Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology

Nomenclature

a

constant

Al 2 O 3

aluminum oxide

A sr

irradiance factor

B 0

magnetic field

B e

Hall number

B i

ion slip number

c

concentration ( mol m − 3 )

c Fx , c Fg

skin friction coefficients

c P

specific heat capacitance ( J k g − 1 K − 1 )

( c ) ∞

ambient concentration ( mol m − 3 )

( c ) W

wall concentration ( mol m − 3 )

D

mass diffusion ( m 2 s − 1 )

E

activation energy number

E 0

activation energy at zero

F , G

dimensionless velocity fields

F w

dimensionless suction velocity

Fe 3 O 4

iron dioxide

H t

heat source number

k

thermal conductivity ( W m − 1 k − 1 )

k e

dimensional chemical reaction parameter

O 1 , O 3 , O 4 , O 2

volume fractions of nanofluid

Cu

copper

Ec

Eckert number

EO

engine oil

FEM

finite element method

GFEA

Galerkin finite element approach

Ha

Hartmann number

hB

hybrid nanofluid

n

power law index number

Nu

Nusselt number

ODEs

ordinary differential equations

Pr

Prandtl number

Q sr

irradiance factor

Re

Reynolds number

Sc

Schmidt number

Sh

Sherwood number

Ti O 2

titanium dioxide

T 0

reference temperature ( K )

tN

tetra-hybrid nanofluid

Tri

tri-hybrid nanofluid

T

temperature ( K )

( T ) ∞

ambient temperature ( K )

( T ) W

wall temperature ( K )

ν

kinematic viscosity ( m 2 s − 1 )

We

Weissenberg number

Y , X , Z

space coordinates ( m s − 1 )

σ

electrical thermal conductivity ( kg m − 3 )

Γ

time relaxation number ( s )

∞

infinity

η

independent variable

Θ

dimensionless temperature

Φ

dimensionless concentration

U , V , W

velocity fields ( m s − 1 )

ρ

density ( kg m − 3 )

