Unsteady Magnetohydrodynamic Mixed Convection Flow over a Rotating Sphere with Sinusoidal Mass Transfer

Document Type : Full Lenght Research Article

Authors

1 Department of Mathematics, National Institute of Technology, Tiruchirappalli, India

2 Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala

Abstract

This paper investigates the unsteady magnetohydrodynamic (MHD) mixed convective fluid flow over a rotating sphere. An implicit finite difference scheme, together with quasi-linearization, is used to find non-similar solutions for the governing equations. The impact of variable physical properties and viscous dissipation are included. It is observed that the skin friction coefficient in the axial direction and the heat transfer coefficient are increasing with an increase in MHD, mixed convection and rotation parameters and with time, whereas the effect is just the opposite for the skin friction coefficient in the rotational direction. The non-uniform slot suction(injection) and the slot movement influence the point of vanishing skin friction to move in the axial direction downstream (upstream).

Keywords

Main Subjects


 

 

 

Semnan University

Journal of Heat and Mass Transfer Research

Journal homepage: http://jhmtr.semnan.ac.ir

Unsteady Magnetohydrodynamic Mixed Convection Flow over a Rotating Sphere with Sinusoidal Mass Transfer

  1. Sahaya Jenifer a, Saikrishnan Ponnaiah [*],a , E. Natarajan b

a Department of Mathematics, National Institute of Technology, Tiruchirappalli, India.

b Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram, India.

 

Paper INFO

 

ABSTRACT

Paper history:

Received: 2021-03-16

Received: 2023-01-04

Accepted: 2023-01-10

This paper investigates the unsteady magnetohydrodynamic (MHD) mixed convective fluid flow over a rotating sphere. An implicit finite difference scheme, together with quasi-linearization, is used to find non-similar solutions for the governing equations. The impact of variable physical properties and viscous dissipation are included. It is observed that the skin friction coefficient in the axial direction and the heat transfer coefficient are increasing with an increase in MHD, mixed convection and rotation parameters and with time, whereas the effect is just the opposite for the skin friction coefficient in the rotational direction. The non-uniform slot suction(injection) and the slot movement influence the point of vanishing skin friction to move in the axial direction downstream (upstream).

DOI: 10.22075/jhmtr.2023.22926.1339

 

Keywords:

Boundary layer;

Heat transfer;

Variable properties;

Mixed convection;

MHD;

Non-uniform mass transfer;

Non-similar solution;

Rotating sphere.

 

© 2022 Published by Semnan University Press. All rights reserved.

 

 

  1. Introduction

The investigation of boundary layer flow and heat transfer over rotating bodies of revolution has several technical applications, including fiber coating, re-entry missile design, and rotary machine design [1]. The sphere being a well-renowned geometry used in engineering devices, many times the motion of spherical models endure rotation and suction/blowing. As a result, understanding the influence of rotation as well as mass transfer on flow over a spinning sphere is critical. Kreith et al. [2] have investigated convection heat transport and flow phenomena and Lee et al. [3] have incorporated forced flow over rotating spheres. The effects of surface blowing on the above-described geometry were investigated by Niazmand and Renksizbulut [4]. Recently, Safarzadeh and Brahimi [5] have established the flow phenomena over the rotating sphere in porous media. Many researchers have worked on flows over rotating bodies such as cylinders, disks, and cones, under different circumstances [6-8].

There is a substantial variation in fluid properties owing to the presence of a temperature gradient across a fluid medium. This temperature variation may be due to heat transfer when the fluid and the surface have dissimilarity in temperature or when there is a loss of heat present in the form of latent energy upon its liberation [9]. Together with these varying physical properties, the process of heat transfer for a variety of objects has already been thoughtfully analyzed by a significant number of researchers [10-16].

The mass transfer through a wall slot holds several tremendous practical implications in thermal protection, fuel injecting system of ramjets, drying theory, galvanizing the innermost section of the boundary layer in adverse pressure gradients, and reducing skin friction on high-speed aircraft [17]. Uniform suction (injection) creates discontinuities at the ends of the slot. An ultimate solution to overcome this is by implementing a non-uniform suction(injection), as discussed by Roy and Nath in [18]. Since then, several researchers have carried out the work on the impact of the non-uniform mass transfer over various two-dimensional axisymmetric bodies [12,13,15-17,19] and that over rotating bodies [11,14,21].

