Modern Touch of a Classical Flow Problem: Thin films of liquid are pervasive and ubiquitous across both the natural and human-made worlds. Thin film flow describes the spreading of a liquid over a solid substrate (static or dynamic) with a free surface, typically air or gas, at the interface. The flow is primarily driven by external forces, such as a translating or rotating substrate, surface tension, or gravitational force [1,2]. To further explore this, consider examples from nature and technology, including rainwater dribbling down a windowpane, blood coursing through tiny blood vessels, tears trickling down a cheek, suspensions oozing over substrates, and lubricant oil coating rotating ball bearings or other spinning objects [3–6]. Studying thin film flow has wide-ranging applications, aiding understanding of phenomena such as spin-coating, gravure printing, absorption columns, mudslides, dry processes, cooling of microelectronics devices, production of glass/wire, wettability, flow over roads/roofs, solar cells, film evaporation, food industry processes, and heat exchangers [7–11].

One application of thin film flow studies is the analysis of complex liquid flows over intricate substrates. For example, it can be utilized to model the surface-tension-driven flow of incompressible fluids, such as the lining of the lungs [12,13]. The complexity of these flows arises from two main factors:

Substrate geometry: This can include flexible substrates like leaves [5] or the intricate lining of the lungs [13], as well as other examples like magma streams or tears on a face [14,15].

Currently, many literature sources exist on both theoretical and experimental studies of thin film flow for Newtonian and non-Newtonian fluids.

The story of thin film flow began in 1916 with the work of Nusselt [23], who presented mathematical equations describing its dynamics. He derived the governing equation for thermal transfer coefficients in condensate thin films, assuming they primarily resist heat transfer. Subsequently, a significant contribution was made in 1974 by Tamir et al. [24], who extended Nusselt’s work by accounting for a constant surface resistance.

While a full historical account is beyond the scope of this discussion due to its substantial length, several noteworthy contributions have explored various aspects of thin film flow:

Kondic et al. [25] examined thin film liquid on an inclined plane, concluding that the inclination angle significantly alters the wetting behavior.

Applications and Experimental Verification: The applications of thin film flow extend beyond theoretical studies. For instance, Mazloomi et al. [31] utilized the lattice-Boltzmann method to investigate thin film flow with thermal sources for surface coating, while Slade [32] explored rivulet formation in thin liquid films. Experimental validation also plays a crucial role in understanding thin film flow. Several studies have focused on this aspect, including:

Ju et al. [33] performed experiments to probe the droplet impinging on thin film flows and highlighted the characteristics of droplet evolution.

Micropolar Fluids with Nanoparticles: It is well known that nanoparticles in liquids can enhance thermal transfer rates and facilitate other valuable industrial phenomena, such as the instant cooling of electronic chips or rapid water heating in geysers and heat exchangers. Notably, reference [10] was the first to theoretically analyze the hydrodynamic and thermal effects of thin film flow in water containing seven different nanoparticles, even developing a closed-form solution for fluid velocity and temperature.

Contribution and Scope: Inspired by the work in [10], this study extends its analysis to explore other fluid types, such as micropolar fluids, which are relevant to blood, colloidal suspensions, and epoxies. It derives a closed-form solution for the thin film flow of a micropolar fluid containing five different nanoparticles over a thermally heated inclined plane.

In various industries, fluids such as tomato ketchup, honey, blood, colloidal suspensions, and mercury, along with other industrial fluids, do not behave ideally like water. These liquids are categorized as non-Newtonian fluids and require distinct models and constitutive equations to describe their flow behavior. The study of the thin film flow of such non-Newtonian fluids has yielded valuable insights [38–42]. To further explore it, let us examine some recent contributions from 2021 to 2023.

Recent Highlights (2021–2023):

2021:

Looking Forward: These examples highlight the active research landscape in non-Newtonian thin film flows. As this field continues to evolve, it offers immense potential for advancements in diverse areas such as bioengineering, microfluidics, and coating technologies.

