Thermodynamic irreversibility effects with Marangoni convection for third grade nanofluid flow

In this study, an analysis was performed to investigate the thermal and mass transport of radiative flow of a third-grade nanofluid with magnetohydrodynamic. The analysis concerns two-dimensional flow around an infinite disk. Heat transport is studied via heat generation/absorption, thermal radiation and Joule heating. Chemical reaction with activation energy is also considered. The nanofluid characteristics, including Brownian motion and thermophoretic diffusion, are explored via the Buongiorno model. Entropy analysis is also conducted. Moreover, the surface tension is assumed to be a linear function of concentration and temperature. Through adequate dimensionless variables, governed PDEs are non-dimensionlized and then tackled by ND-solve (a numerical method in Mathematica) for solutions purposes. Entropy generation, concentration, velocity, Bejan number and temperature are plotted as functions of the involved physical parameters. It is noticed that higher Marangoni number intensify velocity however it causes a decrease in the temperature. Entropy rate and Bejan number boost for large value of diffusion parameter.


Introduction
In the current scenario, nanofluids become operative coolant media in many industrial and technological processes. Due to high thermal efficiency, nanofluid got incredible popularity among researchers and scientists. The idea of adding nanoparticles in traditional fluid for advancing its thermal conduction was given by Choi [1]. He was the first to propose that normal fluids can be replaced by more thermally effective fluids called nanofluids. Normal fluids have limited applications due to their poor thermal conductivity. The introduction of nanofluids brings a lot of new applications in industrial and mechanical research. Buongiorno [2] presented the mathematical model for nanofluid flow based on thermophoresis and Brownian diffusion. Alsaedi et al. [3] conducted a study on a hybrid nanofluid that is enclosed between two coaxial cylinders. Entropy generation and melting phenomena during flow of nanofluid is examined by Alsaadi et al. [4]. Kandasamy et al. [5] examined the MHD flow of nanomaterial via the Buongiorno model. Sheikholeslami and Shehzad [6] studied nanofluid flow saturated through a porous medium with mixed convection. Muhammad et al. [7] provided a study on a fourth-grade nanofluid subjected to both stagnation point and convective boundary conditions. The latest work in this area can be seen in Refs. [8][9][10][11][12][13].
The Marangoni effect is due to the surface tension gradient, and it is the mass transfer between two fluid interfaces. Temperaturedependent aforementioned phenomena are known by thermo-capillary convection. James Thomson introduced this mechanism in 1855. It is widely used in the field of artwork. Soap film stabilization, convection cells or Benard cells, etc. are common applications of the Marangoni effect. Sreenivasulu et al. [14] expressed radiation impact in the MHD flow of viscous material with thermosolutal Marangoni convection. Hayat et al. [15] studied Marangoni convection and thermal radiation in flow of nanomaterial. Zhao et al. [16] examined the Soret and Dufour effects of fractional magneto hydrodynamic Maxwell fluid. Mahanthesh and Gireesha [17] consider Marangoni convection, joule heating, thermal radiation and viscous dissipation during the flow of Casson nanomaterial. Zhuang and Zhu [18] examined Marangoni convection in power-law nanoliquids. The energy flux generation via concentration gradient is initially presented by Dufour and called the thermal diffusion (Dufour) effect. Similarly, generation of mass flux by temperature gradient is referred as thermo-diffusion or Soret effect. Soret and Dufour impacts in MHD viscous fluid flow due to rotating cone with radiation effect is explored by Khan et al. [19]. Studies regarding the heat and mass transport are addressed in Refs. [20][21][22][23][24].
To efficient devices, scientists and researchers are constantly looking for procedures to control energy consumption. The main objective is to minimize heat loss to maximize the efficiency of machines. Entropy optimization plays a vital role in the improvement of the performance of many devices in various engineering and industrial sectors. Bejan [25] initially introduce the idea of entropy optimization. Govindaraju et al. [26] investigated the entropy optimization of MHD nanomaterial flow past a stretchable surface. Hayat et al. [27] investigated the entropy generation of Ree-Eyring nanomaterial flow bounded by rotating disks with activation energy. A few observations regarding entropy optimization are given in Refs. [23,[28][29][30][31][32].
In the present work, our main intention is to examine the transportation of Marangoni convection in third-grade nanofluid with Dufour and Soret effects. Joule heating, thermal radiation, and internal heat generation describe heat transport features, while the chemical reaction is considered with activation energy. Governing equations (PDEs) are transformed by using appropriate variables. Entropy rate and Bejan number are considered. Effects of different flow parameters on quantities of interest are explored graphically.

Mathematical formulation
Marangoni convective inclined flow of third grade nanofluid is considered by an infinite disk. Buongiorno model for nanofluid is accumulated by Brownian and thermophoretic diffusion. Soret and Dufour's effects are considered. Current density is given by J = σ(E + V × B). Electric field strength is neglected. The heat transmission rate is explored via heat source/sink and thermal radiation.
Induced magnetic field and Hall effects are neglected due to the low magnetic Reynolds number. Constant magnetic field B 0 is applied normally to flow direction. Geometry of flow model is given in Fig. (1).
After implementing boundary layer assumptions and problem-related assumptions, we get (see Refs. [19,33]) ∂u ∂r Surface tension is defined as [34,33] where Note that γ C , γ T , and σ 0 are positive constants.
Considering the following transformations Using these transformations in Eqs. (1)- (8), and applying first order truncation, we get with Related parameters are

Entropy analysis
Entropy is given as Fig. (1). Flow geometry.
where Bejan number = Entropy via heat and mass transfer Total entropy , where where heat flux (q w ) is In dimensionless form

Mass transfer rate
Sherwood number is expressed as where the mass flux (J w ) is In dimensionless form

Solution methodology
The PDEs associated to the problem are non-dimensionalized via adequate variables. The non-dimensional PDEs are then tackled through ND-solve. To ensure convergence of ND-solve, an appropriate numerical method with fine grid, initial and boundary conditions are required. Note that in above equations, f, θ, ϕ, S G and Be are functions of ξ and η.For solutions purpose we have taken ξ=1 and treated f, θ, ϕ, S G and Be as funtions of η (see all plots).

