. Through this paper, we may discern that Eckert number is capable of producing heat in the

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flow field. Mukhopadhyay [10] reported the influence of free convective heat transfer of the

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flow caused by a nonlinear surface. Hayat et al. [11] discussed the magnetohydrodynamic

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flow of non-Newtonian Oldroyd-B fluid type. The combined impacts of frictional heat and slip on viscous fluid flow along a vertical cone were explored by Nadeem and Saleem [12] using HAM. Mekheimer [13] worked on couple stress fluid flows with the action of Lorentz

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force caused by magnetic parameter. Then after, many authors [14-16] inspected various non-

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Newtonian fluids past a stretched surface. Animasaun [17] reported the heat and mass transport behavior of Casson fluid with chemical reaction of higher order. Very recently,

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Sandeep et al. [18] studied the 3D non-Newtonian fluid flow in presence of cross diffusion

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and stated that diffusion thermo parameter has propensity to regulate the mass transfer performance. The research on the flows of non-Newtonian liquids with tiny solid particles

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(having magnetic properties) has salient applications such as designing heat exchangers, coating of fibers and cooling processes. So research on non-Newtonian nano/ferrofluids is

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developing owing to their enormous advantages in heat and mass transport process. Entropy generation in Casson nanofluid flow in presence of Brownian motion was characterized by Qing et al. [19]. Ali and Sandeep [20] used Cattaneo-Christov heat flux theory to explore the non linear radiation effect on Casson-ferrofluid. In general, the working capacity of the machines will also depend upon the effectiveness of the heat transfer rate of the liquids used. Customarily the factories will use water, transformer oil, kerosene as the heat transfer liquids which do not posses sufficient thermal conductivity as the industries needed. Since the solid materials exhibit good thermal conductivity, a scientist namely Choi of ANL has made an attempt by adding nano sized solid particles in the base fluids and discovered that thermal conductivity of the conventional fluids

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ACCEPTED MANUSCRIPT can be increased with the addition of solid nanoparticles like graphene, ferrite, titanium oxide etc. From that pioneering work on words many investigations were perpetrated by the

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researchers of mechanical engineering owing to the vast applications of nanofluids (any fluid

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with solid nano particles) in heat transfer mechanism. Among them Baby and Ramaprabhu

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[21] is one to examine the heat transfer of liquids by dispersing grapheme nano particles. Then after, Khan and Gorla [22] presented a numerical treatment for describing the heat and mass transfer effects on non-Newtonian nanofluid flows across a convective surface by

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making use of finite difference scheme. Recently, Das [23] has given a comparative study

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between base fluids and nanofluids graphically. In this paper the author used copper and aluminium oxide nano particles. Nadeem et al. [24] comparatively explored the flow behavior

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of fluids with nanoparticles. Siddiqa et al. [25] found a mathematical model to characterize

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the bio convective flow of a nanofluid past a wavy cone. In general, ferrofluids (fluids with magnetic nano particles) possesses more thermal

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conductivity if compared with ordinary nanofluids. So Rudraiah and Sekhar [26] worked on the flow of magnetic nanofluids by using Galerkin expansion method. The slip flow of a fluid

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with ferro nano particles was discussed by Qasim et al. [27]. The heat transfer of ferrofluids past a flat surface was characterized by Khan [28] in his recent study. The induced magnetic field impacts on the stagnation point flow of ferroliquids were investigated by Babu et al. [29]. The flow of non-Newtonian fluid consisting of solid nano particles past an exponentially stretching cylinder was described by Malik et al. [30]. Mustafa and Khan [31] have given a strange mathematical model to address the flow of Casson nano fluids. Massoudi and Christie [32] worked on the transport process of third grade fluid within a pipe under the action of viscous dissipation. Amin [33] deliberated on the flow past a cylinder with Joule heat and dissipation. Through this paper the author stated that Eckert number helps to boosts the fluid motion. Partha et al. [34] derived a model to discuss the moment of MHD

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ACCEPTED MANUSCRIPT fluids with heat transfer. Chen [35] considered the problem of power-law fluid flow with heat source. Mustafa et al. [36] performed a numerical study to explore the impacts of non linear

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radiation on nanofluid flows over a melting surface. Devi and Raj [37] examined the

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influence of nonlinear radiation on the flow caused by a shrinking sheet. Mean while the

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same effect on the flow caused by a shrinking surface was examined by Mahanthesh [38]. An effective numerical technique, namely finite element method was adopted by Madhu and Kishan [39]. Bhattacharya [40] conferred the concept of unsteady flow across a shrinking

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surface. Ramana Reddy et al. [41] extended the work of Das [23] and discussed the influence

making use of perturbation technique.

