Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition

Nomenclature

T ∼

temperature of the HNF

C ∼

concentration of nanoparticles

C ∼ s

wall concentration

q r

radiative heat flux

k *

Rosseland mean absorption coefficient

σ *

Stefan-Boltzmann constant

Q *

non-uniform energy source

A , B

heat source and heat sink parameters

x , y

Cartesian coordinates

μ unf

dynamic viscosity of the HNF

ρ unf

density of the HNF

k unf

thermal conductivity of the HNF

σ unf

electrical conductivity of the HNF

( ρ C p ) unf

heat capacity of the HNF

ξ H , ξ i

hall and ion slip parameters

B 0

magnetic field strength

δ

material parameter

ϱ

second material parameter

d , a

arbitrary constants

u w

wall velocity

u slip

velocity slip

D T

thermophoretic diffusion coefficient

v f

kinematic viscosity of the base fluid

k f

thermal conductivity of the base fluid

∀

slip coefficient

h

height of the domain

d h d t

rate of change of height

1 Introduction

Liquid thin film flow (LTFF) is a key concept in fluid dynamics, describing the movement of a thin liquid layer over a solid or fluid surface. This flow is governed by a balance of forces, including surface tension, viscous forces, gravity, and external influences such as thermal or electromagnetic fields [1,2,3]. The significance of LTFF lies in its widespread industrial and technological applications. One of the primary applications is in the coating and printing industry, where the uniform deposition of liquid films ensures the quality and performance of products such as paints, adhesives, and electronic displays. Thin film coatings are essential in microelectronics, photovoltaic cells, and biomedical implants, requiring precise control over film thickness and stability [4,5,6]. In thermal and energy systems, LTFF plays a crucial role in heat exchangers, cooling mechanisms, and evaporation-based technologies. The efficient heat transfer in thin liquid films is pivotal in enhancing the performance of condensers, refrigeration units, and solar thermal collectors. The study of thermophoretic particle deposition in thin films has also gained attention in optimizing energy conversion and environmental applications. Biomedical and microfluidic applications further underscore the importance of LTFF. In lab-on-chip devices, controlled thin film flows facilitate drug delivery, chemical reactions, and diagnostics. Biological thin films, such as tear films in ophthalmology, exhibit complex rheological properties that require advanced modeling techniques for medical advancements. Zhang et al. [7] explore the thermal and dynamic performance of a squeeze-film flow between two rotating disks. The analysis combines numerical simulation and experimental validation to understand how thermal transport influences the fluid system. They employ the finite-volume technique to solve the governing equations and confirm accuracy via testing for grid convergence and verifying energy conservation. The results provide insights into how thermal impact and surface geometry influence the performance and dynamics of lubrication systems in rotating machinery. This recent work of Zheng et al. [8] introduces a laminar two-dimensional (2D) model for LTFF, validated through experiment outcomes. It incorporates aluminum oxide nanoparticles (NPs) into water, ethylene glycol, and lithium bromide-water solution to increase thermophysical properties. Jiang et al. [9] investigate the thermal distribution, entropy generation, and performance of unsteady LTFF magnetohydrodynamic (MHD) nanofluid flow containing alumina and water over an inclined surface.

Partially ionized fluids (PIFs) are a class of fluids that contain both neutral and charged particles, typically composed of ions, electrons, and neutral atoms or molecules [10]. Unlike fully ionized plasmas, where all atoms are stripped of their electrons, PIFs exhibit a balance between neutral and charged species, leading to complex interactions influenced by electromagnetic forces, collisions, and thermal effects. These fluids are commonly found in astrophysical, industrial, and engineering applications, including fusion reactors, atmospheric phenomena, and space propulsion systems [11]. PIFs have critical applications in various scientific and engineering fields. In aerospace engineering, they play a vital role in ion thrusters and plasma propulsion systems, where controlled ionization enhances thrust efficiency. In astrophysics, they are essential in understanding stellar atmospheres, planetary ionospheres, and interstellar medium dynamics [12,13]. In nuclear fusion technology, partially ionized plasmas are significant in magnetically confined fusion devices (such as tokamaks and stellarators), where neutral particle interactions influence plasma stability and energy confinement [14]. Similarly, in industrial applications, PIFs are used in plasma-assisted manufacturing, semiconductor processing, and advanced material coatings [15]. The concept of PIFs in fluid dynamics is important since it fills a gap between fluid mechanics and plasma physics. They supply conceptual information needed on how electromagnetic forces influence fluids and their effectiveness in creating high technologies and estimating in nature. In addition, the investigation of the PIFs contributes toward the construction of new computational approaches and experimental procedures, which in turn provide more realistic forecasts of system behavior [16,17,18]. Iqbal et al. [19] used the homotopy analysis method to computationally investigate the Cattaneo–Christov heat flux and the behavior of gyrotactic microorganisms in the flow of a partially ionized hybrid nanofluid (HNF). Their result shows that radial velocity is reduced with increasing values of the magnetic field, Hall effect, ion slip, and material parameters. Also, the transverse velocity is enhanced with increasing values of magnetic field, Hall effect, ion slip, and material parameters. Jazaa et al. [20] presented a non-Newtonian Powell–Eyring fluids model that revealed the influence of PIF, slip, magnetic, and unsteadiness on the enhancement of thermal transport. The entropy-optimized performance of flow and thermal transport of a partially ionized, magnetically influenced Prandtl fluid enriched with three kinds of nanomaterials drifting across a convectively heated cylinder were studied by Wang et al. [21]. They observed that the Hall and ion slip effects reduce entropy formation.

