Investigation of entropy generation in power law fluid-filled bulging enclosures: effects of flow and heat transfer

Abstract

This paper presents a complete investigation into the phenomenon of dual convection flow within a bulging enclosure along with non-Newtonian power-law fluids and employs the Galerkin finite element method for simulation. The main objective is to study the heat and mass transfer and entropy generation within the enclosure. Various factors, such as the Rayleigh number, power law coefficient, Lewis number, and effects of magnetic inclination are evaluated for their influence on flow dynamics and heat distribution. The study also investigates the relationship between these parameters and entropy generation, providing insights into the irreversible processes within the system. A comparative analysis of averaged Nusselt and Sherwood coefficients reveals distinct thermal and fluidic behaviours across varying power-law indices, emphasizing differences between shear thickening, shear thinning, and Newtonian fluid behaviours. The findings provide valuable insights into non-Newtonian power-law fluids in complex flows, enhancing our understanding of flow and heat transfer in bulging enclosures.

KEYWORDS:

• Power law fluid
• mixed convection
• MHD
• entropy generation
• Galerkin finite element method

Nomenclature

$X,Y$	=	Dimensionless horizontal and vertical coordinates
$x,y$	=	Horizontal and vertical coordinates with dimensions (metres)
$U$, $V$	=	Dimensionless velocity components along X and Y
$u,v$	=	Velocity component along x and y-axis
$T$	=	Temperature in Kelvin (K)
$c_p$	=	Specific Heat $\left(\text{J}.\text{k}{\text{g}}^{-1}.{\text{K}}^{-1}\right)$
$K_e$	=	Thermal conductivity (effective) $\left(\text{W}.\text{m}{\text{K}}^{-1}\right)$
$C$	=	Concentration (non-dimensional)
$P,p$	=	Dimensionless and dimensional fluid pressure
$B$,$B_0$	=	Magnetic field and its strength in Tesla
$\mathrm{Sh},S{h}_{\mathrm{avg}}$	=	Local and Averaged Sherwood number
$\mathrm{Nu},N{u}_{\mathrm{avg}}$	=	Local and Averaged Nusselt number
$\mu$	=	Kinematic Viscosity $\left({\text{m}}^2.{\text{s}}^{-1}\right)$
$\gamma$	=	Angle of Inclination (degree)
$\mathrm{Pr}$	=	Prandtl number
$\mathrm{Ra}$	=	Rayleigh number
$\theta$	=	Temperature (non-dimensional)
$\rho$	=	Fluid density $\left(\text{kg}.{\text{m}}^{-3}\right)$
${\varrho }_e$	=	Thermal diffusivity (effective) $\left({\text{m}}^2.\text{se}{\text{c}}^{-1}\right)$
$\mathrm{NEL}$	=	Number of elements
$\mathrm{Le}$	=	Lewis number
$\mathrm{DOF}$	=	Degree of freedom

1. Introduction

Non-Newtonian fluids, with their unique characteristics and behaviours, have emerged as a vital realm of research with practical applications spanning engineering, food science, materials science, pharmaceuticals, and industry. Natural substances display a variety of characteristics, featuring non-Newtonian fluids present in materials like mucus, blood, and plant sap, emphasizing their distinct behaviours and practical applications. Research on fluids that exhibit non-Newtonian behaviour drives innovation across biomedicine, cosmetics, and geology, fostering the development of new technologies and methods. Particularly in biomedicine, the analysis of non-Newtonian fluids like synovial fluid and blood holds significant importance, providing valuable understandings of essential biological processes. To study non-Newtonian liquids thoroughly, they are classified into shear thickening and shear thinning classes based on their behaviour. The viscosity of shear-thinning fluids decreases with an increase in shear rate, whereas the viscosity of shear-thickening fluids increases as the shear rate goes up. Experts in the field of rheology have introduced multiple models to depict the classification. Among these models, the power-law fluid is the most widely recognized and comprehensive, capturing the essential features comprehensively. Ozoe and Churchill [1] analysed convection within a horizontal cavity involving power-law fluid flow. They investigated the hydrodynamic stability and circulation zones by measuring them with the model parameter. In their study, Kaddiry et al. [2] determined the value of the power-law exponent that accurately captures the flow characteristics of both pseudoplastic and dilatant materials. This assessment was carried out by examining the flow within a four-equal sides cavity. Kim et al. [3] analysed the dynamic buoyancy-driven convection of a power-law exponent within an isothermal square enclosure by considering various temperature conditions at the enclosure boundaries. Later on Horimek et al. [4] introduce laminar natural convection in various heated square enclosures for a power-law fluid, considering clockwise and counterclockwise inclinations. Simulations are performed using a finite volume code, revealing increased disturbances in dynamic and thermal fields with higher Rayleigh numbers and lower rheological index, improved heat exchange coefficient, and identification of an optimal angle for counterclockwise inclination influenced by the parameters. Ternik et al. [5] utilized the finite volume method to model hydrodynamic and thermal fields, investigating the behaviour of power-law fluids while using the Boussinesq approximation. Their study deepened the understanding of fluid dynamics and heat transfer, offering insights into the hydrodynamic and thermal characteristics of such systems. Turkyilmazoglu et al. [6] utilize finite element simulations to analyse a modified lid-driven cavity flow with a dual-sectioned moving lid, examining its impact on vortex formation and fluid dynamics for potential mixing and purification applications. Sajoudi et al. [7] investigated convection in a quadrilateral with a parallel side cavity filled with motionless power-law fluid, with adiabatic lower and upper surfaces. Natural convection inside the cavity containing the non-Newtonian fluid described by power-law model, induced by differentially heated side walls studied by Turan et al. [8]. Khezzar et al. [9] introduced a study on stable two dimensional natural convection occurring within rectangular cavities that were filled with non-Newtonian fluids, particularly those described by power law and Boussinesq models. The researcher discussed the effects of varying parameters on flow patterns and thermal flow rates. An overview of recent progress in the study of power-law non-Newtonian fluids, considering multiple physical aspects and diverse flow generation domains, is presented in Refs [10–16].