The boundary-layer flows are found to be both unsteady and non-similar in nature. The unsteadiness and non-similarity that occur may be due to the body’s curvature or the velocity profiles at the boundary or the surface mass transfer, or perhaps an amalgamation of all the factors mentioned above. A vast majority of the researchers restrained their works to unsteady self-similar flows or steady non-similar flows due to mathematical complexities. A brief review of methods to find a non-similar solution for steady flows and the references of apposite works done up till 1967 has been stated in [21]. In the past two decades, many researchers worked on a non-similar solution for steady/unsteady flows over various shapes of non-spinning bodies [12,13,15,17,22-24]. In the case of rotating bodies, authors in [25-27] presented self-similar solutions for steady/unsteady flow over a rotating sphere, whereas in [10,11,28] have given non-similar solutions.

The inclusion of the effect of MHD and mixed convection has received keen attention recently. An enormous number of researchers have analyzed the effect of mixed convection on steady or unsteady fluid flow over various non-spinning bodies [17,29-32] and over rotating bodies [1,33-39]. On the other hand, the effect of MHD on steady or unsteady fluid flow over two-dimensional axisymmetric bodies has been observed by Sathyakrishna et al. [40], and over a rotating sphere has been studied in [10,27,41,42]. The above studies were focused on analyzing the flow problem with either mixed convection or magnetic field. The combined effect of MHD and mixed convection on a steady fluid flow over a sphere, rotating sphere, wedge, and the vertical elastic sheet has been studied in [11,12,19,43], respectively. Recently, Ghani and Rumite [44] have worked on the MHD mixed convection flow over a solid sphere by using the Keller-box method.

Further taking unsteadiness into account, Chamkha et al. [22,45] showed the combined effect of MHD and mixed convection of fluid flow at the forward stagnation region of a rotating sphere in the presence of chemical reaction and heat source and at different wall conditions. Mahdy et al. [26] have investigated the same with an analysis of entropy generation due to non-Newtonian Casson nanofluid. Recently, Jenifer et al. [46] obtained non-similar solutions for an unsteady MHD mixed convective flow over a stationary sphere with mass transfer. Gul et al. [47] have worked on the stagnation point flow of blood-based hybrid nanofluid over a rotating sphere with the inclusion of mixed convection and a time-dependent magnetic field. Considering an impulsively rotating sphere, Calabretto et al. [48] have explored the effects of unsteadiness and Mahdy et al. [49] have further extended the study to homogeneous - heterogeneous reactions in MHD mixed nanofluid flow. Numerous writers have recently researched flow through spinning spheres while taking into account phenomena like double diffusive convection, magnetophoresis and joule heating. [50-52].

From the literature review, the sinusoidal mass transfer in the case of unsteady rotating sphere is not studied so far. The novelty of this work lies in finding non-similar solutions under the combined effects of the following circumstances.

  • Temperature-dependent viscosity and Prandtl number
  • MHD mixed convective flow over a rotating sphere
  • Unsteady (accelerating) flow model
  • Sinusoidal suction/injection through a slot
  • Viscous dissipation and Joule heating

This study finds its applications in flows over rotating axisymmetric bodies where the flow is time dependent and the boundary layer can be controlled by implementing the above-mentioned factors. The governing equations are transformed with the help of nonsimilar transformations and the corresponding nonsimilar solutions are obtained by using implicit finite difference method along with quasilinearization technique. Important flow parameters such as skin friction and heat transfer coefficients are analyzed for various values of the effects taken into account. The fluid considered here is water due to its extreme practical applications in engineering.

  1. Mathematical Formulation

The coordinate system and flow model over a heated sphere is presented in Figure 1. It is assumed that the sphere rotates with angular velocity  (a time-dependent function) with its rotation axis parallel to . A constant magnetic field  is enforced perpendicular to the sphere’s surface. The mixed convective flow is supposed to be in the upward direction, and the sphere rotates in direction.

