For a wide range of miniature devices, including sensors, actuators, and microfluidic systems, the performance and dependability of controlling the interfacial balance of fluids within micro-porous MEMS systems is crucial. In such restricted spaces, fluid conduct is dominated by capillary forces, floor tension, and viscous interactions because of the high floor-area-to-volume ratio. Complex interfacial dynamics are introduced by the presence of microporous materials, especially when multiple immiscible fluids or viscoelastic media are involved. Avoiding interruptions like fingering instabilities, bubble entrapment, or channel blockage—which can impair device performance or lead to failure—requires maintaining a stable interface. According to research, controlling fluid interface behavior and decreasing instabilities in micro-structured domains may be possible by adjusting elements such as pore geometry, surface wettability, electric fields, and external vibrations [1–6]. The use of electrowetting, surface patterning, and dielectrophoretic control to dynamically tune interfacial anxiety and enhance stability in MEMS-based completely microfluidic channels has been investigated in recent research [7, 8]. These discoveries are critical to developing precision fluid manipulation in MEMS technologies used in power microstructures, lab-on-a-chip devices, and biological diagnostics.

Tian and colleagues were the first to incorporate fractal geometry into MEMS design, creating fractally modified models including graphene-based parallel plate systems. Their research demonstrated that the fractal dimension of porous media affects pull-in stability and can be adjusted, establishing a strong basis for fractal analysis in micro porous MEMS fluid and electro-mechanical interactions [9]. The min-review article explores periodic properties of micro-electro-mechanical systems using various methods, introduces fractal MEMS systems, and discusses future prospects, focusing on recent developments [10]. The examines the static and dynamic behavior of graphene cantilever beam resonators under electrostatic actuation. It presents a nonlinear static problem solution and calculates the generalized stiffness coefficient for a lumped cantilever model under tip loading. The study focuses on the dynamic pull-in phenomenon and the influence of excitation frequency on the system’s dynamic response, emphasizing the importance of frequency selection in designing stable graphene-based MEMS resonators [11].

Bingham fluids (BF), a class of viscoelastic materials, exhibit yield stress behavior, meaning they behave as solids until a critical stress is exceeded, and flow like viscous fluids [12, 13]. The Bingham model is characterized by two main factors: yield shear stress and viscosity. When the yield stress is exceeded, the material shows quasi-Newtonian flow; nevertheless, when the deviatoric stress tensor magnitude is smaller than the yield stress, the material remains rigid. The yield stress minus the applied shear stress dictates the deformation rate in thin-film flow. Bird et al. [14] categorized materials exhibiting yield stress and provided preliminary characterizations in fundamental flow fields. Bingham fluids are widely used in engineering and geophysical applications because of their specific properties, including drilling muds, suspensions, and biological fluids [15].

Bingham fluid flow stability, particularly in porous media, is crucial for industrial operations such as enhanced oil recovery and groundwater remediation [16, 17]. External influences, such as magnetic fields or pressure gradients, can destabilize the interface of immiscible Bingham fluids, resulting in complex interfacial dynamics [18]. The combination of viscoelasticity, porosity, and surface tension significantly impacts the initiation and evolution of instabilities in these fluids [19]. Integrating Bingham fluid dynamics with magnetohydrodynamics (MHD) yields new insights into the behavior of viscoelastic fluids under magnetic fields, with important implications for industrial and biological applications [20]. Bingham fluids have been extensively studied in particle physics due to their unique viscoelastic character, which is beneficial in both natural and industrial applications. Real-world fluids include blood, filth, ice, lubricating oil, fresh concrete, polymers, and paint. They are further classified into two types: Bingham plastic and Bingham pseudoplastic fluids, which are non-Newtonian but exhibit differing yield stress features.

A physically consistent particle-based BF simulation method was developed to understand BF dynamics better [21]. In addition, mathematical modeling has been applied to study BF flow in porous media using a multi-membrane pumping mechanism [22]. More research investigated the ion-slip effects in MHD events when Bingham fluids flowed between two porous plates under suction conditions [23].