The investigation of HNFs is therefore an enhanced class of nanofluids, which comprise two or even more peculiar NPs dispersed in the base fluid for improving heat and fluid characteristics [22,23,24]. These nanofluids, augmented by mixing numerous kinds of NPs including metallic (gold, silver, iron, zinc oxide [ZnO]), non-metallic (graphene, CNTs), and ceramic (titanium dioxide, graphene oxide), possess advanced thermal conductivity, thermal transport efficiency, and stability than the regular nanofluids. HNFs offer numerous benefits over single-particle nanofluids, including low thermal performance enhancement, instability, and agglomeration [25]. On the one hand, the benefits expected from HNFs are countless. First, it can increase the thermal conductivity and the heat exchanging capability. Second, the HNFs have improved rheological features involving viscosity regulation of a flow. The HNFs are employed in numerous areas due to the fact that they are involved in controlling fluid dynamics and thermal transport [26]. They have been employed in car manufacturing, electronics, aerospace, and energy application sectors. For example, they are used in automobile cooling systems to raise the efficiency of engines, in electronics cooling systems to increase the dissipation of heat from high-density devices and in solar thermal systems to increase the efficiency of energy harvest and storage [27,28,29]. They are also employed in nuclear reactors, heating, ventilation, and air conditioning systems, and systems salient to biomedical uses like drug delivery and hyperthermia. Furthermore, they play an important function in the designation of energy savings and the optimization of the sustainability of industry processes. The combination of using carbon nanotubes and metal oxide NPs makes HNFs ideal for high-performance cooling and heating fluids for industries, which in future will determine the future of fluid dynamics and thermal engineering [30,31,32]. The work of Mishra and Pathak [31] inspects the hydro-thermal performance of molybdenum disulfide–silicon dioxide/water HNF and molybdenum disulfide–silicon dioxide–graphene oxide/water ternary HNF flowing over an inclined cylinder, considering heat generation/absorption effects. The outcomes describe that the thermal profiles are higher for the ternary HNF than HNF for increasing values of thermal radiation parameter and Eckert number. Also, the velocity profiles of both HNFs experience a reduction when higher values of suction/injection parameters are applied. The natural convection thermal transport of alumina–multiwalled carbon nanotube (Al₂O₃–MWCNT)/water HNF prepared via a two-step method at varying volume concentrations (0.00–0.20%) was studied by Scott et al. [32]. The numerical investigation of the flow and thermal performance of MgO–Al₂O₃/water HNF within a Darcy–Forchheimer–Brinkman framework under the influence of a horizontal magnetic field was conducted by Mebarek-Oudina et al. [33]. The outcomes show the increased pattern of the thermal and dynamical behavior of the HNF when augmenting the concentration, shape factor, porosity, and Darcy number; however, it also engenders increased formations of irreversibilities in the system that were revealed to enhance with the permeability and the great properties of the nanofluid. Nevertheless, this thermally enhanced pattern is shown to degrade with strong Hartmann values, which also reduced both thermal and viscous entropies.