 

Figure 1. Flow model

The variation of temperature between the free stream and the sphere’s surface is assumed to be less than . Within this temperature limit considered, the properties of water, such as density  and specific heat  vary up to a maximum of 1%, and this minute variation allows the use of  and  as constants. On the other hand, properties such as viscosity  and thermal conductivity  vary significantly with temperature, and so does the Prandtl number . Both  and  have an inverse linear relationship with temperature as specified in [23].

 

(1)

with  

(2)

The boundary layer flow is governed by the following equations:

 

(3)

 

 

(4)

 

(5)

 

 

(6)

Initial conditions:

 

 

(7)

Boundary conditions:

 

 

 

 

(8)

The transformations to convert the equations (4)-(6) and the conditions (7) and (8) into a non-dimensional form are as follows:

 

 

 

 

 

 

(9)

The above transformations satisfy (3) identically, and the non-dimensional forms of (4)-(6) are given below

 

 

 

 

(10)

 

 

(11)

 

 

 

(12)

with the boundary conditions

 

 

 

(13)

where

 

 

 

 

 

 

 

 

 

 

 

 

(14)

The velocity distribution at the boundary layer’s edge is written as,

 

 

(15)

Hence  and  can be written as expressions in  as follows.

 

 

 

(16)

where

 

 

(17)

The following suction/injection distribution at the wall is taken as a sinusoidal function. It exhibits a nonuniform mass transfer only in the interval  which can endure a slow mass transfer at the slot’s ends without breaking its continuity.  Here, is the mass transfer parameter with  indicating suction and  indicating injection through the slot. The parameter  determines the slot length which is fixed at  whenever mass transfer is applied in this paper.

 

.

(18)

The value of surface mass transfer  is given by

 

 

 

(19)

It is convenient to write the equations in  instead of .   and  are related by

 

(20)

where

 

(21)

Substituting equations (20) and (21) in the equations (10), (11) and (12), we obtain the dimensionless equations,

 

 

 

 

(22)

 

 

(23)

 

 

 

 

(24)

The boundary conditions become

 

 

.

(25)

The skin friction coefficients in the  and directions and the heat transfer coefficient can be written as

 

 

 

(26)

where,

 

 

 

(27)

  1. Method of Solution

Quasilinearization is a technique introduced by Bellman and Kalaba [53], that can linearize nonlinear initial boundary value issues and is considered an extension of the Newton Raphson approach in functional space. This approach not only linearizes the original nonlinear equation, but it also gives a series of functions that converge to the nonlinear problem's solution. After quasilinearizing the highly nonlinear equations (22)-(24), the following set of linear partial differential equations are obtained.

 

 

 

 

 

 

 

 

 

(28)

Here, the superscript and  denote the previous and current iterations and the coefficients are as follows









 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

         

 

 

 

 

 

 

 

 

All the coefficients above are known values from -th iteration. With step sizes in their respective directions, the linearized partial differential equations in (28) are discretized using central difference scheme in  direction and backward difference scheme in  directions and the linear difference equations are written in the following matrix form [54]

 

 

(29)

where the coefficient matrices are as follows

 

 

 

 

 

where

 

The system of tridiagonal blocks (29) is then solved by using Varga’s algorithm [55] for  in  direction, which is discretized into  subintervals, with fixed  and the forward marching continues in  direction. The above-mentioned process repeats for the subsequent steps in  direction.

The convergence of the solution at each step is assumed to be achieved when the maximum absolute difference between the current and previous iterations is less than the tolerance value, which is set at . Here, the step sizes are taken as  and .  is considered to be 6.

  1. Results and Discussion

The precision of our study is ensured by comparing the obtained solutions with those available in the literature in both steady and unsteady cases.

In the case of steady flow, the effect of rotation  on the skin friction parameters  and the heat transfer parameter,  are presented in Figure 2, and the effect of mixed convection parameter  on skin friction coefficient in direction  is shown in Figure 3.

The results are compared with those of Roy and Saikrishnan [14] and Chen et al. [29], respectively.

Also, in the case of unsteady flow, the impact of MHD parameter  on  at times  are presented in Figure 4 and are compared with those of Sathyakrishna et al. [40]. All the above-mentioned studies agree with our results.