Analysis
Effects of the Marangoni number, Marangoni ratio parameter, magnetic, radiation, heat generation, diffusion, and other involved influential flow parameters on the velocity field, temperature distribution, concentration, Bejan number and entropy rate are presented below. Table 1 is created to provide naming for the various parameters and expressions involved in our study.

Velocity
Velocity variations over various flow variables are presented in Figs. (2)(3)(4)(5). Here, the influence of the magnetic parameter on the velocity field is illustrated in Fig. (2). Velocity reduces against higher magnetic parameter. It is due to the enhancement of Lorentz forces in fluid flow which acts as a resistance force to fluid flow and thus the velocity of the fluid decays.   Fig. (2). M vs Velocity.

Temperature
Temperature variations for different involved parameters are depicted in Figs. 6-11. Fig. (6) displays impact of magnetic parameter on temperature. Temperature (θ(η)) rises for higher magnetic parameter (M). Physically, for larger (M) due to strong resistive forces (Lorentz force) cause resistance to liquid flow and results in more collision between the fluid particles thus temperature boosts for higher (M). Temperature variations for Marangoni number (Ma) is displayed in Fig. (7). A reduction in temperature (θ(η)) occurs for higher estimations of Marangoni number (Ma). Impact of radiation on temperature is displayed in Fig. (8). Higher (Rd) results in temperature enhancement. Physically, for an augmentation in radiation causes a decline in mean absorption coefficient which enhances thermal flux and thus temperature improves. Influence of heat generation (δ) on temperature distribution is illustrated in Fig. (9). (θ(η)) increases against higher heat generation parameter (δ). When a heat source is present, it releases thermal energy into the Fig. (4). Ra vs Velocity. Fig. (3). Ma vs Velocity. Fig. (5). sin ∝ * vs Velocity.
K. Muhammad et al. system, which raises the internal energy of the fluid and leads to an increase in temperature. The amount of temperature increase will depend on the amount of heat released, the heat capacity of the fluid, and the size of system. A rise in temperature can be seen for higher values of Dufour number (Du) see Fig. (10). Effects of thermophoresis parameter (N t ) on (θ(η)) is given in Fig. (11). A rise in temperature occurs for large values of thermophoresis parameter (N t ).

Concentration
Figs. 12-17 expresses the impacts of different flow parameters of concentration. Fig. (12) represents the effect of Marangoni number (Ma) on concentration. Clearly, concentration decays against higher Marangoni number (Ma). Fig. (13) illustrate variation of concentration for Schmidt number (Sc). Concentration reduces for higher Schmidt number. Since, mass diffusivity decays for an   K. Muhammad et al. increment in Schmidt number and as a result reduces. Fig. (14) depicts effect of (k 1 ) over concentration. (ϕ(η)) decreases against higher chemical reaction parameter (k 1 ). Fig. (15) witnesses' outcome of Brownian motion parameter (N b ) for concentration. Reduction in concentration occurs against higher Brownian motion parameter (N b ). Brownian diffusion is the process by which particles in a fluid move randomly due to collisions with other particles. This can lead to a decrease in concentration of the particles over time, as the particles become dispersed throughout the fluid. Fig. (16) elucidate influence of Soret number (Sr) for concentration. Concentration enlargement occurs for higher Soret number (Sr). Fig. (17) display variation of concentration against thermophoresis parameter (N t ). An increment in concentration is observed for higher thermophoresis parameter (N t ).

Bejan number and entropy generation
The entropy generation rate and Bejan number outcomes for various parameters are displayed in Figs. 18-25. Fig. (18) depicts impact of magnetic parameter on entropy rate (S G (η)). An increment in magnetic parameter (M) results in entropy rate enhancement. Physically, more resistive forces (Lorentz forces) generate for higher magnetic field. Which enhances system disorder and consequently entropy rate boosts. Variations of magnetic field for Bejan number (Be) is displayed in Fig. (19). Reduction in Bejan number occurs for larger magnetic parameter (M). Figs. 20 and 21 are intended to display impact of radiation on (S G (η)) and (Be). Both entropy rate and Bejan number show similar behavior against higher radiation parameter (Rd). Physically an increment in (Rd) results in more radiation emission which increases disorder in system and as a result entropy rate increase. Figs. 22 and 23 display effect of Brinkman number (Br) on entropy rate (S G (η)) and Bejan number (Be). Reverse behavior of entropy rate and Bejan number is noted against higher

Final remarks
In the present work, our main intention was to examine the irreversibility analysis with the transportation of Marangoni convection in third-grade nanofluid with Dufour and Soret effects. Joule heating, thermal radiation, and internal heat generation described heat transport features, while the chemical reaction and activation energy was considered to determine mass transport characteristics. The