η max

Independent variable at maximum domain

1 Introduction

Because of their applications, non-Newtonian fluids have been the subject of in-depth research by numerous scholars. Jamshed et al. [1] numerically computed the magnetized Powell Eyring nano liquid with the utilization of a single-phase model. Finite element analysis based on cross fluid induced by vertical disk exposed to tetra hybrid nano-size particle mixture dispersion was discussed by Sohail et al. [2]. Waseem et al. [3] analyzed the entropy approach concerning magnetized hybrid nanosize particles by invoking the optimal homotopy analysis method technique. Moreover, Waseem et al. [4] also analyzed the entropy production appertaining to 3D expanded magnetically driven hybrid nanofluid influenced by thermal radiation along with heat generation impacts. Statistical analysis and morphological features concerning rheological materials dispersed in the transmission of silica, titanium, alumina, and also ethylene glycol with the utilization of Galerkin methodology were captured by Li et al. [5]. Entropy theory and Powell–Eyring hybrid nanofluidic with heat dissipation were reported by Aziz et al. [6]. A comparative study proffered by Jamshed et al. [7] based on inclined magnetically driven Powell-Eyring Nano liquid induced by nonuniformly rotating expanded flat surface. A theoretical investigation was conducted by Avramenko et al. [8] based on fluid occurrence and also heat transfer across the boundary layer subjected to surface. Efficiency appertains to thermal energy as well as solute particles with the utilization of hybrid nanosize particles influenced by chemical reaction along with activation energy exposed to the parabolic surface, which was studied by Chu et al. [9]. In this study, the finite element approach was opted. Akram et al. [10] explored the hybridized consequences concerning thermal as well as concentration convection based on peristaltic transmission magnetized Powell–Eyring nano liquids configured by slanted an asymmetric channel. Aziz et al. [11] modeled and also simulated the 3D magnetized spinning water-reliant multi wall carbon nanotubes nano fluidic flow demeanor by invoking convective prior conditions as well as slip impacts. Patil et al. [12] have examined the unsteady magnetized in terms of thermal radiation and chemical response nano Powell-Eyring liquid close to the stagnation point bounded by a convectively heated extending sheet. A 3D Casson fluid model under consideration of temperature-reliant conductance via shooting technique was discussed by Sohail et al. [13]. Moreover, Joule heating along with viscous dissipation impacts also contributed to this study. Radial magnetized non-Newtonian fluidic occurrence by utilizing an advanced heat flux model influenced by Joule heating was reported by Sohail et al. [14]. Waqas et al. [15] numerically analyzed the convected occurrence based on Oldroyd-B nano liquid in the presence of motile organisms together with thermal radiation configured by a spinning disk. Song et al. [16] discussed the advanced Darcy law and also nonlinear thermal radiation applications rely on bio-convected occurrence referring to micropolar nano liquid bounded by rotating disk. Thermal conductivity performance is relevant to the convected flow demeanor of cross nano fluid comprising swimming microorganisms induced by a cylinder exposed to melting phenomenon, which was numerically evaluated by Imran et al. [17]. Binary-band nonreciprocal thermal radiation under coupling of optical Tamm conditions within the magnetophotonic multilayers was reported by Wu et al. [18]. Owing to the attractive broad applications of borophene, Super-Planckian thermal radiation was examined by Zhou et al. [19] across the borophene sheets. The Williamson fluid model was studied by Khan et al. [20] in terms of blood-carrying gold nano-size particles subjected to curved surfaces under significant effects of thermal radiation, suction, and also Lorentz force. Computational analysis was performed by Imran et al. [21] appertains to nanosized particles shape toward the hybrid nano fluidic occurrence owing to the plate with solar radiation. Moreover, Imran et al. [22] also studied the physical attributes based on bio-convection toward the nano liquid flow demeanor across a paraboloid of revolution bounded by horizontal surface exposed to motile organisms. Entropy production was scrutinized by Farooq et al. [23] across the magnetized nano liquid occurrence under consideration of nonlinear thermal radiation. Buoyancy convected flow demeanor and also heat dissipation of nano size liquids across the annulus possessing a thin baffle were studied by Pushpa et al. [24]. The entropy approach together with the thermal transmission of Cu-water nano-fluid within the penetrating lid-driven cavity in terms of magnetic field was discussed by Marzougui et al. [25]. A statistical approach was conducted by Shafiq et al. [26] in terms of dual stratification across the stagnation point Walter’s B Nano fluidic occurrence through a radial Riga plate. Numerical as well as series solutions were obtained by Mushtaq et al. [27] relying on Powell–Eyring liquid via exponentially extending sheet. Figure 1a displays several varieties of solar thermal collectors, such as parabolic troughs, linear Fresnel reflectors, compound parabolic concentrators, flat plate collectors, collectors with evacuated tubes, and concentrating solar dishes. Abrar et al. [28] looked into the micro-rotation effects of a tangent hyperbolic hybrid nanofluid flowing across a porous sheet. By utilizing the shooting approach in conjunction with (RKF-45), one can achieve the numerical simulations to leading differential equations. The physical meaning of the essential characteristics is obtained via graphic results. Samad Khan et al. [29] investigated the entropy produced by the micro-rotation of Casson’s nanofluid and the impact of a magnetic field flowing through a nonlinear stretched surface. When designing a mechanical system that depends on heat transport, entropy minimization is crucial. An exponentially expanding convective wall is used to study the rheology of silver and alumina nanoparticles by Abrar et al. [30,31]. Internal heat generation and absorption impact when a magnetic field is applied. The impact of an angled magnetic field (MG) on the formation of entropy in a Jeffery nanofluid flow across a stretching surface with viscous dissipation was examined by Akhtar et al. [32]. In a flexible circular duct that is axisymmetric within the walls of the cilia, Abrar et al. [33] explored the peristaltic transfer of nanofluid. In all cases when solid nanoparticles are taken into consideration, the temperature profile drastically reduces as the radiation parameter rises. Li et al. [34] estimated the aspects of the Eyring liquid with the suspension of partial ionization on rotating disks. They have assumed thermophoretic deposition (particles) resolved numerically with double diffusion. Das et al. [35] studied thermos-diffusion and thermal energy on the surface visualizing thermal transport and thermal radiation. Hall currents, magnetic force, and Hall currents were assumed. Ali et al. [36] analyzed the character of ternary hybrid nano-martial in Casson nanofluid considering Coriolis force, Hall currents, buoyancy force, and rotating plate were assumed. Different technologies of solar radiation consider temperature ranges as shown in Figure 1a.

Figure 1

(a) Different technologies of solar radiation consider temperature ranges. (b) Physical model and coordinates system.

In published works, investigation regarding three-dimensional flows in tetra-hybrid nanofluid is not documented well. This artice is organized as follows: The model’s formulation is covered in Section 2. Section 4 presents the analysis of the results, while Section 3 covers the methodology of the numerical approach. Section 5 discusses important points.

