Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature

1 Introduction

Thin film flow describes the movement of a fluid layer whose thickness is much smaller. Thin film flows involve the motion of liquid or fluid films with small thickness compared to their lateral dimensions. It has many applications in mechanical engineering as well as manufacturing processes like coating of glass, fluidization of reactor, coating of wire and fiber, etc. The concept of thin film over the starched surface has derived by Wang [1]. Qasim et al.’s [2] investigation revealed that thin liquid films exhibit the best transfer of mass and heat compared to thicker liquids. This characteristic makes them more suitable for applications where minimal heat is required for maintaining the stability of the thin film. Researchers are currently exploring the use of hybrid nanoparticle fluid films as a coating material for glass surfaces, with the aim of developing solar panels. In the near future, car windows are expected to serve as solar panels, providing energy for both cooling and heating purposes. This innovation offers the potential to seamlessly integrate renewable energy generation and effective thermal management into everyday items. The applications of magnetic and electric fields play a vital role in fluid film dynamics, particularly in microfluidic systems, where electric fields are used to manipulate and disintegrate thin films stretched between suspended microscopic droplets. Sun et al. [3] numerically analyzed thin liquid film flow on a spinning disk, considering the effects of nonlinear radiations and a thermal source. Their study highlights how these thermal factors influence heat transfer and flow behavior on rotating surfaces. Thin film flow over an inclined plate has numerous applications, including paint finishing in stable dispersions, chemical engineering processes, and surface coating technologies. Various factors like electric as well as magnetic factors are applied to stabilize the uniform dispersion of fluid films for enhancement the glassy texture of the final product. The performance of liquid film upon an inclined surface has been discussed by Kapitza et al. [4]. Dholey [5] has extended and refined the idea of Kapitza et al. [4] by employing a similar geometrical view and has examined the instability of fluid using magnetic and electric effects.

Flow on a stretching sheet describes the fluid motion that occurs when a flat surface is continuously stretched. This phenomenon finds applications in various industrial processes, like polymer processing and coating deposition. Controlling this flow is vital for optimizing manufacturing processes and improving products’ quality. Various metallurgical, mechanical, and designing processes, such as paper fabrication and drawing, glass blowing, and assembling of plastic and elastic sheets, exhibit the flows of different fluids upon stretching sheets. The flow upon stretching the surface was first discussed by Crane [6] for investigating different flow characteristics. Sudarmozhi et al. [7] discussed the melting thermal effects on fluid flow past an elongating sheet with thermophoresis effects. Rosca and Pop [8] examined the time-based flow of fluid passing a permeable shrinking/stretching curved sheet. Their findings revealed that as the permeability factor increased, the fluid motion declined. Their results suggest that the permeability of the sheet influences the flow characteristics, affecting the rate of fluid motion on the stretching sheet. Abbas et al. [9] inspected magnetized radiative fluid flow over a curved expanding sheet and revealed that growth in magnetic factor has augmented the thermal distribution and has weakened the velocity characteristics. Imtiaz et al. [10] investigated the magneto-hydrodynamic (MHD) time-dependent fluid flow on a curved extending sheet with the influence of both homogeneous and heterogeneous reactions. Sanni et al. [11] examined the viscous fluid flow progression past a curved as well as nonlinear stretched surface. Hayat et al. [12] scrutinized numerically the nanofluid flow induced by nonlinear stretching and curved surface associated with mass and thermal convective constraints. Okechi et al. [13] debated on viscous fluid flow on a curved and exponentially elongating surface. Further investigations about the concept of stretching sheet flows can be explored in previous studies [14,15,16].