The non-Newtonian Powell-Eyring model (P-EM) fluid is a constitutive equation employed for the characterization of the transient flow performance of non-Newtonian fluid, in which the viscosity depends on the rate of shear stress [34]. Several industrial and biological fluids like polymers, suspensions and biological fluids have more complicated flow behavior than Newtonian fluids and cannot be characterized by a simple Newtonian model [35]. In this respect, the P-EM is considered an improvement of the existing literature on fluid dynamics since it effectively combines both Newtonian and non-Newtonian performances of these materials [34]. P-EM is its potential for the prediction of liquid performance of shear rates. In contrast to other models, like Bingham plastic and power-law models, the present P-EM has no singularity at either high or low shear rates. The usage of the P-EM is suitable for utilization in a wide range of industries. In chemical engineering, it is applied in developing processes of polymer solutions, paints, and food products. However, in the field of biomedical, it contributes to the prediction of the flow characteristic of blood, which displays non-Newtonian properties that are essential to modeling circulation and designing products related to the circulatory system in the human body [36]. Also, the model is essential for the description of the flow of the drilling fluids and the crude oil, which can have rather complicated rheological behavior. The work of Hashim et al. [37] considers blood as a P-EM with hematocrit-dependent viscosity. They combined single-phase and two-phase mixture models to examine the thermal and mass transport properties of HNFs, highlighting the nominal toxicity and medicinal significance of ZnO–iron(iii) oxide NPs. This exploration of thermal and mass transport phenomena in fluids with varying thermophysical features, focusing on P-EM with distinct characteristics, was studied [38]. However, they examine the performance of this shear-thinning fluid interacting with an exponentially stretchable surface, considering influences such as thermal radiation. The effects of the radiation and thermal performance of a molybdenum disulfide and graphene oxide NPs on electromagnetic radiation absorption and enhanced thermal conductivity, driven by industrial demands for advanced non-Newtonian P-EM fluids were studied [39]. Entropy analysis of unsteady magnetohydrodynamic thin liquid film flow of Maxwell nanofluids with variable fluid properties was studied [40]. A renovated Jaffrey–Hamel flow problem and new scaling statistics for heat and mass fluxes with the Cattaneo–Christov heat flux model were studied [41]. Analytic solutions for a liquid film on an unsteady stretching surface were studied [42]. Heat transfer in a liquid film over an unsteady stretching surface with viscous dissipation in the presence of an external magnetic field was studied [43].

Fluid thin films are liquid layers with thicknesses ranging from nanometers to micrometers, exhibiting unique physical behaviors due to surface tension, viscosity, and external forces such as gravity, electromagnetic fields, and thermal gradients. These films play a critical role in numerous natural and industrial processes, influencing lubrication, coating technologies, biological systems, and heat transfer mechanisms. Fluid thin films have diverse applications in engineering and science. In coating and surface engineering, they are fundamental to processes such as painting, printing, and thin-film deposition in microelectronics. In thermal management systems, they enhance heat dissipation in cooling technologies, including microfluidic heat exchangers and phase-change materials. In biomedical applications, fluid thin films are crucial for drug delivery systems, tear film stability in ophthalmology, and bio-inspired microfluidics. Additionally, in aerospace and energy sectors, they are utilized in advanced lubrication systems, solar thermal energy collection, and fuel management in spacecraft. The study of fluid thin films continues to evolve with advancements in nanotechnology, smart coatings, and interfacial science, making them a key focus in modern applied physics, materials science, and engineering disciplines. Entropy generation and heat transfer in boundary layer flow over a thin needle moving in a parallel stream in the presence of nonlinear Rosseland radiation were studied by Afridi and Qasim [44]. Minimization of entropy generation in MHD mixed convection flow with energy dissipation and Joule heating: utilization of Sparrow–Quack–Boerner local non-similarity method was studied by Afridi et al. [45]. Comparative study and entropy generation analysis of Cu–H₂O and Ag–H₂O nanofluids flow over a slender stretching surface was studied by Afridi and Qasim [46]. Theoretical analysis of MHD Maxwell two-phase nano-flow subject to viscous dissipation and the chemical reaction was studied by Afridi et al. [47].

This study aims to investigate the enhancement of thermal transport and the effects of viscous dissipation in a time-dependent 2D thin liquid film composed of non-Newtonian HNF. The research focuses on the influence of solar radiation, Hall current, and second-order slip conditions on heat and mass transfer properties, with an emphasis on entropy generation and fluid transport efficiency. This study aims to formulate the governing equations for the time-dependent thin-film flow of a non-Newtonian HNF while incorporating the effects of solar radiation, Hall current, and second-order slip conditions. The HNF consists of magnesium oxide (MgO) and ZnO NPs suspended in water as the base fluid, ensuring enhanced thermal transport properties. The research seeks to transform these governing equations into a system of dimensionless ordinary differential equations (ODEs) using similarity variables to facilitate analysis. The study intends to solve the transformed ODEs numerically using the collocation method to obtain velocity, temperature, and entropy generation profiles. Additionally, it aims to analyze the influence of key physical parameters, such as the Brinkman number, magnetic field intensity, slip conditions, and solar radiation parameter, on the thermal and flow behavior of the HNF thin-film system. Furthermore, the research focuses on comparing the thermal transport performance and entropy production characteristics of the MgO–ZnO/water HNF with conventional nanofluids, emphasizing its advantages in improving heat transfer efficiency. Lastly, the study explores potential applications of the findings in biotechnology, medical therapies, and advanced energy storage systems, providing insights into optimizing thermal management in these fields.

