Effects of Combined Heat and Mass Transfer on Entropy Generation due to MHD Nano ﬂ uid Flow over a Rotating Frame

: The current investigation aims to explore the combined effects of heat and mass transfer on free convection of Sodium alginate-Fe 3 O 4 based Brinkmann type nano ﬂ uid ﬂ ow over a vertical rotating frame. The Tiwari and Das nano ﬂ uid model is employed to examine the effects of dimensionless numbers, including Grashof, Eckert, and Schmidt numbers and governing parameters like solid volume fraction of nanoparticles, Hall current, magnetic ﬁ eld, viscous dissipation, and the chemical reaction on the physical quantities. The dimensionless nonlinear partial differential equations are solved using a ﬁ nite difference method known as Runge-Kutta Fehlberg (RKF-45) method. The variation of dimensionless velocity, temperature, concentration, skin friction, heat, and mass transfer rate, as well as for entropy generation and Bejan number with governing parameters, are pre-sented graphically and are provided in tabular form. The results reveal that the Nusselt number increases with an increase in the solid volume fraction of nano-particles. Furthermore, the rate of entropy generation and Bejan number depends upon the magnetic ﬁ eld and the Eckert number.


Introduction
Most of the conventional liquids such as saltwater, liquid metal, plasma, etc. are conducting fluids that have over the years captured immense attention of renowned researchers to the study of the dynamics of these fluids because of their significant engineering applications like MHD generators, flow meters, metal purification, metallurgy, geothermal energy extractions, and polymer technology. Some studies [1][2][3] involve fluid flow analysis of the electrically conducting fluids. However, Hall current effect is significant for a strong magnetic field and a low density [4]. In the pioneering work of [5], the Hall current effect was taken into consideration to examine the magnetohydrodynamic flow of a viscous ionized gas passing through parallel plates. Further studies of Hall effects have been communicated by [6] who reviewed the peristaltic flow of a Jeffrey non-Newtonian fluid over vertical walls in the presence of porous medium and Hall influence. Muthucumaraswamy et al. [7] studied the unsteady flow of a viscous fluid over an exponential plate accelerating due to density difference with Hall effects and thermal radiation.
Hydromagnetic fluid flow problems are essential in the field of earth science, Meteorology. Interestingly, the Hall current induces both the primary and secondary flows in fluid governed of Coriolis force. Very recently, Krishna et al. [8] have been analyzed the mixed convection laminar flow of hydromagnetic viscous rotating fluid flow over a porous vertical sheet with Hall effects. Hall current influence on the unsteady flow of an oscillating fluid over an exponential slip plate with chemical reaction is investigated by [9]. They concluded that the Coriolis force and Hall current tend to augment the fluid velocity in the secondary flow direction whereas, in the primary flow direction, Ion-slip current enhanced. Given these applications, Hall current effects on rotating magnetohydrodynamic have been studied in various flow geometries, for example [10,11].
Above mentioned literature was performed in the fluid flow models of a conventional fluid flow of an electrically conducting fluid. Still, fluids with the inclusion of nanometer-sized particles (nanofluid) behave quite differently from that of the traditional fluid in several vital aspects. A report from the current trend in research has shown that heat transfer is enhanced in the thermal system through the embedded nanoparticle into conventional liquids. The applications exist in a solar receiver, nuclear reactor, microbial fuel cell, thermal storage, biomedical applications, heat exchangers, industrial cooling medium. Authors have established several results about nanofluid flow in various geometries, for example, see Ali et al. [12,13].
In energy management, minimizing entropy production in a thermal system cannot be overemphasized because of its limited percentage of energy available as heat. It is, however, imperative to improve the amount of energy available for work through entropy generation. Some relevant articles that analyzed the flow and heat transfer using the 2nd law of thermodynamics are [14][15][16][17]. Opanuga et al. [18] have examined the Hall current and ion-slip on a steady flow of micropolar fluid through an infinite vertical channel with entropy generation.
The objective of the current analysis is to examine the rate of entropy optimization on MHD Brinkmantype nanofluid flow over a vertical rotating plate with the influence of radiation and chemical reaction. It is, therefore, pertinent to examine the effect of this feature because entropy production occurs in moving fluid with high temperature. To the best of our knowledge, the present study has not remained investigated. By applying suitable transformations, the governing equations of the model are converted to non-dimensional form and then solved by employing the Runge-Kutta-Fehlberg scheme. Effects of all the pertinent parameters on velocity, temperature, nanoparticle concentration, skin friction coefficient, Nusselt number, Sherwood number, entropy generation, and the Bejan number profiles are shown through graphs and extensively discussed.