Nanofluid flow demonstrates the movement of a liquid containing suspended nanoparticles. Nanofluid flow finds applications in heat exchangers, electronics cooling, energy systems, and biomedical engineering, where enhanced thermal conductivity and improved heat transfer properties offer advantages. Hobiny and Abbas [17] discussed dual-phase lag bio-thermal modeling through living tissues affected by laser irradiations. Optimizing nanofluid flow enables advancements in various industries, enhancing performance and efficiency in diverse engineering and scientific applications. Various investigators have conducted different investigations [18,19,20] for the augmentation of the thermal performance of pure fluid. Dinarvand et al. [21] employed copper and alumina nanoparticles in a two-step method to enhance drug delivery through the microcirculatory system over a permeable and stretched sheet. Shojaie Chahregh and Dinarvand [22] discussed blood-based hybrid nanofluid flow through an artery to facilitate drug delivery and blood rotation within the respiratory system. Their study focused on developing a fundamental model to understand the rotational behavior of blood and the distribution of medicines in the respiratory system. Chamkha et al. [23] investigated heat and mass transfer using variable temperature and concentration conditions, considering the combined effects of chemical reactions and nanoparticle shape. They found that fluid velocity decreased, while thermal distribution increased with higher nanoparticles’ concentration. Khan et al. [24] studied heat transfer in bio-convective hybrid nanofluid flow on a heated moving needle and deduced that the Nusselt number has increased with a surge in Eckert number, thermophoresis, and Brownian motion effects. Gul et al. [25] examined Marangoni convection in MHD blood-based carbon nanotubes nanofluid flow and noted a reduction in velocity and augmentation in thermal profiles as nanoparticle concentration increased. Kamis et al. [26] explored the influence of suction on unsteady hybrid nanofluid film flow over a stretched sheet.

The transfer of mass in various species mostly occurs when there is a variation in the concentration of these species. Due to this concentration gradient, the migration of species from higher to lower concentration zones has been noticed. Such phenomena can be found in the diffusion of nutrients inside tissues, thermal insulation, and the handling of foods. In the past few decades, the transmission of mass subject to chemical reaction has remained an active field of research. Hobiny and Abbas [27] conducted experimental investigations to analyze the thermal responses of cylindrical biological tissues under laser irradiation, providing insights into heat distribution and temperature rise. Their study contributed to understanding laser–tissue interactions, which is vital in medical applications like laser surgery, therapy, and tissue engineering. Marin et al. [28] studied a nonlinear bio-heat model for skin tissues with the impacts of external heat sources. Eid et al. [29] explored the Prandtl number using the impact of high-order chemical reactions on the motion of nanofluid and concluded that the magnetic effect has augmented the thermal distribution and skin friction. Dhlamini et al. [30] inspected mixed convective and binary chemically reactive flow of nanofluid and revealed that with progress in chemical reaction factor, the concentration of fluid has augmented. Saeed and Abbas [31] examined experimentally the utilization of the finite-element approach for a non-linear bio-heat model for spherical tissues.

This research work develops a mathematical model for hybrid nanofluid flow on an inclined surface. The following points highlight the novelty of this study:

• The work of Qasim et al. [2] has been expanded by incorporating inclined and stretching surfaces to investigate the unsteady thin film fluid flow, which has not been discussed yet in the presence of nonlinear mixed convection.

• The inclusion of an inclined surface introduces an additional factor that influences the fluid dynamics and flow behavior.

• Furthermore, the incorporation of stretching surfaces adds another dimension to the analysis, considering the effects of surface deformation on fluid flow.

• Nonlinear mixed convection has been added to the proposed model in mathematical form along with the impacts of a variable magnetic field.

• By incorporating variable temperature and concentration, the research provides valuable insights into the thermal and concentration gradients that affect the fluid flow, particularly in the vicinity of the inclined surface.

• Homotopy analysis method (HAM) has been employed to solve the proposed mathematical model.

2 Formulation of the problem

A thin plate is considered with θ being an angle of inclination of the sheet with the horizontal axis as exhibited in Figure 1. The following assumptions are made:

• The sheet behaves as elongating with velocity U w = b x ( 1 − α t ) − 0.5 with b ( 1 − α t ) − 0.5 an operative rate of elasticity that exhibits an enhancing behavior on the domain 0 ≤ α < 1 such that a , b > 0 and b is the elasticity constant.

