American Journal of Computer Science and Mathematics
Viscoelastic and Fractional Second Grade Fluid Dynamics in Stretching Surface and Microfluidic Flow Systems: A Unified Continuum Framework for Memory, Elasticity, and Magnetohydrodynamic Effects
The study of viscoelastic and second grade fluids has become a cornerstone of modern continuum mechanics due to its immense relevance in polymer processing, biological transport, microfluidics, and magnetohydrodynamic technologies. Unlike Newtonian fluids, viscoelastic fluids exhibit time dependent memory and elastic recoil, enabling them to store and release mechanical energy in ways that fundamentally alter flow patterns, boundary layer behavior, and particle transport. This work presents a comprehensive theoretical and analytical synthesis of second grade and fractional second grade fluid dynamics in the context of stretching surfaces, microfluidic channels, porous media, and magnetohydrodynamic environments. By integrating classical second grade models with fractional calculus based memory representations, this study constructs a unified continuum description capable of representing both short term elasticity and long term hereditary behavior. The analysis is grounded exclusively in authoritative research spanning microfluidic filament dynamics, propulsion in viscoelastic media, particle migration, MHD effects, and fractional derivative modeling. Emphasis is placed on how viscoelastic stresses modify momentum diffusion, interfacial stability, boundary layer formation, and energy transport across a wide range of physical configurations. The theoretical framework is expanded to cover unsteady shear flows, porous substrates, thermal gradients, slip boundary effects, and nonuniform magnetic fields, enabling a multidimensional perspective on non Newtonian transport. A detailed interpretation is provided for how second grade and fractional order effects generate flow instabilities, modify wave propagation, alter drag and propulsion mechanisms, and influence heat and mass transfer in stretching sheet and channel systems. The results demonstrate that the coupling between elastic stress relaxation, fractional memory, and magnetic damping leads to flow regimes that cannot be captured by conventional Newtonian or integer order models. These regimes include delayed boundary layer development, enhanced particle migration, non monotonic velocity profiles, and memory induced damping or amplification of disturbances. The work further establishes that fractional order parameters act as physically meaningful measures of fluid memory, enabling precise tuning of transport behavior in microfluidic and industrial applications. The article concludes by identifying how these models provide a theoretical foundation for next generation polymer processing, biofluid transport, and MHD based microdevices, while also addressing their limitations and future extensions.
Copyright (c) 2026 American Journal of Computer Science and Mathematics
Transient Magnetohydrodynamic and Thermal Transport Behavior of Viscoelastic Second Grade Fluids in Porous and Stretching Flow Regimes
The study of viscoelastic and non Newtonian fluids has remained one of the most intellectually demanding and practically relevant areas of fluid mechanics for several decades because of the intrinsic coupling between fluid microstructure, memory effects, and macroscopic transport phenomena. Among the various constitutive models developed to describe such materials, the second grade fluid model has been especially significant due to its capacity to represent normal stress differences and elastic effects while still allowing mathematically tractable analysis. In many engineering and geophysical applications, these fluids flow through porous structures, interact with magnetic fields, and experience strong thermal gradients, making the problem even more complex and multidimensional. This research presents a comprehensive theoretical and analytical investigation of transient and steady flow behavior of second grade viscoelastic fluids under magnetohydrodynamic influence, thermal radiation, and porous medium effects in configurations such as stretching sheets, oscillating and impulsively started plates, and stagnation point flows. Drawing strictly on established developments from prior literature, the present article integrates classical boundary layer theory, transient Stokes problems, and modern non Newtonian modeling to develop a unified framework for understanding how viscoelasticity, magnetic forces, thermal diffusion, and porous resistance jointly determine velocity, temperature, and stress fields. The analysis is deeply grounded in earlier foundational work on impulsive and oscillatory plate motions in viscous fluids, as well as later extensions to second grade fluids, thermal transport, and porous media. Particular attention is devoted to the subtle interplay between elastic memory and diffusive momentum transport, which leads to markedly different transient responses compared with Newtonian fluids. The role of thermal radiation, variable thermal conductivity, viscous dissipation, and convective boundary conditions is examined in detail to reveal how heat transfer mechanisms are altered by fluid rheology and magnetic damping. By synthesizing a large body of prior results, this study demonstrates that second grade fluids exhibit richer and more sensitive transient and steady behavior than classical viscous fluids, especially in magnetohydrodynamic and porous environments. Velocity overshoots, delayed boundary layer development, enhanced or suppressed heat transfer, and strong coupling between flow and thermal fields emerge as robust qualitative features. The results also highlight how stretching surfaces, peristaltic motion, and natural convection geometries further amplify or modify these effects. The work provides a coherent theoretical foundation that can be used by engineers and applied scientists to predict, control, and optimize transport processes involving viscoelastic fluids in industrial, biomedical, and geophysical systems.