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of inclined magnetic field and hall current on a nanofluid flow over a convective surface by

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Slenderness is an important attribute of any object. In engineering and industrial

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processes, we often come across multitudinous flow situations past the geometries of variable thickness. Paraboloid of revolution is nothing but the upper half surface of the flow domain.

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For example needle is an object having slenderness. To our utmost knowledge Lee [42] is one among the first few researchers, who studied the flow over a slender object (thin needle).

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Then after, Davis and Werle [43] succeeded in giving a numerical solution to investigate the flow behavior along the upper part of the flow geometry. Due to the recent advanceme in nano technology, Makinde and Animasaun [44-45] and Animasaun [46] introduced a mathematical model to figure out the flow of nanofluids along the paraboloid of revolution in presence of bioconvection and nonlinear radiation and mentioned that increase in temperature ratio leads to depletion in heat transfer rate. Dual solutions for the nanofluid flow caused by a variable thickness surface were reported by Sulochana and Sandeep [47]. To fill the gap of Animasaun and Sandeep [48], in this study, we analyzed the nonlinear radiation effects on MHD ferrofluid flow past a revolving paraboloid by considering non-Newtonian model. We incorporated the impacts of viscous dissipation and

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ACCEPTED MANUSCRIPT convective surface in the energy equation and boundary conditions respectively. To the best of authors’ knowledge, no records were found in the literature on combined influences of

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nonlinear radiation and viscous dissipation especially on non- Newtonian ferrofluid flows. A

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comparative study is also made between Casson and Casson ferrofluids. The effects of flow

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governing parameters on velocity and temperature distribution are explained with the assistance of plots. 2. Formulation of the problem

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The two-dimensional flow of Casson-ferrofluid across an upper part of convective surface revolving paraboloid is considered. The Cartesian coordinate system is adopted by supposing

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that x  axis is in the direction of the horizontal surface and y  axis is orthogonal to the

r

is the velocity power index) is exerted on the flow as depicted in Fig.1.

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related constant and

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surface. The magnetic field of strength B( x)  B0 ( x  q)( r 1)/2 (Here q is a stretching sheet

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x.

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The temperature near the surface is Ts ( x)  T  D ( x  q)(1r )/2 , which varies with the distance

Fig.1 Schematic representation

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ACCEPTED MANUSCRIPT We made the below said assumptions on the fluid motion. The fluid motion is laminar and time independent.



The stretching velocity of the fluid layers is us  D ( x  q)r , here A is a constant.



The structure of the magnetic nano particles is presumed to be spherical.



Reynolds number presumed to be insignificant, so that Hall effects and induced

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

magnetic fields are disregarded.

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With the suppositions divulged above, the flow equations will be coupled and nonlinear. So, the boundary layer equations reporting the physical situation of the flow depending on the

       y  x

   0, 

(1)

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    x  y

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model suggested by Animasaun and Sandeep [48] are provided below.

2   T  T   2T 16   T 3 * T  1       k    (1   )    2 ff ff   y 2 3 y  k * y   y x x y   y 

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 C p  ff

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3     2   2  0.5(r  1)  1    ff     (1   )  g (  ) x ( T  T )     ff 0 T ff  2 y 3  ff x   y xy x y    2    ff B( x) ,  y



2

 Q* Ts ( x)  T  exp  Ny 0.5(r  1)( x  q) r 1 U 00.5bf0.5



(2)

    (3)  , 

The relevant boundary conditions on the flow will be

u  D( x  q)r , v  0,  kbf

T  hbf (Ts ( x)  T ), at y  D( x  q)0.5(1r ) y

u  0, T  T , as y  

(4) (5)

Since y  D ( x  r )1r is the starting point of the flow, eqn. (4) is not imposed at y  0 . In Eqs. (2)-(5) the suffixes ff and bf are taken to differentiate between ferrofluid and base fluid respectively. Further,  is the stream function, u 

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  and v   represent fluid x y

ACCEPTED MANUSCRIPT velocity along the flow direction and orthogonal to the flow directions respectively.