• The variable temperature and concentration at surface of elongating sheet are described as T s ( x , t ) = T 0 − T ref ( b x 2 / 2 υ ) ( 1 − α t ) − 3 2 and C s ( x , t ) = C 0 − C ref ( b x 2 / 2 v ) ( 1 − γ t ) − 3 / 2 , where T 0 , T ref and C 0 , C ref are slit and constant reference temperatures and concentrations such that 0 < T ref < T 0 and 0 < C ref < C 0 .

• The term ( b x 2 / υ ( 1 − α t ) = x U w / υ ) describes the local Reynolds number that is based upon U w (surface velocity).

• Variable magnetic effects with strength B 0 ( x , t ) = B 0 ( 1 − α t ) − 0.5 have been considered in the transverse direction of flow.

• The thickness of the film is h ( t ) that is positioned on the inclined sheet.

• The temperature of sheet varies from T 0 at the slit which is in direct proportion to x 2 , where the amount of temperature drops with growth in time.

Figure 1

Graphical view of the flow problem.

With the help of the above assumptions, the leading equations are

The flow constraints at the physical boundaries are

The collection of variables that can undergo transformation is described as follows:

The resultant equations in dimensionless form are given as

with related boundary conditions as follows:

The aforementioned physical parameters M , S , Ec , Pr , γ , Sc , Gr , Gr ⁎ , G c , Gc ⁎ , N t , and N b are mathematically described as

The thermophysical properties of nano/hybrid nanofluids are outlined in Table 1, while their corresponding numerical values are presented in Table 2. These tables provide a comprehensive description and quantification of the properties exhibited by the nano/hybrid nanofluids under consideration.

3 Solution method

The solution of Eqs. (7)–(9) subject to Eq. (10) has been evaluated by using HAM [34,35]. The presented approach is a semi-analytical method that exhibits rapid convergence, making it suitable for solving nonlinear differential equations. To apply this approach, some starting values are necessary. For the current problem, these starting values are provided as follows:

with

Such that

where a i ( i = 1 → 7 ) are fixed values.

4 Results and discussion

This work investigates thin film flow with the effects of a magnetic field in association with Cu and CuO nanoparticles past an inclined plate. The thickness of the film is taken as a variable to sustain the uniformity of the thin film for sliding on the surface, while the dragging terminology is taken as nonlinear for the flow of film down the plate. To optimize the production of entropy and to reduce the fluid’s friction, the second law of thermodynamics is employed. Additionally, the current model supplements by incorporating the collective role of mass and heat transportation. Some emerging factors have been collected in the modeling of the problem; the influence of these factors will be deliberated in the following paragraphs.

4.1 Velocity distribution

The variations in velocity characteristics in response of different values of emerging factors are shown in Figure 2(a)–(i). Figure 2(a) illustrates the impact of magnetic forces on fluid motion. As the Lorentz force increases for growth in M , it generates a resistive force in the fluid against the velocity of nanoparticles, resulting in heightened resistance experienced by the fluid particles. During this process, the fluid velocity drops as M escalates. The impression of thickness factor β upon fluid motion is depicted in Figure 2(b). The observed trend in this figure reveals that as thickness of fluid film upsurges, the fluid particles move more slowly along the inclined plate, indicating a reduction in their downward motion. Indeed, as the values of β increase, greater cohesion is observed among the fluid particles. This increased cohesion results in reduced fluid motion. Figure 2(c) and (d) show the influences of thermal and nonlinear thermal Grashof numbers ( Gr , Gr ⁎ ) upon fluid motion. In these figures, an augmenting behavior has been observed in fluid flow characteristics for intensification in the values of ( Gr , Gr ⁎ ). Similarly, for growth in mass Grashof and nonlinear mass Grashof numbers ( G C , G C ⁎ ) , there is an expansion in the velocity panels as shown in Figure 2(e) and (f). Physically, this can be understood as for upsurge in Grashof number the viscous force weakens due to which the liquid particles observed minimum constraints in flow direction. This phenomenon is more remarkable in case of nonlinear factors Gr ⁎ as depicted in Figure 2(d) and for Gc ⁎ as shown in Figure 2(f). The impact of ϕ 1 , ϕ 2 (volume fractions) on fluid flow characteristics is observed in Figure 2(g). It is noticed that with growth in ϕ 1 , ϕ 2 there is retardation in fluid motion. Actually, for surge in ϕ 1 , ϕ 2 , the fluid behaves in more dense manners due to which fluid’s nanoparticles experience more resistance in the direction of their motion. In this process, the fluid motion of nanoparticles upon the inclined sheet becomes less as illustrated in Figure 2(g). The influence of λ over fluid motion is shown in Figure 2(h) where a deteriorating behavior of f ′ ( η ) has been noticed in response of growth in the values of λ . Since λ describes the connection between relaxation and retardation times, so with upsurge in λ there is more resistive force to fluid particles transportation. In this process, the fluid flow characteristics declines as shown in Figure 2(h). The impression of unsteadiness factor S over fluid’s motion is illustrated in Figure 2(i) with retarding behavior in the flow characteristics. Since, with growth in S, more friction is created in the line of fluid motion; hence with growth in S, the fluid motion declines as depicted in Figure 2(i).