Magnetohydrodynamics (MHD) is a field of fluid mechanics dedicated to the study of how electrically conductive fluids interact with magnetic fields, with significant applications in industries that benefit from improved performance and operational capabilities of electric generators, advanced pumps, and cooling systems. Magnetohydrodynamics (MHD) is of significant importance across various scientific and engineering fields, as it provides valuable insights into the workings of different natural occurrences and artificial systems. Alam et al. [17] discussed mass and heat transfer in a continuous flow under MHD and examined the application of thermophoresis on an inclined flat region. Turkyilmazoglu et al. [18] have derived precise solutions for MHD fluid flow within a triangular pipe, revealing the impact of the Hartmann number on modulating velocity profiles and flow dynamics under a uniform magnetic field. Kandasamy et al. [19] investigated the impact of heat source and thermal stratification impact on mass and heat transfer transfer within MHD flow. The significant role of magnetic fields is broadly acknowledged across various domains encompassing both medical and engineering sciences. Zain et al. [20] employed numerical analysis along with the Galerkin Least-Squares method in their research to explore how an MHD affects blood flow within a stenosed artery which is bifurcated. The structure of the artery was depicted as a bifurcated passage featuring stenosis with overlapping shapes, and the fluid's non-Newtonian attributes were characterized using a generalized power law model. The research indicates that applying magnetic field results in a 39% reduction in flow reversal for shear-thinning fluids, 26% for Newtonian fluids, and 27% for shear-thickening fluids while reducing adverse pressure and sharp wall shear stress. These results highlight the potential of power-law fluids and MHD thrombosis and promoting uniform flow. Turkyilmazoglu et al. [21] investigate how a uniform magnetic field influences the hydrodynamic properties of microscale plug flows, particularly affecting vortex dynamics and flow resistance. Hussain et al. [22] discussed computational modelling of MHD mixed convection and entropy in a wavy elbow-shaped cavity featuring a rotating cylinder, utilizing a power law hybrid nanofluid. The GFEM was used to compare Nusselt (Nu) numbers for different parameters to analyse the thermal and hydrodynamic behaviour. The results show that a higher power law index increases the Nu number, clockwise rotation enhances thermal convection, and a higher aspect ratio increases the thermal transfer rate, alongside a detailed entropy generation analysis. Kardgar [23] discussed the entropy generation in non-Newtonian hybrid nanofluid in a porous cavity with the effect of magnetohydrodynamic. The researcher concludes that an increase in porosity leads to a rise in both the Bejan number and thermal entropy generation. Yasin et al. [24] used a computational approach to study mixed convection in a parallelogram-shaped cavity, focusing on the effects of magnetohydrodynamics (MHD). The research findings indicate that at higher Richardson numbers, free convection becomes the dominant mode whereas at low Richardson numbers, forced convection prevails of heat transfer.

Based on the reviewed literature, it is concluded that research on the heat transfer and flow behaviour of non-Newtonian power-law fluids in a bulging cavity is limited. This study aims to investigate entropy generation, heat and mass transfer phenomena, and the behaviour of non-Newtonian fluids with shear thickening and shear thinning properties in a bulging enclosure. This study also explores different power-law indices (n = 0.7, 1, and 1.3) to illustrate varied fluid behaviours, and it also examines the effect of an inclined magnetic field. The analysis of flow, thermal, and solutal areas within the enclosure uses the Galerkin Finite Element Method. Through the investigation of these parameters, valuable understandings regarding the behaviour of non-Newtonian fluids in intricate flow configurations, encompassing processes such as entropy generation, mass transfer, and heat transfer.

2. Mathematical formulation

This study focuses on a two-dimensional bulging enclosure having length L and height H containing a power law fluid, as illustrated in Figure 1. This research investigates the characteristics of flow and heat transfer within the enclosure. The configuration considered involves minimum temperature and concentration on the right-side wall and maximum temperature and concentration on the left-side wall, while the remaining walls are adiabatic. Hence, this study presents a novel approach to exploring the non-Newtonian model that describes the properties of shear-thinning and shear-thickening fluids. A magnetic field of constant magnitude B is applied in a horizontal direction. The fluid equations are simplified using the Boussinesq approximation [25], where the fluid density is regarded as a function of temperature.

Figure 1. Sketch view of the problem.