Thermal and Flow Dynamics of Viscoelastic and Non Newtonian Fluids in Porous and Stretching Sheet Environments Under Radiative and Magnetohydrodynamic Influences
The transport of heat and momentum in non Newtonian and viscoelastic fluids has become one of the most intellectually rich and technologically relevant areas in modern fluid mechanics. This is largely because a vast range of industrial, geophysical, and biomedical processes involve fluids whose rheological behavior deviates significantly from the Newtonian assumption. Polymer melts, biological fluids, lubricants, slurries, and suspensions exhibit elastic memory, shear dependent viscosity, and stress relaxation effects that profoundly modify their flow and heat transfer characteristics. When such fluids interact with porous structures, oscillating boundaries, stretching sheets, magnetic fields, radiation, or temperature dependent material properties, the resulting transport processes become strongly coupled, nonlinear, and sensitive to multiple physical mechanisms. The present article develops a unified theoretical and interpretive framework for understanding these interactions by synthesizing and critically extending the body of knowledge represented in the provided references. The central objective of this work is to examine how viscoelasticity, micropolarity, second grade and Maxwell type constitutive behaviors interact with porous media, radiative heat transfer, thermal conductivity variations, magnetic forces, and moving or stretching boundaries. Drawing strictly from the cited literature, the article reconstructs the intellectual evolution of these models from early studies on Stokes oscillatory flow and plane wall motion to modern investigations of magnetohydrodynamic and radiative flows over stretching sheets and through porous substrates. The methodological emphasis is placed on analytical and semi analytical solution strategies, including exact solutions, similarity transformations, perturbation approaches, and approximate analytical schemes such as homotopy based methods. Rather than reproducing equations, the study provides detailed conceptual explanations of how these techniques operate and what their physical implications are. The results discussed reveal that non Newtonian rheology fundamentally alters velocity distributions, thermal boundary layer thickness, wall shear stresses, and heat transfer rates. Elastic memory tends to delay momentum diffusion, while second grade effects can either enhance or suppress flow depending on the competition between normal stress differences and viscous dissipation. Variable thermal conductivity introduces strong nonlinear coupling between temperature and velocity fields, while thermal radiation acts as an effective heat source that thickens thermal boundary layers. Magnetic fields generate Lorentz forces that dampen motion and convert kinetic energy into thermal energy, thus modifying both flow resistance and temperature profiles. Porous media impose additional drag and alter buoyancy driven convection, particularly when combined with non Newtonian stress responses.