The stability of Bingham fluid flow has been investigated in several scenarios. For example, stability criteria in laminar Bingham-Poiseuille flows have been examined, notably in the case of fluid sheets descending sloping planes [24]. Studies on oblique channel flows have demonstrated that the Bingham parameter has a stabilizing influence on liquid motion. Furthermore, a mathematical model was developed to characterize different BF inputs within a channel [25].

Yield stress fluids interact with porous media, increasing complexity due to nonlinear rheology and medium heterogeneity [26]. Avalanches were triggered at one end of the system to examine the statistical properties of non-flowing surfaces, emphasizing the yield stress and plastic viscosity of Bingham fluids. Furthermore, numerical solutions have been developed to calculate velocity and temperature fields in time-varying Couette-Poiseuille flows of Bingham fluids [27, 28].

Electromagnetic effects have also been investigated in BF research, specifically the role of the Hall current, which is influenced by electron transitions between Landau levels induced by an electromagnetic wave’s electric and magnetic fields [29]. Researchers have also looked at the transport of viscoelastic liquids across uneven microchannels, considering viscosity changes and porous media [30]. Further study has focused on viscoelastic liquids’ energy transport characteristics and inflow behavior, providing insights into their complex dynamics in constrained spaces [31].

Fractal analysis in fluid mechanics has been utilized to understand and study various natural systems, including fluid mechanics and geophysical geometrical formations [32]. Because of the complex and frequently aberrant behaviors found in such environments, research into fluid features inside fractal spaces has received a lot of attention [33]. Fractal spaces, with non-integer dimensions and self-similar features, have different fluid dynamics than regular Euclidean spaces [34]. Fractal geometries exhibit anomalous diffusion, which occurs when particles’ mean squared displacement (MSD) deviates from the linear dependence observed in normal diffusion. Fractals’ complicated and convoluted paths impede or facilitate fluid movement in non-uniform ways. Studies have shown that the medium’s fractal structure in percolation clusters significantly impacts diffusion rates, resulting in sub-diffusive behavior. Restricted spatial characteristics in fractal environments reduce molecular mobility, resulting in lower MSD values compared to unconstrained situations. This behavior is commonly observed in several systems, including biological cell plasma membranes, where anomalous diffusion is used to study membrane architecture [35].

Furthermore, fractal dimensionality affects energy dissipation rates in fluid flows, influencing stability and turbulence characteristics. Research on the thermodynamics and pair correlations of fractal liquids shows that the non-Euclidean structure necessitates new theoretical approaches for accurate exploration. These findings are essential in disciplines like porous media flow, where fractal geometry influences transport and mechanical properties. Recent studies have investigated engineered fractal fluid transport systems for several applications [36, 37]. Feng’s study develops a two-scale fractal-fractional oscillator model for porous media vibration systems, using He’s frequency formula and Ma’s modification. The model reveals the fractal dimension significantly influences attenuation behavior, demonstrating the versatility of fractal-based analytical techniques in dynamic and structural systems [38].

The two-scale fractal theory to calculate the fractal dimensions of porous concrete, focusing on the effects of porosity and pore size on strength. It proposes mathematically reliable formulations and dimensionless models for concrete properties. The theory also uses nano/micro particles’ size and distribution for strength prediction, providing new insights into optimal concrete design [39].

A fractal pore-scale model is used to study fluid flow, heat conduction, and gas diffusion through saturated porous material. The results show strong correlations between conductivity properties and pore structure changes. This method provides insights into transport processes, oil and gas resources, energy storage, carbon dioxide sequestration, and fuel cell applications [40]. Using box-counting techniques to analyze aggregate size distributions and determine fractal gradation, Gao et al. [41] showed that the fractal dimension of recycled concrete aggregate significantly influences key mechanical properties—such as workability, strength, and durability.