 (  C p ) sp (  C p ) ff  (  C p )bf 1      (  C p )bf 



 ff   bf 1  

 3 (  1)  (  2)  (  1) 

 ff  bf 1   

2.5

is

electrical

  

is

the

conductivity

heat

(where

capacity,

   sp /  bf

),

   is the dynamic viscosity, ( T ) ff  bf 1     sp  is the volumetric  bf  

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particles)

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  is the density (Here  is the volume fraction of solid nano 

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

 sp bf

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

 ff  bf 1     

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 K  2 K 2  thermal expansion coefficient and knf  kbf  1  is thermal conductivity (where  K1   K 2 

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* K1  ksp  2kbf and K2  kbf  ksp ) of the ferrofluid.  is the Casson parameter, k is the

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mean absorption coefficient, g 0 is the gravitational acceleration acting on the fluid,  * is the

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Steffen Boltzmann constant, T and T respectively denote the temperature in the flow field and faraway temperature respectively, Q* is the dimensionless heat source parameter and

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hbf is the convective heat transfer coefficient of the base fluid.

We introduce the below said stream function fulfilled.

 such that the continuity equation (1) is

  2(r  1)1bf A( x  q)r 1 F ( ),

(6)

Here  is the similarity variable given by

  0.5(r  1) Abf1 ( x  q)r 1 y,

(7)

The dimensionless temperature  ( ) can be defined as

( ) 

T  T or T  T (1  r  ) Ts ( x)  T

(8)

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ACCEPTED MANUSCRIPT In Eqn. (8), r  Ts / T is the temperature ratio parameter. Substitution of antecedent similarity transformation Eqs. (6)-(8) along with the expressions of

2M N2 F  r 1 2r    N1  FF   F 2   Grb   (Grs  Grb )     0, r 1  

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r 1    Rd (1   r  )3    N3 Pr  F   F    (1   1 ) Ek F 2 r 1   2Q Pr 3Rd ( r  1)(1   r  )2 2  h Exp( N )  0, r 1

(9)

(10)

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N

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(1   1 ) (1   ) 2.5 F  

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 ff , ( C p ) ff ,  ff , k ff ,  ff , (T ) ff in eqs. (2)-(5) yield the following equations.

Since y  D ( x  r )1r is the starting point of the flow, eqn. (4) is not imposed at y  0 . So

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  (0.5) D A(r  1)bf1 .

in en (7) yields a new similarity variable

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the substitution of y  D ( x  r )1r

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The dimensionless boundary conditions subject to the new variable  are

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dF  1 r    1, F     ,    (1  ), d  1 r  dF  0,   0, d

at   0,

(11)

as   ,

(12)

In Eqs. (9) and (10)   sp  N1  1       bf  

,

  3 (  1) N 2  1    (  2)  (  1) 

,

 (  C p ) sp N3  1      (  C p )bf 

  

,

g0 ( T )bf (Ts  T )  bf B02  K1  2 K 2  , is the magnetic parameter, Grb  is the N4    M 2 2 r 1 A ( x  q ) A  K   K bf  1 2  volumetric coefficient of thermal expansion of base fluid, Grs 

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g0 ( T ) sp (Ts  T ) A2 ( x  q)2 r 1

is the

ACCEPTED MANUSCRIPT volumetric coefficient o f thermal expansion of Fe3O4 nano particles, Rd 

 (C ) A2 ( x  q)2 r is the Eckert number, Pr  bf p bf is the (C p )bf (Ts  T ) kbf

hbf kbf

1 m 2 bf ( x  b) 2 is the convection parameter. m 1 A

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

Q* ( x  q)r 1 is the space dependent internal heat source parameter and Abf (C p )bf

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Prandtl number, Qh 

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radiation parameter, Ek 

16 *T3 is the 3k *kbf

The transmuted Eqs. (9)-(10) with the boundary restrictions (11)-(12) are coupled with the

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domain [ , ) . Hence, we consider f ( )  f (   )  F ( ) ,  ( )   (   )  ( ) to avail the computations in the interval (0, ) .