As discussed earlier, this study focuses on exploring the hydro- and thermo-dynamical effects of thin film flow in micropolar fluids containing nanoparticles. These fluids hold significant techno-scientific value and exist not only in the laboratory but also in the natural world. Blood, which represents the micropolar nature with its cellular and platelet components, is noted in [54,55]. Colloidal suspensions and liquid crystals, characterized by the presence of colloids, are also part of this category [56]. Rigid-rod epoxies, with their rod-like molecular structure and polymeric suspensions, further expand the diverse scope of micropolar fluids [22,57].

Micropolar fluids are distinguished from Newtonian and non-Newtonian fluids through their ability to sustain body couple and couple stress in addition to conventional body force and Cauchy stress. The Cauchy stress tensor in a micropolar fluid arises from the microscopic, needle-like particles it contains. While Newtonian fluids require only three degrees of freedom for analysis, micropolar fluids necessitate six, reflecting their more complex internal structure.

The literature on the thin film flow of micropolar fluids remains surprisingly sparse. A notable study from Yusuf et al. in 2021 [58] investigated this phenomenon over a static inclined porous substrate, primarily focusing on entropy analysis and employing numerical tools based on the differential transformation method. While this study is noteworthy, it leaves several intriguing questions unanswered, thus hindering a comprehensive understanding of micropolar fluid behavior in real-world situations.

Unanswered Questions and the Research Focus: These unaddressed questions become the driving force for the present research. For instance, how will the fluid respond to a heater within or near the substrate? What happens when the substrate becomes dynamic, moving at a uniform or non-uniform speed? What impact do different nanoparticles, incorporated as additive colloids, have on the flow dynamics? These and many more questions will guide this research, promising valuable insights into the world of micropolar fluids.

The Power of Nanoparticles: Nanoparticles play a vital role in shaping the landscape of modern technology [59–62]. Gold (Au) and silver (Ag) nanoparticles are extensively used in separation science, facilitating efficient separation processes [63–65]. Solutions of gold nanoparticles even reduce thermal resistance by 13% compared to pure water. These nanoparticles also hold vast potential in cancer treatment, where their heat-generating capability aids in photodynamic therapy.

Molybdenum disulfide MoS2 shines in the field of electronics and logic devices due to its tunable semi-conductive properties. It also serves as a solid lubricant additive in greases and gear oils, enhancing their performance by reducing friction and wear, lowering operating temperatures, and increasing load-carrying capacity. Additionally, MoS2 helps dampen vibration and noise, leading to reduced energy consumption. Its antioxidant and anti-corrosion properties further contribute to extending the life of lubricants [66–68].

Aluminum oxide (Al2O3) nanoparticles are prominent in the medical field as antibacterial agents owing to their low toxicity towards eukaryotic cells [69]. Silicon dioxide (SiO2) nanoparticles, on the other hand, improve combustion efficiency and reduce emissions in dual-fuel engines. SiO2-based nanofluids, particularly those in methanol, can increase peak heat release rate and peak pressure by up to 4.3% and 8.6%, respectively, while also enhancing brake-specific fuel consumption and brake thermal efficiency under high loads [70,71].

The Current Study: This study explores the thin film flow of a micropolar fluid (base fluid) containing five distinct types of nanoparticles. This flow occurs over an inclined substrate moving with a uniform speed (U) under the influence of gravity. Section 2 meticulously formulates the governing equations for this scenario. Section 3 then tackles the derivation of a closed-form exact solution for the corresponding boundary value problem. Section 4 shows the obtained results and provides insightful commentary on their implications. Finally, Section 5 wraps up the study by summarizing the essential findings and their significance.

In order to validate the calculations herein, Fig. 2b is set for fluid speed if K1=0 in Eq. (19) particularly for the Newtonian fluid. This figure agrees well with Fig. 2b of [10]. This satisfaction provided motivation to study the fluid speed for K1≠0, micropolar fluid, which is shown in Fig. 2a for different values of flow parameter α1.