Experimental two-phase invasion percolation flow patterns were observed in hydrophobic micro-porous networks designed to model fuel cell-specific porous media. The inlet channels were invaded homogeneously, and fractal breakthrough patterns were analyzed to quantify flooding density and geometrical diversity. Fractal analysis confirmed that the experiments fall within the flow regime of invasion percolation with trapping. The fractal dimension, D, was proposed as a parameter for modeling liquid water transport in the GDL [42].

A modified Lindstedt-Poincaré transformation is used in this study to evaluate the interfacial stability of MHD Bingham fluids in fractal space. This study expands on standard stability analysis by using fractal behavior to comprehend better the complex dynamics of fluids in porous and irregular media. The study focuses on developing a nonlinear governing equation for interfacial displacement and translating it to a more manageable form using advanced linearization techniques. The modified Lindstedt-Poincaré transformation is employed in this study to create the system’s fractal behavior and to offer a more explicit specification of the stability conditions that govern the interface. The effect of fractal parameters, magnetic fields, and viscoelastic properties on stability and frequency dynamics is thoroughly studied. This technique advances our theoretical understanding of interfacial dynamics in MHD Bingham fluids while also providing insights into engineering, geophysics, and industrial processes in which non-Newtonian fluids interact with electromagnetic fields in complex situations.

A planar contact separating two immiscible Bingham fluids in a permeable medium will be investigated to understand our study’s physical model better. The fluids are entirely saturated in a fractal porous structure and inhabit the regions y < 0 and y > 0. These fluids are subjected to external magnetic fields, as illustrated in Figure 1. A uniform magnetic field is applied perpendicular to the interface, expressed as B ̲ j = 0 , ε j H 0 j , where j = 1,2, and ε represents the magnetic permeability, B is the magnetic induction vector, and H 0 represents the magnetic field. Given the assumption of a small magnetic Reynolds number, the induced magnetic field (MF) is neglected, indicating that fluid movement has minimal influence on the applied MF. The Bingham BF exhibit yield stress behavior, meaning they behave as solids under low stress and transition to viscous flow once the applied stress surpasses a critical threshold. The porous medium follows a fractal geometry, where porosity and permeability depend on scale. The upper and lower fluids may possess distinct rheological and magnetic properties, which directly impact system stability. Fluid movement in the porous medium is governed by Darcy’s law, which is particularly relevant in slow, creeping flow conditions. This empirical law establishes a relationship between the liquid’s viscosity, the medium’s permeability, and the flow rate. Physically, Darcy’s law serves as a crucial link between liquid dynamics and physiological processes, where gravitational acceleration influences the system, represented as g ̲ = − g e ̲ y , with e ̲ y being the unit vector along the vertical -axis.

Interfacial instability can occur due to a variety of variables, including shear effects produced by velocity differences, magnetic field fluctuations that alter pressure distribution, and fractal effects that influence wave propagation. Capillary action and biological systems with fluid movement determined by viscosity and medium permeability are examples of such instabilities. In these instances, permeability is usually assumed to be constant for both non-Newtonian fluids, making the mathematical analysis easier. The study employs a two-dimensional Cartesian coordinate system (x,y), with the x-axis horizontally aligned between the two fluid layers and the y-axis vertically orientated. The fluids in the lower and top layers have different densities ρ 1 and ρ 2 and magnetic permeabilities μ 1 and μ 2 . The liquid layers are propelled in a positive x-direction stream by μ 1 and μ 2 , respectively.