Given the assumptions mentioned above, the equations in the dimensional form are written as [26–30]: (1) $\begin{array}{rl}& \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0\end{array}$(1) (2) $\begin{array}{rl}& u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }x}+\frac{1}{\rho }\left(\frac{\mathrm{\partial }{\tau }_{\mathrm{xx}}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{\mathrm{xy}}}{\mathrm{\partial }y}\right)+{\xi }_x\end{array}$(2) (3) $\begin{array}{rl}& u\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }y}+\frac{1}{\rho }\left(\frac{\mathrm{\partial }{\tau }_{\mathrm{xy}}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{\mathrm{yy}}}{\mathrm{\partial }y}\right)+{\xi }_y+{\mathrm{\Lambda }}_y\end{array}$(3)

The force term (magnetic field and Boussinesq approximation) are, $\begin{array}{rl}\left[\begin{array}{c}{\xi }_x\\ {\xi }_y\end{array}\right]& =\varrho B_0^2\left[\begin{array}{cc}-\sin \gamma & \sin \gamma \\ \cos \gamma & -\cos \gamma \end{array}\right]\left[\begin{array}{c}\mathrm{usin}\gamma \\ \mathrm{vcos}\gamma \end{array}\right]\end{array}$ $\begin{array}{rl}{\mathrm{\Lambda }}_y& =\mathrm{\rho g}\left[{\text{β}}_c\left(c-c_l\right)+{\text{β}}_T\left(T-T_l\right)\right]\end{array}$${\mathrm{\Lambda }}_y$ is called buoyancy effects, $({\text{β}}_T,{\text{β}}_c)$ are thermal and solutal expansions and $\varrho$ is thermal diffusivity. (4) $\begin{array}{rl}& u\frac{\mathrm{\partial }T}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }T}{\mathrm{\partial }y}=\alpha \left(\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}\right)\end{array}$(4) (5) $\begin{array}{rl}& u\frac{\mathrm{\partial }c}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }c}{\mathrm{\partial }y}=D\left(\frac{{\mathrm{\partial }}^{2}c}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}c}{\mathrm{\partial }{y}^2}\right)\end{array}$(5) $\begin{array}{rl}& {\tau }_{\mathrm{ij}}=2{\mu }_a{D}_{\mathrm{ij}}={\mu }_a\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)\end{array}$ $\begin{array}{rl}& {\mu }_a=K{\left\{{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }y}\right)}^2+2\left[{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right)}^2\right]\right\}}^{\frac{n-1}{2}}\end{array}$The magnetic diffusivity is D, and the diffusion coefficient is $\varsigma$.

The boundary conditions are (6) $\begin{array}{r}\begin{array}{l}u=0,v=0,T=T_h,C=C_h\left(\text{For}\text{hot}\text{side}\right)\\ u=0,v=0,T=T_c,C=C_c\left(\text{For}\text{cold}\text{side}\right)\\ u=0,v=0,\frac{\mathrm{\partial }T}{\mathrm{\partial }n}=\frac{\mathrm{\partial }C}{\mathrm{\partial }n}=0\left(\text{For}\text{rest}\text{of}\text{the}\text{wall}\right)\end{array}\end{array}$(6) Non-dimensional equations:

The non-dimension parameters are: (7) $\begin{array}{r}\begin{array}{rl}U& =\frac{\mathrm{uL}}{\alpha },V=\frac{\mathrm{vL}}{\alpha },X=\frac{x}{L},Y=\frac{y}{L},\\ P.\rho {\alpha }^2& =p.L^2,\left(c_h-c_l\right).C=\left(c-c_l\right),\\ P_r& =\frac{K{L}^{2-2n}}{\rho {\alpha }^{2-n}},\\ \left(T_h-T_l\right)\theta & =\left(T-T_l\right)\mathrm{Ra}=\frac{\rho {\text{β}}_{T}g{L}^{2n+1}\mathrm{\Delta T}}{K{\alpha }^n},\\ \mathrm{Ha}& =\mathrm{BH}\sqrt{\frac{\varrho }{\mu }},\mathrm{Le}=\frac{\varsigma }{D},Pr=\frac{v}{\alpha }\end{array}\end{array}$(7) The dimensionless form of Equations (1)–(6) are as follows: (8) $\begin{array}{rl}\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}& =0\end{array}$(8) (9) $\begin{array}{rl}U\frac{\mathrm{\partial }U}{\mathrm{\partial }X}+V\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}& =-\frac{\mathrm{\partial }P}{\mathrm{\partial }X}+P_r[2\frac{\mathrm{\partial }}{\mathrm{\partial }X}\left(\overline{{\mu }_a}\frac{\mathrm{\partial }U}{\mathrm{\partial }X}\right)\\ & +\frac{\mathrm{\partial }}{\mathrm{\partial }Y}\left(\overline{{\mu }_a}\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+\frac{\mathrm{\partial }V}{\mathrm{\partial }X}\right)\right)]+{\xi }_X,\end{array}$(9) (10) $\begin{array}{rl}U\frac{\mathrm{\partial }V}{\mathrm{\partial }X}+V\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}& =-\frac{\mathrm{\partial }P}{\mathrm{\partial }Y}+P_r[2\frac{\mathrm{\partial }}{\mathrm{\partial }Y}\left(\overline{{\mu }_a}\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)\\ & +\frac{\mathrm{\partial }}{\mathrm{\partial }X}\left(\overline{{\mu }_a}\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}+\frac{\mathrm{\partial }V}{\mathrm{\partial }X}\right)\right)]+{\xi }_X,\end{array}$(10) $\begin{array}{rl}\left({\xi }_X,{\xi }_Y\right)& =\left(H{a}^2{P}_r\right(\sin \mathrm{\gamma cos}\mathrm{\gamma V}-{\sin }^2\mathrm{\gamma U}),\\ & H{a}^2{P}_r\left(\sin \mathrm{\gamma cos}\mathrm{\gamma U}-{\cos }^2\mathrm{\gamma V}\right)\\ & +P_r\mathrm{Ra}\left(\mathrm{NC}+\theta \right))\end{array}$ (11) $\begin{array}{rl}U\frac{\mathrm{\partial }\theta }{\mathrm{\partial }X}+V\frac{\mathrm{\partial }\theta }{\mathrm{\partial }Y}& =\left(\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^2\theta }{\mathrm{\partial }{Y}^2}\right)\end{array}$(11) (12) $\begin{array}{rl}\mathrm{Le}\left(U\frac{\mathrm{\partial }C}{\mathrm{\partial }X}+V\frac{\mathrm{\partial }C}{\mathrm{\partial }Y}\right)& =\left(\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{X}^2}+\frac{{\mathrm{\partial }}^{2}C}{\mathrm{\partial }{Y}^2}\right)\end{array}$(12) $\begin{array}{rl}\overline{{\mu }_a}& =\{{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }X}+\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}\right)}^2\\ & {+2\left[{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}\right)}^2+{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)}^2\right]\}}^{\frac{n-1}{2}}\end{array}$