Figure 2

(a–i) Impact of various emerging factors on velocity distribution.

4.2 Thermal distribution

The effect of numerous factors on thermal distribution is depicted in Figure 3(a)–(f). In Figure 3(a), it has been revealed that the growing values of magnetic factor M support the thermal transportation. Actually, for upsurge in M , there is more resistive force among fluid particles in the opposite direction of fluid motion. Due to these resistive forces, the skin friction augments and ultimately thermal energy enlarges that supports the growth in temperature distribution. The influences of volume fraction ϕ 1 , ϕ 2 upon energy distribution are illustrated in Figure 3(b) with an enlarging behavior as perceived in heat profiles. Actually, with growth in ϕ 1 , ϕ 2 the dense behavior of fluid augments for which additional constraints is practiced by the fluid particles. In this phenomenon, the fluid motion has declined while the energy distribution has augmented as explained in Figure 3(b). Growth in Eckert number Ec also supports the thermal distribution as shown in Figure 3(c). The thermal dissipative properties are primarily governed by Ec at the free surface, resulting in the efficient gathering of energy among the fluid particles near boundary on the inclined plate. So the thermal distribution is escalated with higher values of Ec . Figure 3(d) shows the expansion in strength of thermal layer at boarder-line with expansion in S . This states that in steady-state conditions, nanoparticles cool at a faster rate, whereas in time-dependent scenarios, a longer duration is needed for this phenomenon to occur. The influence of Brownian motion factor Nb on thermal panels is shown in Figure 3(e). Clearly with escalation in Nb , the zigzag motion among the nanoparticles escalates as a consequence the kinetic energy is converted to thermal energy. As a result, maximum thermal diffusion occurs that augment thermal distribution as shown in Figure 3(e). The effect of Nt on temperature distribution is depicted in Figure 3(f) with a growing manner in thermal profiles. Since Nt is described mathematically as Nt = τ D B ( T w − T ∞ ) / ν T ∞ , so growth in Nt supports the temperature gradient at free stream of the inclined plate, for which more heat is shifted from high to low temperature zone. In this process, thermal diffusion is at its peak; hence, thermal distribution upsurges with escalated values of Nt as shown in Figure 3(f).

Figure 3

(a–f) Impact of different emerging factors upon thermal distribution.