Because of the minor disturbance for the equilibrium state, a little elevation for the flat interface in the direction of the vertical orientation is described by Equation 1, where t is the time and x represents the horizontal spatial coordinate along the interface. y = ζ x , t , (1)

Surface deflection refers to the rise or displacement of a contact from its initial equilibrium position. This function provides insights into the system’s stability by quantifying how an external disturbance affects the interface, causing it to deviate from its flat, undisturbed state. One approach to defining the increment function as in Equation 2 [33]: ζ x ; t = η t   cos ⁡ k x . (2)

The arbitrary function η t governs the amplitude of the interface disturbance, while k determines the spatial frequency of the disturbance (wavenumber). Studying this relationship enables an examination of the system’s response to perturbations, which is highly valuable for predicting interfacial behaviors in fluid dynamics and material science applications. Assume that the Dirichlet boundary condition stated below Equation 3: ζ 0 , t = η t , (3)

Considering there are two initial conditions (see Equation 4) for the function η t η 0 = A  and  η ˙ 0 = 0 (4) where A refers to the magnitude of the original disturbance. The normal mode technique displays the increase of a disturbance at the contact as Equation 5: ζ x , t = η t e i k x + c . c . (5)

The complex conjugate terms (c.c) are utilized in mathematical analysis to facilitate the study of interfacial responses in two-phase systems. They enable the use of complex exponentials to represent temporal and spatial variations in interface increments, simplifying differentiation and integration in stability analysis. This notation is commonly applied in fluid dynamics, wave phenomena, and stability analysis [33].

The formula S x , y ; t = y − ζ x ; t represents the interface equation. The unit outward normal perpendicular to the surface interface can be described as Equation 6 [43]: n ̲ = ∇ S ∇ S = − i k ζ e ̲ x + e ̲ y 1 − k 2 ζ 2 − 1 l 2 , (6)

Bingham fluids move according to the Bingham plastic model, which describes the connection between shear stress and shear rate in viscoplastic materials. According to this model, a Bingham fluid acts like a rigid body at low shear stresses but becomes a viscous fluid when the applied shear stress surpasses a particular yield stress. The constitutive equation for a Bingham fluid is written as Equation 7 [44]: ς = ς 0 + μ p γ ˙ , (7) where ς is the shear stress, ς 0 yield stress, μ p plastic viscosity, and γ ˙ shear rate. This equation states that the fluid will not deform (i.e., γ ˙ = 0) unless the applied shear stress ς exceeds the yield stress ς 0 . When this threshold is exceeded, the fluid flows with viscosity μ p . Materials that exhibit this tendency include toothpaste, mayonnaise, and certain drilling muds.

The objective at this stage is to analyze the system’s nonlinear characteristic equation, which provides insight into its stability and dynamic behavior. This characteristic equation is derived from Equation 34 and is expressed as: 1 + k ρ * ⁢ ζ + k 2 ⁢ ζ 2 ⁢ ζ t t + a 1 + i b 1 ⁢ 1 + k 2 ⁢ ζ 2 + a 2 + i b 2 ⁢ ζ ⁢ ζ t + a 3 + i b 3 ⁢ ζ + a 4 + i b 4 ⁢ ζ 2 + a 5 + i b 5 ⁢ ζ 3 = 0 , (35)

The coefficients a j  and  b j have been expressed in non-dimensional form and are listed in the Appendix.

Rewriting complex coefficients in polar form provides a more intuitive framework for analyzing oscillations, phase correlations, and stability in systems like those described in Equation 35. Consequently, the complex coefficients in Equation 35 can be expressed in polar form as represents in Equations 36, 37 a j + i   b j = r j   e i   θ j , (36) where the phase θ j and amplitude a j are defined as r j = a j 2 + b j 2 and θ j = tan − 1 b j a j (37)

Thus, it becomes essential to represent the complex coefficients in Equation 35 in a more descriptive form. This allows for a clearer understanding of their behavior and facilitates the analysis of their contribution to the system’s dynamics: 1 + k ρ * ⁢ ζ + k 2 ⁢ ζ 2 ⁢ ξ t t + 1 + k 2 ⁢ ζ 2 ⁢ r 1 e i θ 1 + r 2 e i θ 2 ⁢ ζ ⁢ ζ t + r 3 ⁢ e i θ 3 ⁢ ζ + r 4 e i θ 4 ⁢ ζ 2 + r 5 e i θ 5 ⁢ ζ 3 = 0 . (38)