(13)

r 1    Rd (1   r   )3    N3 Pr  f    f    (1   1 ) Ek f 2 r 1   2Q Pr 3Rd ( r  1)(1   r   ) 2 2  h Exp( N )  0, r 1

(14)

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2M 2r 2   N 2 f   N1  ff   f    Grb   (Grs  Grb )    0, r 1 r 1  

(1   1 ) (1   )2.5 f  

N

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In view of the above mentioned transformations Eqs. (9)-(12) become

with the boundary conditions

df  1 r    1, f     ,   (1   (0)), d  1 r 

at   0,

(15)

df  0,  0, d

as   ,

(16)

The friction factor ( f (0) ) and heat transfer coefficient (  (0) ) are also derived in view of their practical applications in engineering technology. The skin friction coefficient ( C fx ) is defined as

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ACCEPTED MANUSCRIPT C fx 

s

0.5

 2  , 2 2r  bf A ( x  q)  r  1 

(17)

The Nusselt number ( Nux ) is defined as ( x  q)qs  2    , kbf (Ts ( x)  T )  r  1 

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0.5

Nux 

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1 2.5 f (0)  Re0.5 x (1   )(1   ) C fx ,

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Substitution of  s and the transformations yields

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 u  where  s  (1   1 )  ff   is the shear stress along the surface.  y  y  D ( x  q )0.5(1r )

(18)

(19)

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 16 *T 3  T  where qs    k ff  is the heat flux from the surface.   3k *   y  y  D ( x  q )0.5(1r ) 

Therefore, the heat transfer coefficient is given by

Nux Re x 0.5 , N 4  Rd 1 (1   (0)r   (0))3

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 '(0) 

(20)

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Using eqs. (18) and (20), we calculated the friction factor ( f (0) ) and heat transfer coefficient for distinct values of flow regulating parameters and presented in tables 2 and 3. 3. Discussion of the results The solution to the B.V.P represented by the Eqs. (13) and (14) with respect to the boundary conditions (15) and (16) has been constructed through an effective numerical process namely R-K-F integration scheme. Following the problem solution, we focused on determining the impacts of distinct flow regulating quantities on fluid velocity and temperature with the support of computer produced plots. The results are acquired for both contexts viz. i) Casson fluid ii) Casson-ferrofluid flows by proceeding with   0 and   0.15 respectively. The results are acquired by giving the values of flow parameters   0.3 , r  0.5 , Grb  0.5 , 11

ACCEPTED MANUSCRIPT Grs  0.5 , M  1.2 , Rd  0.2 , Bi  0.5 ¸ r  1.1 , Pr  6.8 and Ek  0.5 as input for the complete production of results unless otherwise noted in plots and tabular forms. The results

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are elaborated via figs. 2-17 and tables 2 and 3. Table 1 portrays the thermo physical

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properties of H 2O , Fe3O4 . With the help of table 4, comparison of the precedent work with

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Ref. [48] is also done in order to authenticate the current work.

Table 1. Thermophysical properties of H 2O and Fe3O4 . (see Babu et al. [29]] Properties

 ( Kg m3 )

C p ( J Kg 1 K 1 )

H 2O

997.1

4179

0.613

5.5 x 10-6

Fe3O4

5180

670

9.7

0.74x106

 (S m1 )

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k (W m1 K 1 )

Figs. 2 and 3 subsequently unravel the velocity ( f ( ) ) and temperature (  ( ) ) fields for

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selected values of magnetic field parameter ( M ). From Fig. 2, it is apparent that M has propensity to diminish the flow velocity. But a quite retro gate trend is espied through Fig. 3