2 Thermal fluid flow model description

Consider the transient, 2D motion of a non-Newtonian Powell-Eyring HNF, which exhibits both electrical conductivity and partial ionization, influenced by MHD effects and nanofluid interactions. The investigation incorporates thermal radiation and viscous dissipation phenomena over a dynamically extending sheet. A Cartesian coordinate framework ( x ∼ , y ∼ ), is employed, where the x-axis coincides with the stretching surface at y ∼ = 0, and the y ∼ -axis extends perpendicularly away from it. The sheet undergoes uniform elongation due to equal and opposite forces applied along the x ∼ -direction. To establish the governing equations, specific flow assumptions are introduced, as shown in Figure 1a. The reference temperature and concentration are constrained by the conditions 0 ≤ T ∼ ref ≤ T ∼ 0 and 0 ≤ C ∼ ref ≤ C ∼ 0 . The surface temperature is T ∼ s ( x ∼ , t ) = T ∼ 0 − T ∼ ref d x ∼ 2 2 v f 1 ( 1 − at ) 1 / 2 and concentration at the wall is C ∼ s ( x ∼ , t ) = C ∼ 0 − C ∼ ref d x ∼ 2 2 v f 1 ( 1 − at ) 1 / 2 , where T ∼ s and C ∼ s represent the surface temperature and concentration, respectively. The fluid exhibits partial ionization, and external electromagnetic influences are neglected due to the dominance of inertial forces and negligible electric field effects. The thermal energy equation incorporates contributions from radiative heat transfer, Joule dissipation, viscous heating, spatially varying heat generation/absorption, and the influence of NP slip mechanisms. The nanofluid composition is analyzed using the Tiwari–Das model, which quantifies the volume fraction of MgO-ZnO NPs suspended in a water-based fluid, while the Buongiorno model accounts for the simultaneous impact of thermophoretic and Brownian diffusion. NPs are assumed to be uniformly distributed, maintaining thermal equilibrium with the surrounding base fluid. Both single-phase and two-phase formulations are considered, allowing for possible slip interactions between fluid molecules and suspended nanomaterials.

Figure 1

(a) Model configuration. (b) Thermophysical values for MgO, ZnO, and water-based fluid. (c) Minimized residual error.

Following the highlighted assumptions, the governing boundary layer equations are described as [40,41]

Continuity equation

Momentum conservation equation

Energy conservation equation

Species concentration equation (mass transfer)

Generalized Ohm’s law:

Table 1 displays the expressions of the above equation.

The solar radiation ( q r ) is given as

The non-uniform energy source expression is

Here, A represents the heat source, and B is the heat sink. The governing equations are described as

The boundary slip constraints are

Here, the velocity components in the x ∼ -direction ( u ˜ ) and in the y -direction ( v ˜ ), wall temperature ( T ∼ s ), Hall parameter ( ξ H ) , ion slip parameter ( ξ i ), surface temperature, wall velocity ( u ∼ w ), thermophoretic diffusion coefficient ( D T ), temperature ( T ∼ ), second material parameter ( ϱ ), magnetic field strength ( B ∼ ), arbitrary constant ( d , a ), and the material parameter ( δ ). The thermophysical properties of nanofluid for μ unf , ρ unf , σ unf , k unf , and ( ρ C p ) unf are [45]

The thermophysical values for the MgO and ZnO NPs dispersed in a water-based fluid are displayed in Figure 1b.

The following transformations are [15,32,44]:

Following the fulfillment of the mass conservation equation, the velocity components assume the following form:

where

We obtain

Subject to:

The parameters associated with the velocity, temperature, and mass transfer rate are defined as follows:

Here, viscoelastic is ω = u ∼ w 2 d 2 ν f γ , solar radiation Nr = 16 σ * T ∼ 0 3 3 κ * k f , Brownian motion N b = τ 1 D B ( C ∼ s − C ∼ 0 ) ν f , thermophoresis N t = τ 1 D T ( T ∼ s − T ∼ 0 ) ν f T ∼ ∞ , viscosity is ϖ = 1 δ ϱ μ f , magnetic M a = σ f B 0 2 ρ f d , concentration difference E 2 = ( C ∼ s − C ∼ 0 ) C ∼ 0 , Eckert Ec = μ w 2 ( ρ c p ) f ( T ∼ s − T ∼ 0 ) , slip N 1 = O a ν f , Schmidt Sc = ν f D B , Prandtl Pr = ( μ c p ) f k f , temperature difference D 1 = ( T ∼ s − T ∼ 0 ) T 0 , unsteadiness S = d a , diffusion χ 1 = R D B ( C ∼ s − C ∼ 0 ) k f .

3 Entropy production analysis

The entropy generation is presented by

The dimensionless entropy production is

4 Numerical solution