As the flow is generated by moving upward the inclined substrate over it, the fluid is flowing, and the probability of reverse flow downward naturally is dominant. This fact is reflected in Fig. 2a as a negation of speed substrate-rider fluid. This reversal of fluid dominates with the rise in the flow parameter α1. This phenomenon aligns with the physical situation because α1is linearly proportional to the angle of inclination θc of substrate by virtue of Eq. (17a). However, the critical points at which the reverse flow takes place rely on various factors, in which one of the factors is flow parameter α1. The range of flow parameter is α1∈[0,2) for which rider fluid-film moves with the moving substrate for both the Newtonian fluid (K1=0) as well as the micropolar fluid (K1≠0). Nevertheless, the speed of reverse flow increases for the micropolar fluid, which is not surprising because the effects of micropolarity are enhanced, as shown in Figs. 2a and 2b. The threshold value α1=2 is common for both the Newtonian fluid (K1=0) and the micropolar fluid (K1≠0) at which the reverse flow initiates caused by the gravitational force, provided that a no-spin boundary condition β~0=0 is imposed. When a spin boundary condition β~0≠0 is encountered, then the threshold value of α1 varies such that α1=1 if β~0=0.6, as shown in Fig. 3. Additionally, the following facts for the flow attitude are also observed based on Eq. (18) generally and particularly in Table 2.

where (24)u1=u2+K1β~0λ1(eλ1+1) (25)u2=1+α1K1−2+K1α1(eλ1−1)λ1(eλ1+1)(K1−2) (26)T1=β1[2+2K1−K122+56α1−5K1α1]+α1β3Sinh2(λ1)[−α1Sinh2(λ1)3+K12α14] +3K1α1[K1−2]4+4α12β3K1e2λ1λ13[1−Cosh(λ1)−Coth(λ1)+Cosh(λ1)Coth(λ1)] +α1β1K1λ12[−12−Coth(λ1)2+2Sinh(λ1)+K14+K1Coth(λ1)4−K1Sinh(λ1)] +2α12β3K1e2λ1λ12[Sinh(λ1)−Cosh(λ1)]+α1β1K1λ1[32+Coth(λ1)2−3K14−K1Coth(λ1)4] T2=K12α1β3[−1+K12+β~0−K1β~0+K12β~0]+3β1K1λ13[1−K1+K124]−α1β3K1[K1−2]λ1−K12β~0β32λ1[4−4K1+K12] +8α1β3K1e2λ12λ13[K1−2][−1+Cosh(λ1)+Coth(λ1)−Cosh(λ1)Coth(λ1)] +β1K14λ12[4−4K1+K12][1+Coth(λ1)−4Sinh(λ1)]−2α1β3K1e2λ1λ12[(K1+2)Sinh(λ1)−(K1−2)Cosh(λ1)] (27)+β1K1λ1[−3−Coth(λ1)+3K1+K1Coth(λ1)−34K12−K12Coth(λ1)4] T3=β1[2+2K1−K122+56α1−5K1α1]+3α1β1K14λ13[K1−2]+α1β1K1λ1[32+Coth(λ1)2−3K14−K1Coth(λ1)4] (28)+α1β1K1λ12[−12−Coth(λ1)2+2Sinh(λ1)+K14+K1Coth(λ1)4−K1Sinh(λ1)] T4=3β1K1λ13[1−K1+K124]+β1K14λ12[4−4K1+K12][1+Coth(λ1)−4Sinh(λ1)] +β1K1λ1[−3−Coth(λ1)+3K1+K1Coth(λ1)−34K12−K12Coth(λ1)4] (29)+β1K1λ1[−3−Coth(λ1)+3K1+K1Coth(λ1)−34K12−K12Coth(λ1)4] F1=β1[−4+4K1−K12+43α1−23K1α1]+α12β3[−43+K12(1−Sinh(2λ1))2Sinh2(λ1)]+α1β1K1λ1[2−K1] (30)−2α12β3K1Cosh(λ1)λ1[1+Coth(λ1)]+α1β1K1Tanh(λ1)λ12[K1−2]+2α12β3K1eλ1λ12 (31)F2=β1{2−K1−23α1+K1α1λ12[−λ1+Tanh(λ12)]} F3=α1β3K12Sinh2(λ1)[1−Sinh(2λ1)][K1−2]+[−β1K1λ12+β1K1Tanh(λ12)λ12][4−4K1+K12]−2α12β3K1Cosh(λ1)λ1[1+Coth(λ1)]+β~02β3K12Sinh2(λ1)[4−4K1+K12][1−Sinh(2λ1)] (32)+β~02β3K12Sinh2(λ1)[4−4K1+K12][1−Sinh(2λ1)] (33)F4=K1α1β1λ12[−λ1+Tanh(λ12)][K1−2] (i)