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for temperature distribution. The cause for this phenomenon is that the action of magnetic field on the flow establishes the Lorentz force. This force gives rise to heat energy, as a

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consequence it offers a resistance to the flow movement. The variations in the curves of f ( ) and  ( ) corresponding to distinct values of thermal expansion coefficient of base fluid ( Grb ) is depicted in Figs. 4 and 5. Here we notice an opposite trend to the results of Figs. 2 and 3 with the gain in Grb . Customarily, buoyancy force will take place due to the temperature difference near and far away the surface. Owing to this logic we see the results of this kind. Figs. 6 and 7 are created with an interest of describing the nature of fluid velocity and temperature for mounting values of Eckert number ( Ek ) or dissipation parameter. Increasing values of Ek maximizes the friction at the flow surface. So, it is well acceptable truth that dissipation parameter leads to a hike in kinetic

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It is perceptible from Figs. 8 and 9 that radiation parameter ( Rd ) has proclivity to

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amplify the velocity as well as temperature of the fluid. As we elevate the radiation

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parameter, it supplies heat energy to the flow. On account of this we detect a growth in momentum and thermal boundary layers. Fig. 10 is constructed to scrutinize the behavior of

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 r on fluid velocity. It is understood that velocity is an increasing function of  r . Figs. 11 and 12 enables one to clinch that temperature profiles are much influenced by temperature

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ratio parameter (  r ) and Biot number (  ). According to its definition, higher values of  r means that temperature near the surface is higher than that of away from the surface. Biot

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number is a convective parameter which increases the heat near the surface. Consequently,

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we notice a hike in thermal boundary layer thickness with growing values of  r and  . Fig.

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13 unravels the graph of Prandtl number ( Pr ) versus fluid temperature. It is manifest that escalating the magnitude of Pr shrinks the thermal boundary layer. As the thermal diffusivity is inversely proportional to Pr , the capability of transferring the heat will be also minimized.

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So, we discern a fall in temperature profiles. Figs. 14 and 15 expose the nature of velocity and temperature profiles for multiple magnitudes of velocity power index parameter ( r ). From the output of these figures, it is understandable that velocity and temperature are accelerating functions of r . Usually boosting the values of r improves the slenderness of the flow surface and correspondingly fluid motion will be also developed. This phenomenon dispenses heat energy in the flow. In view of these reasons velocity and temperature profiles will rise with r . Figs. 16-17 reveal the domination of Casson parameter (  ) on velocity and temperature fields. It is clear that an inflation in  diminishes the velocity and temperature

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become greater.

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Tables 2 and 3 have been built to report the characteristics of friction factor and heat

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transfer coefficients for i) Casson fluid and ii) Casson-ferrofluid flows respectively. It is visible that Casson fluid display better heat transfer rate compared to Casson-ferrofluid. Also the skin friction is very high for Casson fluid flow. It is absorbing to grasp that both the fluids

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exhibits same behavior for f (0),  (0) under the impact of all involved flow parameters.

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Biot number and thermal expansion coefficient of the base fluid have propensity to increase both friction factor and heat transfer coefficient where as an exact opposite trend is perceived

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with magnetic field parameter. It is worth to specify that skin friction coefficient is an

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increasing function of radiation, temperature ratio and velocity power index parameters but Nusselt number is a decreasing function of them. Eckert number is capable of enhancing

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friction coefficient but dwindles the local Nusselt number.

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parameter M

Fig.4 Velocity versus thermal expansion coefficient of the base fluid Grb

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Fig.2 Velocity versus magnetic field

Fig.5 Temperature versus thermal expansion coefficient of the base fluid Grb

Fig.3 Temperature versus magnetic field parameter M

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Fig.8 Velocity versus the radiation parameter Rd

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Fig.6 Velocity versus Eckert number Ek

Fig.9 Temperature versus the radiation parameter Rd

Fig.7 Temperature versus Eckert number

Ek

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Fig.12 Temperature versus Biot number

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Fig.10 Velocity versus temperature ratio parameter  r

Fig.11 Temperature versus temperature ratio parameter  r

Fig. 13 Temperature versus Prandtl number Pr

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Fig.16 Velocity versus Casson parameter