The variations in the skin friction coefficients in the directions and the heat transfer coefficient  at various streamwise locations due to the MHD parameter  with are presented in Figure 5 and Figure 6 for both steady and unsteady cases. From the figures,  enhances from zero, hits maximum value and then declines as  increases. With an increase in  and , both  and  increase, whereas the effect is just the opposite on . The reason for this is the magnetic field  induces a magnetic force which in turn creates a supporting force in the meridian direction and an opposing force in the rotational direction. Hence, increasing  accelerates the flow in direction and decelerates the flow in direction and thus resulting in the enhancement of  and reduction in . For fixed ,  decreases monotonically as  increases. In the case of steady flow, the significance of the MHD parameter is not pronounced on  and . However, the effect becomes significant with time .

 

Figure 2. Comparison of the velocity profiles in directions and temperature profile for a steady flow
with those of Roy and Saikrishnan [14] where
  , constant
viscosity and Prandtl number

 

Figure 3. Comparison of the skin friction parameter in the direction with those of Chen et al. [29] where
  , constant
 viscosity and Prandtl number

 

Figure 4. Comparison of the skin friction parameter in the direction for an unsteady flow with those
of Sathyakrishna et al. [40] where
  ,

 

Figure 5. Effect of the MHD parameter  on the skin friction coefficients in directions for

 

Figure 6. Effect of the MHD parameter  on the heat transfer coefficient for

Figure 7 and Figure 8 depict the influence of the mixed convection parameter  on  and  over time for . The presence of mixed convection parameter  signifies favorable pressure gradient. This results in thinning of momentum and temperature boundary layers. As a consequence, both  and  increase and  decreases as  increases at both times . It is to be noted that the significance of  is more prominent on  than on  because there is no explicit dependence of the mixed convection parameter  in equation (23).

Figure 9 and Figure 10 show the impact of rotation parameter  on  and  for . It is found that increasing rotation parameter  results in an increase of  and  and decrease of  at both  and 2. This is because the fluid entering in the axial direction has been forced outward in the rotational direction due to the centrifugal force and has been replaced by the cooler fluid from the normal direction. This results in accelerating the fluid flow in the axial direction and contracting the thickness of momentum boundary layer in that direction as well as the thickness of the thermal boundary layer. Meanwhile, in the rotational direction, the momentum boundary layer thickens. Also, the effect of  on  is found to be small since  affects it indirectly.

For the steady flow,  vanishes while  does not. As  increases, the point of vanishing skin friction coefficient in direction moves slightly downwards, indicating an ordinary separation. It is also observed that  vanishes for  and the point of vanishing skin friction in that direction moves upstream as  increases. However, this does not imply separation since it is unsteady.

Figure 7. Effect of the mixed convection parameter  on the skin friction coefficients in directions for

 

Figure 8. Effect of the mixed convection parameter  on the heat transfer coefficient for

 

Figure 9. Effect of the rotation parameter  on the skin friction coefficients in directions for

 

Figure 10. Effect of the rotation parameter  on the heat transfer coefficient for

The impact of the viscous dissipation parameter  on  for  has been shown in Figure 11. The heat transfer coefficient  decreases for the change of values of  from 0 to  in both steady and unsteady cases. At , the percentage decrease of  is 129% when  and 413% at  as  changes from 0 to . Also, the occurrence of negative is physically simulated by the reversal of heat transfer direction.

The reason for this can be seen from the temperature profile  at  depicted in Figure 12. For  and , the temperature  gets below zero near the wall. This is because nonzero  emphasizes the presence of the viscous dissipation and  too being nonzero brings on joule heating in the energy equation. Due to the impact of these two heating, the fluid near the wall heats up and its temperature becomes more than , although originally  was higher. This results in the wall being heated up instead of being cooled and hence the heat transfer reversal observed in Figure 11. However, such a phenomenon is not observed in the steady case.  does not show much of a difference in the skin friction coefficients in directions as well as the velocity profiles. Hence, the corresponding figures are omitted in this paper.