2 Mathematical procedure and statement

Thermal and concentration performances of complex fluid on 3D surfaces are carried out. Advanced correlations regarding tetra-hybrid nanofluid are developed using a mixture of copper, Al 2 O 3 , iron dioxide, and Ti O 2 in engine oil (EO). The partially ionizations are inserted into mixtures of nanoparticles. The developed mixture over 3D surface is induced with wall velocity [ a ( X + Y , b ( Y + X ) ] , where b and a are constants. Energy equation contains solar thermal radiation and heat sink/absorption with the suspension of four kinds of nanoparticles, while the term regarding activation energy is added to the concentration equation. The physical development of the model is considered in Figure 1b. A set associated with partial differential equations [37,38] is obtained:

Boundary conditions [38] in terms of momentum associated with all directions are defined as follows:

Boundary conditions in terms of concentration, temperature [38]:

Desired similarity variables [38] are defined as follows:

Dimensionless ordinary differential equation (ODEs) are obtained using Eq. (6), which is defined as follows:

Boundary conditions for Eqs. (7)–(10) are mentioned [37,38] as follows:

Correlations associated with tetra-hybrid nanostructures [2] are defined as follows and their values are mentioned in Table 1:

Skin friction factors in terms of y- and x-directions [38] are represented as follows:

Heat transfer and mass diffusion rates in normal form [38] are predicted as follows:

3 Numerical methodology

Galerkin finite element approach (GFEA) [39,40] is visualized as a powerful tool for obtaining numerical solutions of complex nonlinear ODEs utilized in physics and engineering fields. GFEA provides an effective way and accurate way for approximate solution of complex ODEs. The basic steps of GFEA are listed below. The steps of finite element method (FEM) are addressed in Figure 2a. FEM’s advantages are mentioned as FEM deals with complex problems and with various kinds of boundary conditions. FEM needs less investment in case of resources and time. For a variety of physical models arising in applied sciences, this approach is the most appropriate one. FEM is very good at dealing with derivatives’ discretization. Using numerical simulations, FEM enables the validation of theoretical models and hypotheses.

Figure 2

(a) Basic steps related to GFEA. (b) Flowchart of the numerical approach. (c) An error analysis versus number of iterations.

Step 1: Discretization’s

Dividing the problem domain into smaller components, or elements is the first step. Using up to 300 elements is found to be adequate for simulating the current analysis’s solution. The system of ODEs is represented by the “strong form,” and the “weak form” is produced using a process known as the residual method.

Step 2: Shape functions

In the current analysis, linear shape functions are utilized as essential for approximating solutions. These functions help for obtaining solutions. Although the finite element method uses a variety of shape function types, linear shape functions are used in this analysis. These linear shape functions accurately capture the fundamental behavior of the system while streamlining the approximation process.

Step 3: Weak formulation

Boundary conditions and strong form are associated with Eqs. (12)–(14). Weak forms are required in this process to obtain an approximate solution. All terms are gathered on one side and integrated with more than 300 elements. Grid independent survey is shown in Table 2.

Step 4: Finite element formulation

The current problem’s stiffness elements are obtained in this step. Global stiffness matrices are attained for every element in the end.

Step 5: Assembly process

A crucial component of the finite element method is the assembly process. The assemblage concept is used to formulate stiffness matrices.

Step 6: Solution of algebraic equations

Finally, a numerical solution within 10 − 5 computational tolerance is obtained for the system of linear algebraic equations. The following is a list of the stopping conditions [37]. The flow chart of the current approach is illustrated in Figure 2b.

Here, i is the number of iterations and ℵ is nodal values. An error analysis of d G d η , d F d η , Θ ( η ) , and φ ( η ) is plotted in Figure 2c versus several iterations. It noticed that the error of d G d η , d F d η , Θ ( η ) , and φ ( η ) is decreased when several iterations are enhanced. By increasing values of iterations, error plots of d G d η , d F d η , Θ ( η ) , and φ ( η ) are decreased. Table 3 exhibits validation of the current study with the study [38]. The values of − Θ ′ ( 0 ) are recorded in Table 3 with the change of Pr .

4 Results and discussion

Concentration and thermal features in a complex fluid using charged particles and tetra-hybrid nanoparticles are considered toward a 3D surface. Solar thermal radiation and activation are addressed in the energy equation as well as the concentration equation. Such complex modeling is tackled numerically by using a finite element approach. Detailed discussions are mentioned below.