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Fig.14 Velocity versus velocity power index parameter r

Fig.15 Temperature versus velocity power index parameter r

Fig. 17 Temperature versus Casson parameter 

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ACCEPTED MANUSCRIPT Table 2. Friction factor and heat transfer coefficient of Casson fluid

Grb

Rd

r

Bi

r

Pr



Ek

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0.2 0.8 1.4 0.2 0.5 0.8

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1.5 3.0 4.5

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1.1 1.5 1.8

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0.5 0.8 1.2

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4.0 6.8 11.4 0.2 0.3 0.4

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f (0) -0.8539 -1.1396 -1.3678 -0.7342 -0.6851 -0.6403 -0.7090 -0.7025 -0.6961 -0.7072 -0.6828 -0.6558 -0.7457 -0.7411 -0.7386 -0.7260 -0.6958 -0.6656 -0.6894 -0.7145 -0.7306 -0.6007 -0.7072 -0.7873 -0.7316 -0.6983 -0.6676

 '(0) 0.0917 -0.0682 -0.2052 0.1559 0.1756 0.1930 0.1661 0.1580 0.1482 0.1650 0.1084 0.0328 0.6916 0.9744 1.1276 0.3154 0.2823 0.2353 0.0416 0.1617 0.2564 0.1486 0.1650 0.1752 0.2789 0.0515 -0.1496

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M 2 4 6

1.0 2.0 3.0

Table 3. Friction factor and heat transfer coefficient of Casson-ferrofluid M 2 4 6

Grb

Rd

r

Bi

r

Pr

0.2 0.8 1.4 0.2 0.5 0.8 1.1 1.5 1.8 1.5 3.0 19



Ek

f (0) -0.9898 -1.2730 -1.5065 -0.9112 -0.7990 -0.7046 -0.8525 -0.8391 -0.8264 -0.8493 -0.7999 -0.7459 -0.9156 -0.9057

 (0) -0.0017 -0.1662 -0.3088 0.0462 0.0969 0.1365 0.0733 0.0777 0.0773 0.0734 0.0399 -0.0186 0.6272 0.8803

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1.0166 0.2837 0.2449 0.1906 -0.0546 0.0711 0.1820 0.0485 0.0734 0.0890 0.2240 -0.0637 -0.2850

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4.0 6.8 11.4

-0.9003 -0.9052 -0.8802 -0.8521 -0.7944 -0.8567 -0.9002 -0.7201 -0.8493 -0.9465 -0.9039 -0.8133 -0.7360

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0.2 0.3 0.4

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1.0 3.0 5.0

Table 4. Validation of the current results with Animasaun and Sandeep [48]

f (0) f (0) Animasaun and Sandeep [48] Current values 0.1 -0.8671009 -0.86710 0.2 -0.8624053 -0.86240 0.3 -0.8584863 -0.85848

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4. Conclusions

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Numerical exploration of Casson ferrofluid over an upper convective surface of a paraboloid of revolution subject to viscous dissipation and nonlinear thermal radiation is presented. Adequate similarity transforms are invoked to put the flow governing equations in

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dimensionless form and solved numerically. The underneath points are the main conclusions. Influence of all physical parameters is highly pronounced for Casson-ferrofluid when compared to Casson fluid. 

Blending of solid nanoparticles into the non-Newtonian fluid down shifts the fluid velocity.

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Mounting values of viscous dissipation enhance the fluid temperature but dwindles the heat transfer coefficient.

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Nusselt number rises effectively via greater values of Biot number.

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Flow and thermal boundary layers are non-uniform for Casson and Cassonferrofluids.

H.T. Chen, C.K. Chen, Free convection flow of non-Newtonian fluids along a vertical

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[1]

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ACCEPTED MANUSCRIPT Highlights Computational model for convective heat transfer in flow over revolving paraboloid.

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Magnetohydrodynamic flow embedded ferrous nanoparticles.

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Casson ferrofluid perceives high heat transfer when compared with Casson fluid.

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Mounting values of dissipation parameter dwindle the heat transfer coefficient.

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