4.3 Concentration distribution

The impact of chemical reaction factor γ on diffusions of mass is presented in Figure 4(a). With an upsurge in γ , the diffusions of mass augments and results in the growth of concentration boundary. Through this process, the concentration distribution augment is shown in Figure 4(a). The influence of Schmidt number Sc is given in Figure 4(b). It is noticed from this figure that concentration distribution Φ ( η ) declines with growth in Sc . Actually, Sc is in inverse relation to diffusion of mass, so augmentation in Sc means reduction in diffusion of mass that results in the drop of concentration distribution Φ ( η ) . Figure 4(c) depicts that with upsurge in unsteadiness factor S there is a decline in concentration profiles Φ ( η ) . This can be interpreted as, for upsurge in S more time is required for mass to be diffused as compared to steady case. Hence, with growth in S, there is a decline in concentration distribution Φ ( η ) . Figure 4(d) and (e) show the impacts of N b and N t on concentration distribution Φ ( η ) . Upsurge in N b reduces the profiles of Φ ( η ) , whereas adverse effects have been noticed for growth in N t against the profiles of Φ ( η ) . For growth in N b , there is a decline in nanoparticles volume fraction at free surface that retards the concentration distribution as depicted in Figure 4(d). Conversely, for progression in N t , the diffusion of mass upsurge that results in the growth of concentration distribution Φ ( η ) as illustrated in Figure 4(e).

Figure 4

(a–e) Impact of various factors on concentration profiles.

4.4 Bejan number and generation of entropy rate

Variations in irreversibility generation and Bejan number in the reaction of changes in various substantial factors are discussed in Figures 5 and 6. The influence of M on entropy and Bejan number is shown in Figure 5(a) and (b). It is obvious that with higher M nanoparticles experience more restrain during motion that resulting in more disturbance in the flow pattern of fluid and producing more entropy, as illustrated in Figure 5(a). In this physical phenomenon, the Bejan number enhances as shown in Figure 5(b).

Figure 5

(a, b) Bejan number and entropy vs variations in M .

Figure 6

(a, b) Entropy and Bejan number vs variations in Brinkman number Br .

Figure 6(a) and (b) present the influence of Br on entropy and Bejan number. It has been noticed that for growth in Br , entropy rate has augmented. Actually, Br let out the thermal flow through the conductance of molecules with an association of heat transportation to the viscous thermal flow of fluid particles. Moreover, for augmentation in Br , the thermal conductance of nanoparticles retards continuously that results in augmentation of irreversibility generation rate as shown in Figure 6(a). In this physical phenomenon, it has been observed that the Bejan number experiences a deceleration, as illustrated in Figure 6(b).

5 Conclusions

An endeavor has been made in this study to investigate thin film flow with the effects of a magnetic field in association with Cu and CuO nanoparticles past an inclined plate. To optimize the production of entropy and to reduce the fluid’s friction, the second law of thermodynamics is employed. Additionally, the current model supplements by incorporating the combined role of mass and heat transportation. After implementing similarity variables, the leading equations have been converted to dimensionless form and subsequently solved using HAM. After a comprehensive examination of the work, the following observations have been made:

• Fluid motion slows down with increasing values of the thickness factor, magnetic effects, nanoparticle volume fraction, and unsteadiness parameter. In contrast, it intensifies with higher mass and thermal Grashof numbers in both linear and nonlinear scenarios.

• Thermal profiles rise with increases in magnetic parameter, Eckert number, nanoparticle concentration, unsteadiness, thermophoresis, and Brownian motion factors. Physically, these factors enhance thermal energy generation, reduce heat dissipation, and facilitate nanoparticle-induced thermal transport, causing more noticeable temperature gradients.

• Concentration profiles decline with higher Brownian motion, unsteadiness factors, and Schmidt number, while a surge in chemically reactive factors and thermophoresis parameters leads to an upward trend regarding the concentration profiles.

• Entropy generation is enhanced by magnetic effects and reduced by a higher Brinkman number.

• The Bejan number decreases with increasing Brinkman number but increases under stronger magnetic field effects.

• For nanoparticle volume fractions ranging from 0.01 to 0.03, the Nusselt number shows significant improvement, particularly for hybrid nanoparticles, where it increases from 4.3207 to 6.6983%, demonstrating their superiority over traditional nanoparticles.

• Finally, the results of this study show strong agreement with established findings, confirming the accuracy and reliability of the current work.