Figures 13-16 show the influence of non-uniform mass transfer on  and  for  and , at  and . The effect of suction/injection is examined through two slots , one at  and the other at , but not simultaneously. In the case of slot suction ( ), as the slot starts,  and  increase and hit their maximum before the slot’s end. Contrastingly,  decreases as the slot starts and hits its minimum before the slot's end. As the suction parameter increases, the fluid at the sphere’s surface, which has low velocity, is sucked through the slot and is replaced by the fluid in the subsequent layers with comparatively higher velocity. This augments the velocity gradients in both  and directions at the wall  and thus resulting in increasing  and decreasing .

 

Figure 11. Effect of the viscous dissipation parameter  on the heat transfer coefficient for

 

Figure 12. Effect of the viscous dissipation parameter  on the temperature profile at  for

 

Figure 13. Effect of the non-uniform slot suction  on the skin friction coefficients in directions for
, slots at 5

 

Figure 14. Effect of the non-uniform slot suction  on the heat transfer coefficient for ,
 slots at

 

Figure 15. Effect of the non-uniform slot injection  on the skin friction coefficients in directions for
, slots at

 

Figure 16. Effect of the non-uniform slot injection  on the heat transfer coefficient for ,
 slots at

Moreover, since the fluid being sucked is warmer than the adjacent layers, the more the suction, the steeper the temperature gradient at the wall and hence  is enhanced.

The slot injection’s ( ) effect on skin friction and heat transfer coefficients is qualitatively opposite to that of suction in the slot region. In all the above cases, regardless of , the coefficients
    are   enhanced

as  increases and the impact is more pronounced in the unsteady case, since the flow is accelerating with . For the steady case , the suction/injection doesn’t impact the zero skin frictions in both directions. However, when , the point of zero skin friction in direction moves downstream with an increase in suction parameter ( ). The slot movement in the downstream direction from  to  helps the vanishing point of  move further downstream. Meanwhile, the opposite effect is seen with an enhancement in the injection parameter. It should be emphasized here that zero skin friction in only one direction/both directions does not imply the ordinary/singular separation as the flow considered here is unsteady.

Conclusion

An unsteady MHD mixed convection boundary layer flow problem over a geometry of rotating sphere has been solved numerically, and the observations are as follows.

  • The MHD parameter ( ) affects the skin friction coefficient in direction and heat transfer coefficient noticeably in the unsteady case than it does in the steady case for fixed non-zero values of rotation and mixed convection parameters.
  • The mixed convection parameter ( ) is found to have a prominent effect on the skin friction coefficient in the direction and the heat transfer coefficient than the skin friction coefficient in the direction, in both steady and unsteady cases.
  • For non-zero values of rotation parameter an ordinary separation is noted in the steady case as  vanishes while  does not. It is observed that  vanishes for
       and that point of vanishing moves upstream as the rotation parameter increases.
  • At both times , the more the magnitude of viscous dissipation parameter ( ) the less the heat transfer coefficient as a result of heating due to viscous and joule heating effects. Moreover, the unsteadiness results in drastic decrement in the heat transfer coefficient as dissipation increases.
  • The temperature drops below zero in the vicinity of the sphere’s surface in the unsteady case, indicating the fluid near the surface of the sphere getting warmer instead of colder.
  • For fixed non-zero values of and , non-uniform slot suction or slot movement helps the vanishing skin friction in direction to move slightly downstream, whereas the injection shows the opposite effect.

Nomenclature

 

Dimensionless mass transfer parameter

 

Magnetic field strength

 

Dimensionless rotation parameter

 

Specific heat at constant pressure

 

Skin friction coefficient in the direction

 

Skin friction coefficient in the direction

 

Eckert number (viscous dissipation parameter)

 

Dimensionless stream function

 

Surface mass transfer distribution

 

Dimensionless velocity component in the direction

 

Gravity

 

Dimensionless temperature

 

Grashof number

 

Thermal conductivity

 

MHD parameter

 

 Viscosity ratio

 

Nusselt number

 

Prandtl number

 

Radius of the section normal to the axis of the sphere

 

Radius of the sphere

 

Reynolds number

 

Dimensionless velocity component in the direction

 

Dimensional time

 

Dimensionless time

 

Temperature

 

Dimensional velocity components in directions, respectively

 