Stability Preserving Differential Transformation and Multistage Numerical Frameworks for Stiff and Nonlinear Dynamical Systems in Mathematical Physics and Population Modeling
This research article presents a comprehensive theoretical and methodological synthesis of stability preserving numerical frameworks designed for stiff, nonlinear, and oscillatory dynamical systems that emerge across applied mathematics, physics, and biological modeling. Drawing exclusively upon the corpus of references provided, the study integrates the conceptual foundations of differential transformation methods, multistage numerical algorithms, Runge Kutta schemes, Padé approximation, and stability preserving solvers to establish a unified interpretive and computational framework. The increasing complexity of modern mathematical models in plasma physics, chemical reaction kinetics, stem cell therapy dynamics, nonlinear oscillators, and infectious disease modeling has produced a critical need for numerical solvers that maintain accuracy, stability, and long time reliability. Classical numerical techniques often encounter severe limitations when applied to stiff systems, strongly nonlinear oscillators, or chaotic regimes, particularly in the presence of sharp transients or sensitive dependence on initial conditions. In response to this challenge, contemporary numerical research has developed advanced strategies that blend analytical transformations with multistage discretization techniques, ensuring robust convergence even in the most demanding mathematical environments. The differential transformation method has emerged as a powerful semi analytical framework for constructing approximate solutions to differential equations by transforming them into recurrence relations that can be computed iteratively. However, as shown by Bervillier in his extensive theoretical assessment of the method, its naive application may suffer from divergence, slow convergence, or sensitivity to truncation in stiff regimes. These limitations have stimulated the development of multistage differential transformation approaches, as proposed by Do and Jang and further refined by Aljahdaly and Ashi, in which the solution interval is subdivided into multiple segments, each of which applies the differential transformation locally to preserve accuracy and stability. When combined with Padé approximants, as demonstrated by Domairry and Hatami, the resulting hybrid frameworks significantly improve convergence behavior and enable the resolution of nonlinear boundary value problems and nanofluid flow systems with strong coupling effects.
Copyright (c) 2025 American Journal of Computer Science and Mathematics
Spectral and Transform Based Numerical Frameworks for Nonlinear, Fractional, and Epidemic Differential Systems
This study presents a comprehensive and integrative examination of modern numerical frameworks for solving ordinary, partial, fractional, and epidemiological differential systems using spectral and differential transform based methodologies. Drawing strictly on the referenced body of work, the article investigates how polynomial based spectral schemes, differential transformation methods, and adaptive time stepping strategies collectively form a powerful computational ecosystem for addressing nonlinear and complex dynamical models. Differential equations are fundamental tools for describing heat conduction, epidemic propagation, biological interactions, and wave phenomena, yet the inherent nonlinearity, fractional order behavior, and multi dimensionality of many real world systems continue to challenge traditional numerical solvers. As demonstrated in computational immunology models, ordinary differential equations provide a flexible representation of interacting biological components, but their accuracy depends heavily on the stability and convergence of the underlying numerical method used for simulation (Hoops et al., 2016). Spectral based schemes, including Gegenbauer, Lucas, Legendre, and Chebyshev polynomial families, have emerged as high precision approximators that can dramatically reduce computational cost while maintaining global accuracy for smooth solutions (Sayed et al., 2024; Youssri et al., 2023). In parallel, differential transformation methods have proven to be efficient semi analytical techniques that convert differential models into algebraic recurrence systems, enabling the study of nonlinear epidemic, chaotic, and damped wave models with strong stability and fast convergence (Ahmad et al., 2017; Odibat et al., 2010). This article synthesizes these two dominant traditions into a unified narrative that highlights their complementary strengths. By examining how polynomial based spectral algorithms manage spatial and fractional complexities while differential transforms manage temporal and nonlinear expansions, this study demonstrates how advanced numerical solvers can be structured to address the limitations of classical finite difference and Runge Kutta methods. The literature further shows that modified polynomial bases and alleviated spectral shifts enhance numerical conditioning, making these approaches suitable for boundary value problems, time fractional equations, and coupled nonlinear systems (Sayed et al., 2024; Abd Elhameed et al., 2023). At the same time, multistep differential transform schemes overcome the local convergence limitation of classical transforms and provide reliable long time simulations for chaotic and epidemiological models (Odibat et al., 2010).