The mean Nusselt and Sherwood numbers is in the form. (13) $\begin{array}{rl}N{u}_{\mathrm{avg}}& =\underset{0}{\overset{1}{\int }}\mathrm{Nu}\text{d}Y\end{array}$(13) (14) $\begin{array}{rl}S{h}_{\mathrm{avg}}& s=\underset{0}{\overset{1}{\int }}\mathrm{Sh}\text{d}Y\end{array}$(14)

2.1. Entropy generation

In the studied problem, irreversibility arises due to heat transfer, fluid friction, and a magnetic field. Hence, the total entropy can be expressed as the sum of irreversibility arising from thermal gradients, viscous dissipation, and Magnetic effects, as shown below [27,31,32], (15) $\begin{array}{r}{S}_s=S_F+S_T+S_G\end{array}$(15) Where the irreversibility is due to fluid friction $S_F$, heat transfer $S_T$, and magnetic field $S_G$ is calculated as follow, (16) $\begin{array}{rl}{S}_F& =\frac{{\mu }_a}{T_0}\left[2{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+2{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right)}^2+{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\right)}^2\right]\end{array}$(16) (17) $\begin{array}{rl}{S}_T& =\frac{K}{{T_0}^2}\left[{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}^2\right]\end{array}$(17) (18) $\begin{array}{rl}{S}_G& =\frac{\sigma B^2}{T_0}(\mathrm{usin}\gamma -\mathrm{vcos}\gamma {)}^2\end{array}$(18)

The nondimensional form of entropy generation can be expressed as: (19) $\begin{array}{rl}{S}_s& =S_F+S_T+S_G\end{array}$(19) (20) $\begin{array}{rl}{S}_F& ={\mathrm{\Phi }}_1\left[2{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+2{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right)}^2+{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x}\right)}^2\right]\end{array}$(20) (21) $\begin{array}{rl}{S}_T& =\left[{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }T}{\mathrm{\partial }y}\right)}^2\right]\end{array}$(21) (22) $\begin{array}{rl}{S}_G& ={\mathrm{\Phi }}_{11}\text{H}{\text{a}}^2(\mathrm{usin}\gamma -\mathrm{vcos}\gamma {)}^2\end{array}$(22) $\begin{array}{rl}{\mathrm{\Phi }}_1& =\frac{{\mu }_a{T}_0}{k}{\left(\frac{\alpha }{L\mathrm{\Delta T}}\right)}^{2}R{a}_T\\ & =\frac{{\left\{2\left[{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}\right)}^2+{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)}^2\right]+{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }X}+\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}\right)}^2\right\}}^{\frac{n-1}{2}}}{k}\\ & \times T_0{\left(\frac{\alpha }{L\mathrm{\Delta T}}\right)}^{2}R{a}_T\end{array}$ $\begin{array}{rl}\lambda & =T_0{\left(\frac{\alpha }{L\mathrm{\Delta T}}\right)}^2\end{array}$ $\begin{array}{rl}R{a}_T& =\frac{\rho {\text{β}}_T{g}_y{L}^3\mathrm{\Delta T}}{{\mu }_a\alpha }\end{array}$ $\begin{array}{rl}\mathrm{Ha}& =\mathrm{LB}\sqrt{\frac{\sigma }{{\mu }_a}}\end{array}$ $\begin{array}{rl}{\mathrm{\Phi }}_{11}& =\mathrm{Ra}\{2\left[{\left(\frac{\mathrm{\partial }U}{\mathrm{\partial }X}\right)}^2+{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }Y}\right)}^2\right]\\ & {+{\left(\frac{\mathrm{\partial }V}{\mathrm{\partial }X}+\frac{\mathrm{\partial }U}{\mathrm{\partial }Y}\right)}^2\}}^{\frac{n-1}{2}}\lambda \end{array}$ (23) $\begin{array}{rl}{S}_F& =\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{S}_F\text{d}x\text{d}y,{\text{S}}_T=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{S}_T\text{d}x\text{d}y,\\ {\text{S}}_G& =\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{S}_G\text{d}x\text{d}y,{\text{S}}_S=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}{S}_S\text{d}x\text{d}y\end{array}$(23)