Figure 3a–j reveal influences of F w , c , We , B i , and B e on y- and x-components of velocity. Solid curves on graphs are plotted using influence of the role on B i and B e , while dash-dot curves on the graphs are without impact of B i and B e . It was experienced that the motion is induced by the presence of charged particles (ion slip and Hall currents) is higher than the force induced without impacts of ion slip and Hall forces. Figure 3a and b signify the role of F w with velocity curves (both directions). It was observed that motion associated with nanoparticles is declined when F w is enhanced. The role of power law index parameter with velocity curves is addressed in Figure 3c and d using charged particles. In Figure 3c and d, motion of charged particles along with tetra-hybrid nanoparticles is slow down. The description of the non-Newtonian behavior of some fluids, including polymers, suspensions, and slurries, is the power law index, also known as the flow behavior index. The fluid’s viscosity is a function of the shear rate, or the rate at which it deforms as a result of an applied force, and is called n . Increasing the power law index causes the fluid to become more shear-thickening, which causes its viscosity to rise as the shear rate rises. Because of the higher viscosity, if the fluid is moving through a pipe or a channel, the velocity of the fluid will drop. Figure 3e and f shows influences of Weissenberg number ( We ) on both components regarding the velocity field using the concept of charged particles. A decreasing function is predicted among velocity field and Weissenberg number. A dimensionless quantity is known as the We describe the elasticity of several fluids, including melts and solutions of polymers. The proportion of elastic to viscous forces in a fluid is expressed by the Weissenberg number. The fluid becomes more elastic as the Weissenberg number rises, which can have a major impact on the fluid’s velocity field. Figure 3g and h reveals the role of B i on both components of velocity fields. Here, B i is known as the ion slip parameter. B i is a dimensionless number that describes how important electrical and viscous forces are about one another in a fluid. The ratio of the fluid velocity to the ion drift velocity is described by B i . The ions take on a greater role in directing the fluid flow as B i rises. Through collisions, the ions impart momentum to the fluid, with faster-moving ions imparting greater momentum than slower-moving ions. The fluid velocity may rise as a result of this accelerated momentum transfer. Figure 3i and j reveals impact of Hall parameter on both components of velocity fields. Velocity fields increase when the ion slip parameter is enhanced. The Hall current, which propels a flow of charged particles in a direction perpendicular to both magnetohydrodynamic and electric field, is what causes the velocity field to grow when B e is increased. The Hall effect can also result in the development of plasma vortices or other instability, which can improve the velocity field of the plasma.

Figure 3

(a) Various values of F w on D F D η . (b) Various values of F w on D G D η . (c) Various values of n on D F D η . (d) Various values of n on D G D η . (e). Various values of We on D F D η . (f) Various values of We on D G D η . (g) Various values of B i on D F D η . (h) Various values of B i on D G D η . (i) Various values of B e on D F D η . (j) Various values of B e on D G D η .

Figure 4a–c shows the roles of ( Q sr ), heat source number ( H t ), and suction parameter on temperature profile in the presence of charged particles. Figure 4a–c reveals comparative among charged particles. Figure 4a demonstrates visualization of solar thermal radiation on temperature profile. Heat energy inclines due to the enhancement of solar thermal radiation ( Q sr ). The increasing absorption of solar radiation is one physical factor contributing to the temperature field’s rise when the solar thermal radiation number rises. The thermal role of charged particles is then thermal performance for the absence of charged particles. Figure 4b predicts the behavior of thermal energy against heat source number ( H t ). The heat source number ( H t ), a dimensionless metric, expresses how important heat creation is about conductive heat transport. The ratio of the heat generation rate to the conductive heat transfer rate is indicated by the heat source number. As the number of heat sources rises, the amount of heat generation contributing to overall heat transfer increases, which can have a major impact on the temperature field. The increase in the quantity of heat produced per unit volume is one physical factor contributing to the temperature field’s rise as the number of heat sources rises. Figure 4c presents the contribution of ( F w ) on fluid temperature. Comparative analysis between multiple nanofluids is addressed in Figure 4d. Here, solid curves on the graphs are plotted for visualization of tN, dot curves are due to TRi, hybrid nanofluids are revealed by long dash curves, nanofluids are due to dash dot curves, and pure fluid is shown by black dash curves. Maximum production has been achieved for tN rather than for TRi nanoparticles, hybrid nanofluid, and nanofluid and working fluid.

Figure 4

(a) Various values of Q sr on Θ ( η ) . (b) Various values of H t on Θ ( η ) . (c) Various values of F w on Θ ( η ) . (d) Comparative analysis between several nanofluids.