Steady state velocity at the boundary layer’s edge

 

Dimensional meridional, azimuthal and normal distances, respectively

 

Dimensionless meridional distance

 

Ends of slot

Greek Symbols

 

Dimensionless pressure gradient

 

Volumetric coefficient of thermal

expansion

 

 

Step sizes in  and directions, respectively

 

Transformed coordinates

 

Constant used in the continuous function

of time

 

Mixed convection parameter

 

Dynamic viscosity

 

Kinematic viscosity

 

Density

 

Electrical conduction

 

Dimensional stream function

 

Continuous function of time

 

Angular velocity

 

Slot length parameter

Subscripts

 

Conditions at the edge of boundary layer

 

Conditions at the surface of the sphere

 

Conditions in the free stream

 

 

Partial derivatives with respect to these variables

Acknowledgements

The authors are thankful to the reviewers for their valuable comments. The first author would like to thank National Institute of Technology Tiruchirappalli for supporting through an institute fellowship.

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  • Le Palec, G. and Daguenet, M., 1987. Laminar Three-Dimensional Mixed Convection About a Rotating Sphere in a Stream. International Journal of Heat and Mass Transfer, 30(7), pp.1511–1523.
  • Rajasekaran, R. and Palekar, M.G., 1985. Mixed Convection About a Rotating Sphere. International Journal of Heat and Mass Transfer, 28(5), pp.959–968.
  • Rajasekaran, R. and Palekar, M.G., 1985. Viscous Dissipation Effects on Mixed Convection About a Rotating Sphere. International Journal of Engineering Science, 23(8), pp.789–795.
  • Tieng, S.M. and Yan, A.C., 1992. Investigation of Mixed Convection About a Rotating Sphere by Holographic Interferometry. Journal of Thermophysics and Heat Transfer, 6(4), pp.727–732.
  • Patil, P.M., Benawadi, S. and Shanker, B., 2022. Influence of Mixed Convection Nanofluid Flow over a Rotating Sphere in the Presence of Diffusion of Liquid Hydrogen and Ammonia. Mathematics and Computers in Simulation, 194, pp.764–781.
  • Sathyakrishna, M., Roy, S. and Nath, G., 2001. Unsteady Two-Dimensional and Axisymmetric MHD Boundary-Layer Flows. Acta Mechanica, 150(1–2), pp.67–77.
  • Turkyilmazoglu, M., 2011. Numerical and Analytical Solutions for the Flow and Heat Transfer near the Equator of an MHD Boundary Layer over a Porous Rotating Sphere. International Journal of Thermal Sciences, 50(5), pp.831–842.
  • Kazem, S., Tameh, M.S. and Rashidi, M.M., 2019. An Improvement To The Unsteady MHD Rotating Flow Over a Rotating Sphere Near the Equator via Two Radial Basis Function Schemes. The European Physical Journal Plus, 134(12), p.611.