3. Solution methodology

Exact approaches are helpful when there are no obstructions in fluid flow, but they might be challenging to apply in enclosed spaces. Most of the researchers used numerical approaches, such as FEM, FDM, and FVM. The discrete element GFEM simulates complex and irregular geometries on a flat domain and is one of the most flexible numerical approaches. The fluid, heat, and mass transfer flow inside such enclosures has been the focus of a substantial amount of study that utilizes computational approaches. The discretization of the preceding leading Equations (8)–(12), along with boundary conditions (Table 1), is accomplished using the Galerkin finite-element method (GFEM). To increase the solution's accuracy, a composite mesh made of both rectangular and triangular elements is used, as shown in Figure 2. The basic steps of GFEM are depicted in the flowchart in Figure 3.

Figure 2. Mesh analysis.

Figure 3. Schematic diagram of finite element method.

Table 1. Summary of boundary conditions.

Boundary	Velocity condition	Thermal condition
Left side	$U=0,V=0$	$\theta =1,C=1$
Right side	$U=0,V=0$	$\theta =0,C=0$
Remaining (adiabatic) walls	$U=0,V=0$	${\theta }_n=0,C_n=0$

Table 2 represents the results of the grid sensitivity test conducted for $N{u}_{\mathrm{avg}}$, considering the various parameters $\mathrm{Ha}=25,Pr=6.2,\mathrm{Ra}={10}^4,n=0.7$. The results confirm minimal differences between grid levels l and grid levels 8 and 9. As a result, the simulations were conducted using a grid level of L7, consisting of 8118 elements and 75157 degrees of freedom.

Table 2. Grid performance across different levels.

Level	Number of elements	Degree of freedom	$N{u}_{\mathrm{avg}}$	Error % ($N{u}_{\mathrm{avg}}$)
L1	328	3536	0.43408	0.40
L2	522	5457	0.43234	−0.02
L3	720	7344	0.43242	0.07
L4	1336	13226	0.43213	0.02
L5	1932	18802	0.43203	0.00
L6	3158	29733	0.43204	0.03
L7	8118	75157	0.43192	0.00
L8	20652	187340	0.43190

3.1. Code validation

The validation and verification of the ongoing computations by comparing them with the research of Roy and Basak [33] enhances their credibility. This comparison ensures the accuracy reliability, and contributes to the overall trustworthiness of the current study, thereby strengthening its scientific integrity. To ensure agreement with the present simulations, limitations on physical parameters are imposed, specifically setting n = 1. Velocity streamlines and isothermal contours are graphically shown depending on Pr and Ra. The results are compared with studies by Bilal et al. [26] and Khezzar et al. [9], ensuring consistency and corroborating the results in Table 3. It can be seen that the error did not exceed 0.39% between present study and Khezzar et al. [9].

Table 3. Comparison of code results of average Nusselt number $N{u}_{\mathrm{avg}}$ against (n).

n	$N{u}_{\mathrm{avg}}$ in [22]	$N{u}_{\mathrm{avg}}$ in [8]	Present	Error %
0.6	6.9872	6.9345	6.9136	0.3
1.0	4.6990	4.6993	4.6810	0.39
1.4	3.7870	3.7869	3.7812	0.15

4. Results and discussion

In this section, a comprehensive analysis of numerical study on laminar mixed convection in a two-dimensional bulging enclosure having power-law fluid is provided. The main focus of this research is to employ numerical simulations to investigate the impact of various parameters on the flow and heat profile along with entropy generation within the cavity. The discussion shows the impact of different parameters, such as the power law fluid, Rayleigh number, Lewis number, and inclined magnetic effect on the flow and thermal gradients. Furthermore, the study explores the correlation between these parameters and the produced entropy, offering insights into the non-reversible processes within the system. Moreover, the study examines the behaviour of $N{u}_{\mathrm{avg}}$ and $S{h}_{\mathrm{avg}}$, providing crucial insights into the system's convective energy and mass transfer.

Figure 4 shows the impact of the power law exponent n for $\mathrm{Ra}=1e^4$ (Rayleigh number) velocity distribution, isotherms, and isoconcentration are depicted through streamlines, isotherm patterns, and iso-concentration lines. The first part focuses on the velocity profile, isotherms, and iso-concentration contours for n = 0.7, corresponding to the shear-thinning properties of the power-law fluid model. Compact circular patterns are observed in the flow field, indicating the dominance of inertial forces over viscosity effects. This behaviour results in faster fluid particle movement and circular motion around the fin, suggesting reduced resistance from viscosity and a more fluid-like behaviour. In the second part, representing a Newtonian fluid, the isotherms and isoconcentration contours may show different patterns compared to n = 0.7. The flow behaviour is more influenced by thermal buoyancy forces, resulting in different shapes and distributions of the isotherms and isoconcentration contours. In the third part, corresponding to n = 1.3 shear thickening, elliptical deformation regions are observed on all fin sides, suggesting that viscous forces prevail over thermal buoyancy forces. The presence of these zones suggests that viscosity significantly influences the flow, leading to the formation of elongated regions.