When the complex conjugate of Equation 38 is added to the equation itself, the real part of the coefficients is preserved, while the imaginary part is eliminated. This process ensures that the coefficients become purely real, thereby simplifying the equation’s structure and making it more straightforward to interpret and solve. 1 + k ρ * ⁢ ζ + k 2 ⁢ ζ 2 ⁢ ζ t t + 1 + k 2 ⁢ ζ 2 ⁢ r 1 ⁡ cos θ 1 + ζ r 2 ⁡ cos θ 2 ⁢ ζ t + r 3 ⁡ cos θ 3 ⁢ ζ + r 4 ⁡ cos θ 4 ⁢ ζ 2 + r 5 ⁡ cos θ 5 ⁢ ζ 3 = 0 . (39)

Because this equation is a partial differential equation involving two independent variables, x, and t, but contains derivatives only concerning t, it is advantageous to rewrite Equation 39 using the boundary condition (3). This reformulation simplifies the equation as follows: 1 + k ρ * ⁢ η + k 2 ⁢ η 2 ⁢ η ¨ + 1 + k 2 ⁢ η 2 ⁢ r 1 ⁡ cos θ 1 + η r 2 ⁡ cos θ 2 ⁢ η ˙ + r 3 ⁡ cos θ 3 ⁢ η + r 4 ⁡ cos θ 4 ⁢ η 2 + r 5 ⁡ cos θ 5 ⁢ η 3 = 0 . (40)

The simplified Equation 40 is a nonlinear ordinary differential equation involving a single variable. Its structure resembles that of a Van der Pol equation, characterized by its nonlinearity.

To adapt this nonlinear equation to fractal space characteristics, we assume that Equation 40 possesses a total frequency Ω. Utilizing this assumption, we employ the following definition in Equation 41 [47]: τ α = Ω t , (41) where α is a fraction power defined as 0 < α ≤ 1 . For the specific case of α → 1 the classical Lindstedt-Poincaré transformation arises. As a result, the following transformations are necessary:

Accordingly, the definition of the fraction power (α), the following transformations are required. η ˙ t = ∂ η ∂ τ α ∂ τ α ∂ t → Ω d η d τ α , (42)

Consequently, we have η ¨ t → Ω 2 d 2 η d τ 2 α . (43)

In the situation of α→1, the classical Lindstedt-Poincaré transformation is applicable. Assuming that the derivative concerning the new variable τ α follows He’s fractal derivative formulation [46, 48, 49], this generalization extends the classical derivative to incorporate non-local or fractal-like characteristics. He’s fractal derivative approach allows the fractional derivative of order α\alphaα to capture complex multi-scale processes, such as memory effects or anomalous diffusion, which cannot be adequately described using classical calculus. As a specialized form of fractional derivative, He’s fractal derivative is frequently applied to dynamic systems exhibiting fractal properties. The derivative can take on non-integer orders (α), providing a framework that bridges the gap between purely local (integer-order) and non-local (fractional-order) dynamics.

As a result of transformations Equations 42, 43, Equation 40 takes on the fractal derivative form as Ω 2 1 + k ρ * η + k 2 η 2 d 2 η d τ 2 α + Ω r 1 ⁡ cos θ 1 1 + k 2 η 2 + η r 2 ⁡ cos θ 2 d η d τ α + r 3 ⁡ cos θ 3 η + r 4 ⁡ cos θ 4 η 2 + r 5 ⁡ cos θ 5 η 3 = 0   . (44)

When the transformation in Equation 42 is applied to the initial conditions given by Equation 8, the corresponding fractal initial conditions can be expressed as η 0 = A and d η 0 d τ α = 0 . (45)

Fractal derivatives offer a versatile mathematical framework applicable to a wide range of real-world phenomena, including astronomical events, geophysical fluxes, plasma physics, and industrial processes like inertial confinement fusion. These derivatives are particularly valuable for modeling systems that exhibit memory effects, anomalous diffusion, or multi-scale behaviors—dynamics that traditional integer-order derivatives fail to capture effectively.