  • Vajravelu, K., Li, R., Dewasurendra, M. and Prasad, K.V., 2017. Mixed Convective Boundary Layer MHD Flow Along a Vertical Elastic Sheet. International Journal of Applied and Computational Mathematics, 3(3), pp.2501–2518.
  • Ghani, M. and Rumite, W., 2021. Keller-Box Scheme to Mixed Convection Flow over a Solid Sphere with the Effect of MHD. MUST: Journal of Mathematics Education, Science and Technology, 6(1), pp.97–120.
  • Chamkha, A.J. and Ahmed, S.E., 2011. Unsteady MHD Heat and Mass Transfer by Mixed Convection Flow in the Forward Stagnation Region of a Rotating Sphere in the Presence of Chemical Reaction and Heat Source. In Proceedings of the World Congress on Engineering, 1 pp.133–138.
  • Sahaya Jenifer, A., Saikrishnan, P. and Lewis, R.W., 2021. Unsteady MHD Mixed Convection Flow of Water over a Sphere ‎with Mass Transfer. Journal of Applied and Computational Mechanics, 7(2), pp.935–942.
  • Gul, T., Ali, B., Alghamdi, W., Nasir, S., Saeed, A., Kumam, P., Mukhtar, S., Kumam, W. and Jawad, M., 2021. Mixed Convection Stagnation Point Flow of the Blood Based Hybrid Nanofluid around a Rotating Sphere. Scientific Reports, 11(1), pp.1–15.
  • Calabretto, S.A., Levy, B., Denier, J.P. and Mattner, T.W., 2015. The Unsteady Flow Due to an Impulsively Rotated Sphere, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2181), p.20150299.
  • Mahdy, A.E.N., Hady, F.M. and Nabwey, H.A., 2021. Unsteady Homogeneous-Heterogeneous Reactions in MHD Nanofluid Mixed Convection Flow Past a Stagnation Point of an Impulsively Rotating Sphere. Thermal Science, 25(1 Part A), pp.243–256.
  • Almakki, M., Mondal, H., Mburu, Z. and Sibanda, P., 2022. Entropy Generation in Double Diffusive Convective Magnetic Nanofluid Flow in Rotating Sphere with Viscous Dissipation. Journal of Nanofluids, 11(3), pp.360-372.
  • Das, K., Kundu, P.K. and Sk, M.T., 2022. Magnetophoretic Effect on the Nanofluid Flow Over Decelerating Spinning Sphere with the Presence of induced Magnetic Field. Journal of Nanofluids, 11(1), pp.135-141.
  • Mahmood, Z., Alhazmi, S.E., Khan, U., Bani-Fwaz, M.Z. and Galal, A.M., 2022. Unsteady MHD Stagnation Point Flow of Ternary Hybrid Nanofluid over a Spinning Sphere with Joule Heating. International Journal of Modern Physics B, 36(32), p.2250230.
  • Bellman, R.E. and Kalaba, R.E., 1965. Quasilinearization and Nonlinear Boundary Value Problems, The RAND Corporation, American Elsevier Publishing Company, Inc., New York.
  • Inouye, K. and Tate, A., 1974. Finite-Difference Version of Quasi-Linearization Applied to Boundary-Layer Equations. AIAA Journal, 12(4), pp.558–560.
  • Varga, S., 2000. Matrix Iterative Analysis, Springer-Verlag Berlin Heidelberg.