Figure 4. Comparison of streamlines, isotherms, isoconcentration for power law index (0.7, 1, 1.3) at $\mathrm{Ra}=1e^4$.

Figure 5 illustrates the variation of $N{u}_{\mathrm{avg}}$ and $S{h}_{\mathrm{avg}}$ for pseudoplastic, Newtonian, and dilatant behaviours at different Rayleigh numbers $\mathrm{Ra}=1e^3\text{to}\mathrm{Ra}=1e^5$. The graph reveals that at low Rayleigh numbers, the behaviour of the power law exponent (n = 0.7, 1, 1.3) results in similar flow characteristics. However, for the maximum value of Rayleigh numbers, the $N{u}_{\mathrm{avg}}$ All behaviours converge to nearly the same value. Notably, as the Ra increases, the dilatant behaviour exhibits the maximum $N{u}_{\mathrm{avg}}$. Conversely, n increases, the $N{u}_{\mathrm{avg}}$ decreases.

Figure 5. Variation of averaged Nusselt number.

Table 4 represents the results of the analysis for different parameters, including Ha, Lewis number Le, Rayleigh number Ra, and inclination angle $\gamma$. Each combination of Ha, Ra, Le, and $\gamma$ is associated with a different value of the averaged heat transfer coefficient, which represents the thermal transfer rate. The findings demonstrate that the highest values of the heat transfer coefficient are generally obtained at lower Ha values, and these maximum values occur for specific combinations of Ra, Le, and $\gamma$. It should be noted that the particular values of the $N{u}_{\mathrm{avg}}$ vary depending on the power law fluid. The obtained values of Table reveal that dilatant and pesuplastic fluids tend to exhibit higher average Nusselt numbers than Newtonian fluid. For dilatant fluid, the highest average Nusselt number is observed to be 0.85161, which occurs at Ha = 10, Ra = $1e^4$, and Le = 0.1 with $\gamma =0^{\circ }$. The lowest value in this section is 0.43549, which is observed at Ha = 10, Ra = $1e^4$, Le = 0.1, and $\gamma ={30}^{\circ }$. For Newtonian the highest average Nusselt number is 0.50833, obtained at Ha = 10, Ra = $1e^3$, Le = 0.1, and $\gamma =0^{\circ }$. The lowest value in this section is 0.42991, which is observed at Ha = 40, Ra = $1e^3$, Le = 5, and $\gamma ={30}^{\circ }$. For Shear Thining, the highest average Nusselt number is 0.44208, which occurs at Ha = 10, Ra = $1e^3$, Le = 0.1, and $\gamma =0^{\circ }$. The lowest value in this section is 0.42981, observed at Ha = 40, Ra = $1e^3$, Le = 5, and $\gamma ={30}^{\circ }$.

Table 4. Variation of power-law exponent (n) with Rayleigh number and Hartmann number.

Ha		$\mathrm{Ra}=1e^3$	$\mathrm{Ra}=1e^4$	$\mathrm{Le}=0.1$	$\mathrm{Le}=1$	$\mathrm{Le}=5$	$\gamma =0^{\circ }$	$\gamma ={15}^{\circ }$	$\gamma ={30}^{\circ }$
10	n = 0.7	0.43549	0.85161	0.85161	0.80479	0.67029	0.85161	0.84638	0.81598
20		0.43192	0.70354	0.70354	0.66998	0.57378	0.70354	0.67340	0.60822
30		0.43062	0.56733	0.56733	0.55576	0.5083	0.56733	0.54152	0.49955
40		0.43012	0.49648	0.49648	0.49330	0.47320	0.49648	0.48143	0.45948
10	n = 1	0.43071	0.50833	0.50833	0.50158	0.47509	0.50833	0.50692	0.50322
20		0.43033	0.48665	0.48665	0.48319	0.46513	0.48665	0.48307	0.47484
30		0.43007	0.46713	0.46713	0.46569	0.45527	0.46713	0.46285	0.45437
40		0.42991	0.45351	0.45351	0.45295	0.44749	0.45351	0.44975	0.44316
10	n = 1.3	0.43002	0.44208	0.44208	0.44187	0.43949	0.44208	0.44201	0.44180
20		0.42994	0.44047	0.44047	0.44031	0.43844	0.44047	0.44021	0.43956
30		0.42987	0.43850	0.43850	0.43841	0.43709	0.43850	0.43806	0.43701
40		0.42981	0.43665	0.43665	0.43659	0.43574	0.43665	0.43608	0.43485

When the Rayleigh number Ra is increased to Ra = $1e^5$, it has notable effects on the velocity streamlines for shear thinning shown in Figure 6. Unlike at low Rayleigh numbers, the streamlines become less smooth. Secondly, the isoconcentration and isotherm patterns exhibit different characteristics compared to the lower Rayleigh number case. For Newtonian fluid, the flow behaviour is influenced by thermal buoyancy forces. As a result, the shapes and distributions of the isotherms and isoconcentration contours deviate from those observed at lower Rayleigh numbers. For shear thickening behaviour, the presence of a higher Rayleigh number leads to the formation of elliptic deformation regions surrounding the fin on all sides. These zones highlight the dominance of the viscous effect compared to heat convection forces.

Figure 6. Comparison of streamlines, isotherms, isoconcentration for power law index (0.7, 1, 1.3) at $\mathrm{Ra}=1e^5$.