Several graphs are plotted for the transition curve Equation 67 to illustrate the impact of the fractal dimensional parameter α and other key physical parameters on the system’s stability behavior. The graphs represent the transition curve that separates the stable state from the unstable one. These visual representations help analyze how variations in these parameters influence stability transitions and dynamic responses. The dimensionless numbers for this figure were chosen as follows: A = 1 , ρ = 1.2 , μ = 0.1 , U = 0.1 , ε = 0.01 , H = 10 , H a = 3 , Z = 0.01 , W e = 0.5 , D a = 0.01 , ς 0 = 0.1 , B d = 0.01 , B g = 0.01

Figure 2 illustrates the stability plane Ω 2 − k along with a graph of frequency Ω 2 versus wavenumber k. The stability behavior varies with the fractal dimension α , where the stable region is highlighted, and the unstable region remains unshaded. It is observed that the stable region lies above the transition curve, while the unstable region is positioned below it. The graph of frequency Ω 2 against k shows that as k increases, the transition curve shifts higher, taking the shape of a negative exponential function and expanding the unstable region. Additionally, a decrease in the fractal order α leads to a reduction in the stable area. This effect is more pronounced for very small values of k, where the stability zone significantly shrinks as α decreases. This behavior indicates that an increase in the wavenumber k enhances instability, while a decrease in α also has a destabilizing effect on the system.

Figure 3 illustrates the transition curve Equation 67 in the stability plane Ω 2 − α while analyzing variations in the wavenumber k for the same system considered in Figure 2. It is observed that as k increases, the unstable region expands, particularly in areas corresponding to small values of the fractal parameter alpha. This indicates that higher wavenumbers k contributes to greater instability, with the effect being more pronounced when is small. Additionally, larger values of k are associated with smaller values of alpha, leading to a more unstable response in the system. However, for higher values of alpha, the destabilizing influence of increasing k is minimal. This implies that the fractal parameter has a substantial function in determining stability, with its significance decreasing as it increases.

Figure 4 depicts an examination of the influence of the magnetic Bond number H on the stability behavior for the same system seen in Figure 2, with the wavenumber set to k = 2. The results show that raising H changes the stability behavior, such as wavenumber k, resulting in a destabilizing effect in the system. Specifically, when H increases, the unstable region widens, with the effect being most noticeable for small values of the fractal parameter α . This shows that the magnetic field has a major destabilizing effect in regimes with low fractal dimensions. For larger α values, H has a limited impact on stability, causing only slight destabilization. This means that, while the magnetic Bond number is important in stability management, its effects reduce as the fractal parameter grows, emphasizing the interaction of magnetic forces and fractal dimension in regulating system behavior.

Figure 5 depicts a study of the impact of the Hartmann number Ha on the stability plane for the identical system as in Figure 4, with the magnetic Bond number set to H = 10. This evaluation reveals how changes affect the system’s stability characteristics. An analysis of the graph shows that increasing leads the transition curve to shift downward, thereby expanding the stable region. This downward movement implies a stabilizing impact, allowing more of the system to remain stable. However, for very small values of the fractal parameter alpha, the stability region experiences only a slight reduction in its stabilizing influence. As increases, the stabilizing effect gradually diminishes, indicating that the impact of the Hartmann number is more pronounced in low-fractal-dimension regimes. Furthermore, for higher values of alpha, the stabilizing effect of Ha becomes minimal, suggesting that at larger fractal dimensions, the influence of the Hartmann number on system stability is less significant. This behavior highlights the complex interaction between magnetic forces and fractal properties in determining the stability characteristics of the system.

