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
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
^{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: 20210316 Received: 20230104 Accepted: 20230110 
This paper investigates the unsteady magnetohydrodynamic (MHD) mixed convective fluid flow over a rotating sphere. An implicit finite difference scheme, together with quasilinearization, is used to find nonsimilar 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 nonuniform slot suction(injection) and the slot movement influence the point of vanishing skin friction to move in the axial direction downstream (upstream). 



Keywords: Boundary layer; Heat transfer; Variable properties; Mixed convection; MHD; Nonuniform mass transfer; Nonsimilar solution; Rotating sphere. 


© 2022 Published by Semnan University Press. All rights reserved. 
The investigation of boundary layer flow and heat transfer over rotating bodies of revolution has several technical applications, including fiber coating, reentry missile design, and rotary machine design [1]. The sphere being a wellrenowned 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 abovedescribed 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 [68].
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 [1016].
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 highspeed aircraft [17]. Uniform suction (injection) creates discontinuities at the ends of the slot. An ultimate solution to overcome this is by implementing a nonuniform suction(injection), as discussed by Roy and Nath in [18]. Since then, several researchers have carried out the work on the impact of the nonuniform mass transfer over various twodimensional axisymmetric bodies [12,13,1517,19] and that over rotating bodies [11,14,21].
The boundarylayer flows are found to be both unsteady and nonsimilar in nature. The unsteadiness and nonsimilarity 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 selfsimilar flows or steady nonsimilar flows due to mathematical complexities. A brief review of methods to find a nonsimilar 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 nonsimilar solution for steady/unsteady flows over various shapes of nonspinning bodies [12,13,15,17,2224]. In the case of rotating bodies, authors in [2527] presented selfsimilar solutions for steady/unsteady flow over a rotating sphere, whereas in [10,11,28] have given nonsimilar 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 nonspinning bodies [17,2932] and over rotating bodies [1,3339]. On the other hand, the effect of MHD on steady or unsteady fluid flow over twodimensional 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 Kellerbox 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 nonNewtonian Casson nanofluid. Recently, Jenifer et al. [46] obtained nonsimilar 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 bloodbased hybrid nanofluid over a rotating sphere with the inclusion of mixed convection and a timedependent 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. [5052].
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 nonsimilar solutions under the combined effects of the following circumstances.
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 abovementioned 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.
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 timedependent 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 nondimensional form are as follows:

(9) 
The above transformations satisfy (3) identically, and the nondimensional 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) 
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 abovementioned 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.
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 abovementioned 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 

Figure 3. Comparison of the skin friction parameter in the direction with those of Chen et al. [29] where 

Figure 4. Comparison of the skin friction parameter in the direction for an unsteady flow with those 

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 1316 show the influence of nonuniform 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 nonuniform slot suction on the skin friction coefficients in directions for 

Figure 14. Effect of the nonuniform slot suction on the heat transfer coefficient for , 

Figure 15. Effect of the nonuniform slot injection on the skin friction coefficients in directions for 

Figure 16. Effect of the nonuniform slot injection on the heat transfer coefficient for , 
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.
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.
References
[*]Corresponding Author: Saikrishnan Ponnaiah
Email: psai@nitt.edu