The Lewis number plays a pivotal role in shaping the patterns of streamlines, isotherms, and isoconcentration in natural convection scenarios. Figure 7 demonstrates the changes in these variables resulting from different Lewis numbers. Despite the fluctuations in the Lewis number, the momentum distribution remains the same as it does not affect momentum diffusivity significantly. However, the temperature disparity between the side walls induces a clockwise rotation of the fluid in the system.

Figure 7. Comparison of streamlines, isotherms, isoconcentration for power law index (0.7, 1, 1.3) at Le = 1.

Figure 8 presents the relationship between the mean convective heat transfer coefficient. $N{u}_{\mathrm{avg}}$ and mean mass transfer coefficient $S{h}_{\mathrm{avg}}$ for different power law exponents (shear thickening, Newtonian, and shear thinning) with respect to Le, while considering constant parameters Ha = 20, $\mathrm{Ra}=1e^3$, $\gamma =0^o$. The results demonstrate that the highest values of $N{u}_{\mathrm{avg}}$ and $S{h}_{\mathrm{avg}}$ are observed for Le = 0.1, 1, and 5. As the Lewis number increases, the $N{u}_{\mathrm{avg}}$ decreases for both shear thinning and shear thickening fluids, with an increasing Hartmann number Ha. The decrease in $N{u}_{\mathrm{avg}}$ suggest a decrease in heat transfer rate within the fluid flow system, exhibiting a negative correlation with the Le. The decrease in $N{u}_{\mathrm{avg}}$ can be attributed to the higher Lewis numbers, signifying a slower mass diffusion rate relative to thermal diffusion. On the other side, $S{h}_{\mathrm{avg}}$ representing mean mass transfer coefficient in the system, shows a direct relationship with Le. It can be seen that when Le increases for n = 0.7 and n = 1.3, the $S{h}_{\mathrm{avg}}$ also increases, indicating enhanced mass transfer.

Figure 8. Averaged Nusselt and Sherwood numbers versus Hartmann numbers for different Lewis numbers $\mathrm{Le}=0.1,1,5$. With power law indices $n=0.7\text{and}n=1.3$.

For low values of the Le, the streamlines exhibit a more aligned pattern compared to higher values of Le in Figure 9. It can be seen that increasing the Lewis number leads to a higher concentration of fluid near the heated wall due to slower mass diffusion relative to thermal diffusion. This is manifested by the thicker concentration boundary layer at higher Lewis numbers. The isothermal contours provide insights into the temperature distribution, while the isoconcentration contours demonstrate the flow of fluid from the wall which have minimum temperature towards the wall which have maximum temperature. It is observed that the isotherm patterns and streamline contours show minimal variation with changes in the Le, indicating that the other factor have a more significant impact on thermal distribution and velocity field within the fluid flow system.

Figure 9. Comparison of streamlines, isotherms, isoconcentration for power law index (0.7, 1, 1.3) at Le = 5.

A comprehensive evaluation of different parameters like, Hartmann number Ha, inclination angle $\gamma$, Rayleigh number Ra, Lewis number Le is presented in Table 5. The Table shows results for three behaviours: shear thickening n = 0.7, Newtonian n = 1, and shear thinning n = 1.3. The highest value of the mean Nusselt number is observed at a minimum value of Ha for different conditions of Ra, Le, and $\gamma$. After reviewing Table 5, it can be observed that for dilatant fluid, the highest value of $N{u}_{\mathrm{avg}}$ is 2.03942. The specific values of the mean convective heat transfer number vary depending on the power law exponent. Analysing the table, it can be observed that the highest value of the $N{u}_{\mathrm{avg}}$ for shear thickening n = 0.7 is 2.03942, observed at Ha = 10, $\mathrm{Ra}=1e^3$, Le = 5, and $\gamma =0^o$. The lowest value for shear thickening is 0.42966, occurring at Ha = 40, $\mathrm{Ra}=1e^4$, Le = 0.1, and $\gamma ={30}^o$. For the Newtonian fluid n = 1, the highest $N{u}_{\mathrm{avg}}$ is 0.43057, found at Ha = 10, $\mathrm{Ra}=1e^3$, Le = 1, and $\gamma =0^o$. The lowest value is 0.42966, occurring at Ha = 40, $\mathrm{Ra}=1e^4$, Le = 0.1, and $\gamma ={30}^o$. When shear thinning n = 1.3, the maximum number of average Nusselt number is 0.42978, at Ha = 10, $\mathrm{Ra}=1e^3$, Le = 0.1, and $\gamma ={15}^o$. The lowest value is 0.42966, occurring at Ha = 40, $\mathrm{Ra}=1e^4$, Le = 5, and $\gamma ={30}^o$.

Table 5. The relationship between Nusselt Number and Hartmann number with power-law exponent (n).