Problem Formulation
A magnetohydrodynamic convective flow of Sodium alginate-Fe 3 O 4 based Brinkmann type nanofluid is examined in a vertical rotating frame. The flow is assumed to be incompressible and time-dependent.
Subject to the initial and boundary conditions Defining The dimensionless variables are given by Thus, the governing equations are: The boundary conditions are given by where the parameters are defined by  The local skin friction C f À Á , Nusselt number Nu ð Þ, and the Sherwood number Sh ð Þ are: The wall shear stress is p w the heat transfer rate q w and the rate of mass transfer j w The dimensionless form

Entropy Generation
The expression for the entropy generation of this model may be written as Here, the characteristic entropy S 000 gen ¼ , the dimensionless form of entropy generation yields In dimensionless form, Bejan number may be written as Be ¼ @h @y The physical properties of the base fluid and nanoparticles have been reported in the Tab. 1.

Results and Discussion
The system of partial differential Eqs. (8)- (11) with associated initial and boundary conditions Eq. (12) are solved numerically using a finite difference method. In all cases, we have adopted the following default values of parameters c ¼ m ¼  Fig. 3a elucidates the impact of M on T y; t ð Þ. Inside the thermal boundary layer, the dimensionless temperature increases with the magnetic field. In this plot, higher estimations of M connects presence of Lorentz heating in the flow, the force boost the fluid temperature, and a distinct trend is perceived within 0:4 y 2:0. The impact of the radiation parameter on the dimensionless temperature T y; t ð Þ is presented in Fig. 3b. However, an enhancement in the temperature profile is observed at all points in the  presence of thermal radiation. The reason for this trend is that bigger estimations of Nr produce more heat into the fluid, causing a rise in the temperature.
The effects of the solid volume fraction of nanoparticles and Eckert number on the dimensionless temperature are shown in the Figs. 4a and 4b respectively. As shown in Fig. 4a that enhancement in the solid volume fraction of nanoparticles f leads to an increase in the temperature profile. Moreover, an increase in Ec corresponds to a significant rise in the temperature profile. Fig. 4b demonstrates this behavior. Physically, frictional heating produces more heat with an increase in Ec.
Figs. 5a and 5b illustrate the impacts of the chemical reaction C m ð Þ and Schmidt number Sc ð Þ on the dimensionless concentration, respectively. The behavior of the dimensionless concentration for different values of destructive chemical reaction parameters C m > 0 ð Þis portrayed in Fig. 5a. It is noticed that the dimensionless concentration is a decreasing function of C m . In true sense, the amount of nanomaterials presence in the fluid becomes smaller as the destructive chemical reaction occurs. Meanwhile, Fig. 5b displays the concentration profile decreases rapidly with an increase in the Schmidt number. Physically,  The rate of heat transfer as a function of f is exhibited in Fig. 7. for different values of M ; Ec; n; and c. These plots show that due to the temperature gradient, the heat flux is an increasing function of M . Tab. 3 reports that the Nusselt number increases with an increase in the volume fraction of nanoparticles and the radiation parameter. The higher rate of heat transfer from the moving fluid to the wall with larger values of the Brinkman parameter c. However, as shown in Fig. 7, an augmented m; c; Ec and slow down the heat transfer rate Nu. Physically, enhancement in Ec corresponds to upsurge in the thermal field via dissipation, hence, boosting the heat transfer rate.
The influence of the Schmidt number Sc and chemical reaction parameter C m on the Sherwood number Sh is displayed in Fig. 8 and Tab. 4. It may be noted that with a rise in the concentration gradient, mass transport increases for increasing values of both Sc and C m .   The impacts of governing parameters, including M ; c; Ec , and Gr on the entropy generation rate, N G are shown in Fig. 9. It is noticed that the rate of disorderliness becomes low in the absence of M . However, the magnetic field produces a Lorentz force, which boosts the rate of entropy generation. Also, higher values of Ec escalate the entropy production. It is observed from the same plot that improving the magnitude of c marginally suppressed the rate of entropy generation. Moreover, throughout the fluid system, enhanced Gr suppressed the rate of entropy generation. This is an indication that there is more fluid-particle disorder via augmentation in M ; Ec; c, and Gr.

Conclusions
In this paper, the effects of viscous dissipation and chemical reaction on MHD flow with combined heat and mass transfer of incompressible sodium-alginate based Fe 3 O 4 in a rotating frame have been analyzed. The following are the main results of the present study: The dimensionless velocity u y; t ð Þ increases with the augmentation of Gr and f while it peters out via incremented M . The dimensionless temperature h y; t ð Þ increases with the augmentation of M ; f; and Ec. An increase in the chemical reaction parameter and Schmidt number has shown a declining trend for the dimensionless concentration C y; t ð Þ. Viscous drag decreases due to M ; f, and c while shows the opposite fashion via Gr and m. The rate of heat transfer is decreasing due to the rise in Ec and m. The mass transfer rate increases with an increase inSc while it decreases with f. Rate of Entropy generation and Bejan number shows the opposite trend for M and Ec.
Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.