 

[*]Corresponding Author: Saikrishnan Ponnaiah

   Email: psai@nitt.edu

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  • Hatem, N., Philippe, C., Mbow, C., Kabdi, Z., Najoua, S. and Daguenet, M., 1996. Numerical Study of Mixed Convection Around a Sphere Rotating About Its Vertical Axis in a Newtonian Fluid at Rest and Subject to a Heat Flux. Numerical Heat Transfer, Part A: Applications, 29(4), pp.397–415.
  • Hatzikonstantinou, P., 1990. Effects of Mixed Convection and Viscous Dissipation on Heat Transfer about a Porous Rotating Sphere. ZAMM - Journal of Applied Mathematics and Mechanics, 70(10), pp.457–463.
  • Le Palec, G. and Daguenet, M., 1987. Laminar Three-Dimensional Mixed Convection About a Rotating Sphere in a Stream. International Journal of Heat and Mass Transfer, 30(7), pp.1511–1523.
  • Rajasekaran, R. and Palekar, M.G., 1985. Mixed Convection About a Rotating Sphere. International Journal of Heat and Mass Transfer, 28(5), pp.959–968.
  • Rajasekaran, R. and Palekar, M.G., 1985. Viscous Dissipation Effects on Mixed Convection About a Rotating Sphere. International Journal of Engineering Science, 23(8), pp.789–795.
  • Tieng, S.M. and Yan, A.C., 1992. Investigation of Mixed Convection About a Rotating Sphere by Holographic Interferometry. Journal of Thermophysics and Heat Transfer, 6(4), pp.727–732.
  • Patil, P.M., Benawadi, S. and Shanker, B., 2022. Influence of Mixed Convection Nanofluid Flow over a Rotating Sphere in the Presence of Diffusion of Liquid Hydrogen and Ammonia. Mathematics and Computers in Simulation, 194, pp.764–781.
  • Sathyakrishna, M., Roy, S. and Nath, G., 2001. Unsteady Two-Dimensional and Axisymmetric MHD Boundary-Layer Flows. Acta Mechanica, 150(1–2), pp.67–77.
  • Turkyilmazoglu, M., 2011. Numerical and Analytical Solutions for the Flow and Heat Transfer near the Equator of an MHD Boundary Layer over a Porous Rotating Sphere. International Journal of Thermal Sciences, 50(5), pp.831–842.
  • Kazem, S., Tameh, M.S. and Rashidi, M.M., 2019. An Improvement To The Unsteady MHD Rotating Flow Over a Rotating Sphere Near the Equator via Two Radial Basis Function Schemes. The European Physical Journal Plus, 134(12), p.611.
  • Vajravelu, K., Li, R., Dewasurendra, M. and Prasad, K.V., 2017. Mixed Convective Boundary Layer MHD Flow Along a Vertical Elastic Sheet. International Journal of Applied and Computational Mathematics, 3(3), pp.2501–2518.
  • Ghani, M. and Rumite, W., 2021. Keller-Box Scheme to Mixed Convection Flow over a Solid Sphere with the Effect of MHD. MUST: Journal of Mathematics Education, Science and Technology, 6(1), pp.97–120.
  • Chamkha, A.J. and Ahmed, S.E., 2011. Unsteady MHD Heat and Mass Transfer by Mixed Convection Flow in the Forward Stagnation Region of a Rotating Sphere in the Presence of Chemical Reaction and Heat Source. In Proceedings of the World Congress on Engineering, 1 pp.133–138.
  • Sahaya Jenifer, A., Saikrishnan, P. and Lewis, R.W., 2021. Unsteady MHD Mixed Convection Flow of Water over a Sphere ‎with Mass Transfer. Journal of Applied and Computational Mechanics, 7(2), pp.935–942.
  • Gul, T., Ali, B., Alghamdi, W., Nasir, S., Saeed, A., Kumam, P., Mukhtar, S., Kumam, W. and Jawad, M., 2021. Mixed Convection Stagnation Point Flow of the Blood Based Hybrid Nanofluid around a Rotating Sphere. Scientific Reports, 11(1), pp.1–15.
  • Calabretto, S.A., Levy, B., Denier, J.P. and Mattner, T.W., 2015. The Unsteady Flow Due to an Impulsively Rotated Sphere, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471(2181), p.20150299.
  • Mahdy, A.E.N., Hady, F.M. and Nabwey, H.A., 2021. Unsteady Homogeneous-Heterogeneous Reactions in MHD Nanofluid Mixed Convection Flow Past a Stagnation Point of an Impulsively Rotating Sphere. Thermal Science, 25(1 Part A), pp.243–256.
  • Almakki, M., Mondal, H., Mburu, Z. and Sibanda, P., 2022. Entropy Generation in Double Diffusive Convective Magnetic Nanofluid Flow in Rotating Sphere with Viscous Dissipation. Journal of Nanofluids, 11(3), pp.360-372.
  • Das, K., Kundu, P.K. and Sk, M.T., 2022. Magnetophoretic Effect on the Nanofluid Flow Over Decelerating Spinning Sphere with the Presence of induced Magnetic Field. Journal of Nanofluids, 11(1), pp.135-141.
  • Mahmood, Z., Alhazmi, S.E., Khan, U., Bani-Fwaz, M.Z. and Galal, A.M., 2022. Unsteady MHD Stagnation Point Flow of Ternary Hybrid Nanofluid over a Spinning Sphere with Joule Heating. International Journal of Modern Physics B, 36(32), p.2250230.
  • Bellman, R.E. and Kalaba, R.E., 1965. Quasilinearization and Nonlinear Boundary Value Problems, The RAND Corporation, American Elsevier Publishing Company, Inc., New York.
  • Inouye, K. and Tate, A., 1974. Finite-Difference Version of Quasi-Linearization Applied to Boundary-Layer Equations. AIAA Journal, 12(4), pp.558–560.
  • Varga, S., 2000. Matrix Iterative Analysis, Springer-Verlag Berlin Heidelberg.
  • Receive Date: 16 March 2021
  • Revise Date: 04 January 2023
  • Accept Date: 10 January 2023
  • First Publish Date: 10 January 2023