Ha		$\mathrm{Ra}=1e^3$	$\mathrm{Ra}=1e^4$	$\mathrm{Le}=0.1$	$\mathrm{Le}=1$	$\mathrm{Le}=5$	$\gamma =0^{\circ }$	$\gamma ={15}^{\circ }$	$\gamma ={30}^{\circ }$
10	n = 0.7	0.42972	0.43838	0.43838	0.80484	2.03942	0.43838	0.43787	0.43664
20		0.42968	0.43344	0.43344	0.67002	1.72255	0.43344	0.43286	0.43183
30		0.42967	0.43125	0.43125	0.55578	1.35528	0.43125	0.43091	0.43041
40		0.42966	0.43037	0.43037	0.49331	1.06557	0.43037	0.43020	0.42996
10	n = 1	0.42967	0.43057	0.43057	0.50159	0.98286	0.43057	0.43055	0.43051
20		0.42967	0.43028	0.43028	0.48320	0.92120	0.43028	0.43024	0.43015
30		0.42967	0.43006	0.43006	0.46569	0.83518	0.43006	0.43001	0.42992
40		0.42966	0.42991	0.42991	0.45296	0.74635	0.42991	0.42987	0.42980
10	n = 1.3	0.42966	0.42978	0.42978	0.44187	0.60862	0.4297	0.42978	0.42978
20		0.42966	0.42977	0.42977	0.44031	0.59643	0.42977	0.42976	0.42976
30		0.42966	0.42975	0.42975	0.43840	0.57809	0.42975	0.42974	0.42973
40		0.42966	0.42973	0.42973	0.43659	0.55694	0.42973	0.42972	0.42971

Figure 10 illustrates the correlation between the inclination angle $\gamma$ and the $N{u}_{\mathrm{avg}}$ as well as the $S{h}_{\mathrm{avg}}$, considering non-Newtonian fluids, including a set of parameters $\mathrm{Ra}=1e^3$, Pr = 6.8, and Le = 0.1. The findings indicate that the greatest value of $N{u}_{\mathrm{avg}}$ and $S{h}_{\mathrm{avg}}$ are attained when the inclination angle is at its maximum. For the maximum value of the inclination angle, both the $N{u}_{\mathrm{avg}}$ and $S{h}_{\mathrm{avg}}$ decrease for both shear thinning and shear thickening fluids. The highest value of the mean Nusselt number for shear thickening is 0.88, which is approximately the same as that for shear thinning. However, shear thickening exhibits a higher value for the mass transfer rate than shear thinning.

Figure 10. Averaged Nusselt and Sherwood numbers versus Hartmann numbers for different inclination angle $\gamma =0^{\circ },{15}^{\circ },{30}^{\circ }$. With power law indices $n=0.7\text{and}n=1.3$.

Figure 11 illustrates the entropy generation analysis considering the power law exponent for shear thinning and shear thickening fluids, as well as Ha. The graph represents the components of irreversibility analysis, namely heat transfer, fluid friction, magnetic irreversibility, and total entropy. It can be observed that Ha increases, there is a decrease in entropy generation. This relationship indicates that a magnetic field suppresses the overall entropy production in the system. when the Lewis number is increased, there is a reduction in the range of entropy generation. This implies that higher Lewis numbers result in a more limited span of entropy production in the system. The analysis of irreversibility provides valuable insights into the interplay between the Hartmann number, power law index, and Lewis number, illuminating the impact of these parameters on the overall entropy behaviour in the system.

Figure 11. Irreversibility analysis of (a) heat transfer (b) Magnetic (c) fluid friction (d) Total entropy for varying Lewis numbers $\mathrm{Le}=0.1,1,5$ versus Hartmann numbers With power law indices $n=0.7\text{and}n=1.3$.

Figure 12 illustrates the impact of inclined angle γ affects entropy, including heat transfer, fluid friction, magnetic, and total entropy under different Ha. The thermal entropy exhibits higher values at lower Ha and minimum value at maximum Ha, as it is for fluid friction and total entropy. For non-Newtonian fluids, the fluid friction, thermal entropy, and total entropy decreased, while magnetic entropy is increased when the inclined angle is increased.

Figure 12. Comparison of entropy generation due to heat transfer $S_T$, magnetic irreversibility $S_G$, Fluid friction $S_F$, and total entropy $S_S$ for different inclination angle $\gamma =0^{\circ },{15}^{\circ },{30}^{\circ }$ verses Hartmann number with power law indices $n=0.7\text{and}n=1.3$.

5. Conclusion

This research comprehensively analyses flow and heat transfer characteristics of the two-dimensional bulging enclosure, focusing on non-Newtonian power-law fluids. Numerical simulation is employed to investigate the impacts of various parameters on the thermal distribution, flow pattern, and entropy generation. The results show the noteworthy influence of the power law exponent on flow behaviour, for shear thinning fluids demonstrate accelerated particle movement, whereas shear thickening fluids give rise to elliptic-shaped deformation zones. The Nusselt numbers converge at higher Rayleigh numbers, and the power law exponent affects the heat transfer rate, with shear thinning behaviour resulting in the maximum Nusselt number.

• Shear-thinning behaviours are associated with the highest Nusselt numbers, indicating more effective heat transfer.

• Both shear-thickening and shear-thinning fluids exhibit higher Nusselt and Sherwood numbers than Newtonian fluids, suggesting enhanced convective heat and mass transfer.

• The Lewis number and inclination angle provide insights into momentum distribution, concentration boundary layers, and the interrelation of heat and mass transfer rates.

• The presence of a magnetic field reduces overall entropy production, pointing to more efficient processes.

• The study enriches the understanding of irreversible processes in non-Newtonian power law fluids within complex flow configurations.

• The results have practical implications for developing efficient systems and processes across various fields, enhancing our ability to predict and control fluid behaviour in critical applications.

Disclosure statement

No potential conflict of interest was reported by the author(s).