Figure 5a–c plots the experience of Sc , F w , and K c on mass diffusion with activation energy and without activation energy. Figure 5a estimates the behavior of the Schmidt number ( Sc ) on the mass diffusion profile in the presence of activation energy and absence of activation energy. The relative significance of mass diffusion and momentum diffusion in a fluid is expressed by the dimensionless Schmidt number (Sc). Sc = v / D , where v is the kinematic viscosity and D is the diffusion coefficient, is the ratio of the fluid’s viscosity to the sum of its density and diffusion coefficient. When the Schmidt number rises, it indicates that the fluid’s diffusion coefficient is relatively low when compared to its viscosity. For instance, when the fluid is extremely viscous or when the diffusing species is particularly big or heavy, this may occur. When this occurs, the rate of mass diffusion is decreased because the molecules or particles that are diffusing face more movement resistance as a result of the viscous forces of fluid. Figure 5b captures the experience of the suction number on the mass diffusion profile. In Figure 5b, mass diffusion declines when the suction parameter is enhanced. The role of chemical reaction number on mass diffusion is carried out in Figure 5c. The characteristic time scale of the chemical reaction is substantially shorter than the characteristic time scale of mass diffusion when the number of chemical reactions increases. This may occur, for instance, when the reaction rate is rapid or when the species that is diffusing the energy is highly reactive. As a result of the chemical reaction’s quick consumption or production of the diffusing species, the concentration gradient in these situations is minimized, which lowers the rate of mass diffusion. Figures 6 and 7 show the estimation of the suction number ( F w ) and the Weissenberg number ( We ) on skin friction coefficients. It was observed that the skin friction coefficient increases when F w is increased. The similar behavior is estimated on skin friction coefficient versus enhancement of Weissenberg number ( We ). Figures 8 and 9 demonstrate the impacts of Q sr and H t on the Nusselt number including suspensions of nanoparticles. Nusselt number declines when Q sr and H t are increased. Moreover, maximum thermal enhancement for tetra-hybrid nanofluid rather than nanofluid, tri-hybrid nanofluid and nanofluid in view of skin friction coefficient and Nusselt number. Table 3 presents the comparative study of the present finding with the published data. Table 4 reveals observations of We , n , F w , Q sr , and H t on Nusselt number. It was included that maximum enactment for heat energy has been investigated for Al₂O₃–Fe₃O₄–Cu–TiO₂/SA rather than thermal performance for Al₂O₃–Fe₃O₄–Cu/SA.

Figure 5

(a) Various values of Sc on φ ( η ) . (b) Various values of F w on φ ( η ) . (c) Various values of K c on φ ( η ) .

Figure 6

Various values of F w on divergent velocity with different nanoparticles.

Figure 7

Various values of We on the divergent velocity with different nanoparticles.

Figure 8

Various values of Q sr on Nusselt number with different nanoparticles.

Figure 9

Various values of H t on Nusselt number with different nanoparticles.

5 Conclusions

Multiple features of activation energy and solar thermal radiations past the 3D surface including various types of nanoparticles are addressed here. Charged particles with EO are inserted in the complex fluid. The present complex model is tacked numerically by finite element methodology. Convergence of required problem is ensured in view of Table 2. Concluding remarks of the present model are mentioned as when there are charged particles present, there is more motion and thermal energy than when there are no charged particles present. Mass diffusion in the presence of activation energy is higher than mass diffusion in the absence of activation energy. Maximum cooling process and maximum thermal energy for case tetra-hybrid nanofluid rather than tri-hybrid nanoparticles and nanofluid. With the enhancement of charged particles, motion regarding nanoparticles is enhanced versus enhancement of charged particles. The greater the power law index, suction, and Weissenberg number, the acceleration of nanoparticles is enhanced. The highest thermal energy is achieved versus enhancement of solar thermal radiation and heat sink number, but minimum thermal energy can be achieved when the suction number is increased. Mass diffusion decreases against inclination Schmidt number, suction, and chemical reaction numbers. In residential, commercial, and industrial facilities, solar thermal radiation may be employed to warm water. In distant areas without electricity, refrigeration systems can be powered by solar thermal radiation and utilized to keep food and other perishable products.

6 Future directions

The current model is associated with tangent hyperbolic liquid addition of nanofluids on a 3D surface. Further, the forces of ion slip and Hall currents are observed. Utilizations of FEM for numerical procedure and graphical outcomes are observed. In the future, the current model will be studied using complex rhetorical fluids like Bingham plastic, Herschel-Bulkley, and third and fourth-grade fluids employing a finite element approach. Moreover, the current model with Bingham plastic, Herschel-Bulkley, and third- and fourth-grade fluids in the form of partial differential equations will be simulated with FEM. In the future, new correlations of hybrid nanofluid will be implemented for measuring thermal efficiency.