For the Newtonian fluid (K1=0), Eq. (19) reduces to

It coincides with [10], as attempted for the Newtonian fluid. Further if angle of inclination is zero that implies α1=0 then Eqs. (19) and (34) yield (35)u={1 if β~0=01−β~0K1λ1(e2λ1−1)[e2λ1−λ1y+eλ1y−e2λ1−1] if β~0≠0where λ1=(2−K1)K2. It can be concluded that umicropolar=uNewtonian=1 if α1=0 with β~0=0. (ii)

The fluid speed u decreases with the rise in the micropolarity K1 because of the increase in the colloids of micropolar fluid.

Since there are no aciculate particles that collide in the Newtonian fluid (K1=0), the spin S=0 for it is as depicted in Fig. 4. The spin S lives with the micropolar fluid (K1≠0) merely, and it controls the microrotation of aciculate particles as they collide about their centroids. Fig. 4 is plotted on the basis of Eq. (20). It shows that the spin S depends upon the micropolarity parameter K1, the flow parameter α1, and the spin boundary parameter β~0. The dominancy of rise in the spin is observed in the middle of the substrate because the spin of aciculate particles is linearly proportional to the velocity gradient.

Fig. 5 indicates that the spin rises with the micropolarity parameter K1 for all values of the flow parameter α1 and spin boundary parameter β~0. Further, the spin is also enhanced with the spin boundary parameter β~0. The increase in the spin is also observed as the flow parameter α1increases for all values of the spin boundary parameter β~0 and micropolarity parameter K1.

According to Eq. (21), the temperature θ(y) majorly depends upon the flow parameter α1, the micropolarity parameter K1, the viscous dissipation parameter β3, and the spin boundary parameter β~0 for the micropolar fluid. The variation of the temperature with respect to the aforementioned parameters is depicted in Fig. 6. Fig. 6c is plotted for comparison purposes for the Newtonian fluid (K1=0). The following can be derived when K1=0 is substituted in Eq. (21):

(36)θ(y)_|(K1=0,β3=0)=[−y+y22+(13y−16y3+124y4)α1]β1

Eq. (36) and Fig. 6c both align with Eq. (12) and Fig. 3a of [10]. Figs. 6a and 6b are plotted for micropolar fluid. These figures show that (i) the temperature increases across the channel as the micropolarity increases, due to the collides in the fluid, the thermal flux rises; (ii) the temperature for micropolar fluid is greater than the temperature for the Newtonian fluid consistently by the increase in the kinetic energy of the aciculate particles in the micropolar fluid; (iii) the temperature θ increases with spin boundary parameter β~0 because spin boundary condition reduces the thermal transfer rate; (iv) in contrast, the temperature θ decreases with flow parameter α1 for both micropolar and Newtonian fluids as it is expected due to the rise in the gravitational force; (v) θ_|Micropolar=θ_|Newtonian=β1[y22−y] for α1=0 with β~0=0=β3 for all points of the channel; (vi) If α1=0 with β~0≠0,β3≠0 then the expression for temperature for micropolar fluid will differ slightly as: (37)4θ(y,α1=0)=−2(y−2){2β1−K12β~02β3sinh2[λ1]}−4yK1β~0β1λ1−2K12β~02β3(−1+e2λ1)2λ1{1+e4λ1−e−2(y−2)λ1−e2yλ1}+2(coth[λ1]−1)K1β~0β1λ13{−1+e2λ1−e−(y−2)λ1+eyλ1−2yλ1eλ1}+2(coth[λ1]−1)K1β~0β12λ13{1−2y2λ12+e2yλ1(−1+2yλ1)}