The graph in Figure 6 illustrates the effect of the Ohnesorge number Z on the stability plane Ω 2 − α . The dimensionless parameters used for this figure are identical to those in Figure 5, except that the Hartmann number is fixed at Ha = 30. This analysis provides insights into how variations in Z influence system stability. The stability behavior observed in this graph closely resembles the effects of the Hartmann number presented in Figure 5. Specifically, an increase in Z leads to a stabilizing effect, similar to what is observed with increasing Ha. As increases, the transition curve shifts downward, expanding the stable region and reinforcing the system’s resistance to perturbations. This suggests that the Ohnesorge number plays a crucial role in reducing instability by modifying the fluid’s response to external forces. Overall, the results indicate that, like the Hartmann number, the Ohnesorge number enhances stability by suppressing disturbances, making it an essential parameter in determining the behavior of the system under varying conditions.

Figure 7 shows the fractal time history, showing the fractal solution Equation 60 against variations in the fractal dimensional parameter α. This graph is a schematic representation of the same system as in Figure 2, providing insight into the system’s temporal evolution under different fractal settings. The presence of a dampening feature is one of the most noticeable aspects of this image. The damping effect gets increasingly significant as the fractal parameter increases, showing that the system suppresses oscillatory instabilities more effectively over time. This shows that greater values help to improve stability by lowering fluctuations and mitigating instability. Increasing α has a stabilizing effect, as higher fractal dimensions result in more regulated and damped system responses. As a result, it not only reduces instability but also maintains overall system stability, making it an important parameter in managing the system’s dynamic behavior throughout time.

Figure 8 displays the fractal time history, demonstrating how changes in the magnetic Bond number influence system behavior with the wavenumber set to k = 3. Similarly, Figure 9 shows how the wavenumber grows while the magnetic Bond number remains constant at H = 1. These computations are for the same system as seen in Figure 7, but the fractal parameter is kept constant at α = 0.1. These results provide an important finding about the influence of increasing on the system’s dynamic response. As either value grows, the cycle rate lowers, indicating that the system is becoming unstable. This implies that higher values of or contribute to a decrease in oscillation frequency, resulting in more extreme instability. The decrease in cycle rate demonstrates how the magnetic Bond number and wavenumber influence the system’s temporal evolution, emphasizing their importance in influencing stable behavior under fractal settings.

This study employs sophisticated nonlinear analysis techniques using Fractal analysis to control Interfacial stability of MHD Bingham Fluids in Micro-Porous MEMS Structures. The Harmonic Equivalent Linearization Method (HELM) is an important methodological tool used in this study. It simplifies the analysis by translating the nonlinear dynamical characteristic equation into an equivalent linear form. This transformation improves the analytical analysis and solution to the stability problem, making it easier to anticipate system behavior under various parametric conditions. The governing equations are stated in a fractal framework using the modified Lindstedt-Poincaré transformation, which is critical for capturing Bingham fluids’ complicated interfacial stability properties. This work incorporates fractal adjustments to account for the impacts of non-integer dimensions, providing a more accurate description of the physical system, particularly in porous and uneven media. The fractal framework provides more insight into how microstructural differences affect fluid stability, which is critical for real-world applications involving geometric complexity and diverse structures.

Several major observations emerge from the numerical and analytical findings:

1. Increasing the fractal dimension α improves system stability by adding stronger damping effects, minimizing instability, and maintaining interface stability over time.

The fractal time history study shows that higher alpha values result in stronger damping effects, which increase the system’s robustness to perturbations. The combination of magnetic, hydrodynamic, and fractal properties facilitates the intricate interplay of forces that influences stability behavior in MHD Bingham fluids. Overall, this research advances our understanding of nonlinear stability in viscoelastic fluids under fractal effects and magnetic fields. The findings are relevant to industrial and geophysical applications, particularly those requiring interfacial stability, such as fluid flow in porous media, enhanced oil recovery, and biomedical engineering. Future research could extend these findings by looking into parametric effects and the nonlinear Mathieu equation contributions. Our study develops nonlinear, fractal, and MHD-based modeling techniques for MEMS structures, addressing electrostatic interactions, geometric nonlinearity, and scale-dependent effects. This methodology will be validated and expanded for microscale technologies.