Rheological and Geometric Effects on Laminar Transport in Deforming and Constricted Flow Domains
The study of fluid motion in geometrically complex domains has been one of the most enduring and intellectually rich areas of classical and modern fluid mechanics. Constricted tubes, wavy channels, stretching sheets, and stretching cylinders all represent physical configurations that appear not only in industrial and technological processes but also in biological and biomedical systems such as blood flow in arteries, microcirculatory transport, polymer extrusion, coating flows, and magnetohydrodynamic devices. The present research develops an integrated and original synthesis of laminar flow through constricted and wavy geometries with the theory of non Newtonian and second grade fluids subjected to stretching and magnetic effects. The work is strictly grounded in the foundational and contemporary references provided, which span early biomechanical investigations of arterial flow, thermodynamic analyses of differential type fluids, and modern similarity and analytical treatments of stretching surface problems. By weaving together these traditionally separate strands of literature, this article establishes a unified conceptual and physical understanding of how geometry, rheology, and external fields interact to shape velocity distributions, stress development, heat transfer, and flow stability. The abstract highlights that laminar flow in constricted tubes is not merely a problem of geometric obstruction but a complex interplay between boundary layer separation, shear induced endothelial stress, and non Newtonian rheological response, as originally explored in biomedical contexts by Fox and Hugh, Fry, Forrester and Young, and Haldar. These insights are then extended by incorporating second grade and micropolar fluid models as reviewed by Dunn and Rajagopal and Dunn and Fosdick, allowing for the representation of elastic and memory effects that are essential for blood and polymeric fluids. The stretching sheet and stretching cylinder frameworks pioneered by Crane and elaborated by Ariel, Ishak, Hayat, Liao, and others are reinterpreted as idealized analogues of deforming vessel walls and industrial substrates, making it possible to bridge biomechanics and process engineering in a single theoretical narrative. Magnetohydrodynamic and radiative effects further enrich the model by capturing how electromagnetic forces and thermal gradients modify flow resistance, boundary layer thickness, and energy transport. The methodology of this work is entirely text based and analytical in spirit, focusing on conceptual integration, physical interpretation, and thermodynamic consistency rather than on explicit equations. The results demonstrate that many apparently distinct phenomena reported across the literature, such as flow separation in stenosed arteries, drag reduction on stretching sheets, and stability conditions of second grade fluids, are manifestations of a common set of governing principles. The discussion elaborates the implications of these findings for understanding vascular disease, optimizing industrial stretching processes, and designing magnetohydrodynamic control strategies. Limitations and future directions are explored in depth, emphasizing the need for experimental validation and the extension to more complex rheological and geometric configurations. Overall, the article offers a comprehensive, coherent, and original account of laminar and non Newtonian flow in constricted and stretching geometries, demonstrating that the classical works of the late twentieth century and the modern analytical methods of the twenty first century can be combined into a powerful framework for both science and engineering.
Nonlinear Transport Phenomena in Viscoelastic and Second Grade Fluids over Stretching Surfaces with Variable Thermophysical Properties and Electromagnetic Effects
The transport of momentum and heat in non-Newtonian fluids over stretching surfaces has become one of the most central themes in modern theoretical and applied fluid mechanics due to its relevance in polymer extrusion, metallurgical processing, coating technologies, biomedical transport, and thermal management systems. Among the many classes of non-Newtonian models, second grade and viscoelastic fluids represent a particularly important family because they simultaneously capture elasticity, normal stress effects, and memory while remaining mathematically tractable for boundary layer analysis. In realistic industrial and biomedical environments, however, fluid properties such as viscosity and thermal conductivity are rarely constant. Instead, they vary strongly with temperature, shear rate, and even concentration of suspended particles. Furthermore, thermal radiation, heat sources and sinks, magnetic fields, and electromagnetic forcing often coexist, especially in high temperature polymer processing, liquid metal transport, microfluidics, and magneto-biofluid applications. This study develops a comprehensive, theoretically consistent and physically interpretable framework for the coupled flow and heat transfer of second grade and viscoelastic fluids over stretching sheets and plates when viscosity and thermal conductivity vary with temperature and when electromagnetic and radiative effects are present. Building exclusively on the body of literature provided, the work synthesizes results from stretching sheet theory, viscoelastic boundary layer dynamics, magnetohydrodynamics, variable property transport, and biofluidic microtransport into a unified descriptive methodology. The formulation is grounded in the similarity transformation philosophy introduced for stretching surfaces and extended by later authors to non-Newtonian and magnetized flows. The roles of heat generation and absorption, radiation, viscous dissipation, porous substrates, and electromagnetic body forces are all incorporated conceptually. Rather than presenting equations, this article provides a detailed, narrative-based explanation of how the governing physics, boundary layer structure, and thermodynamic coupling evolve under these complex conditions. Results reported in the literature are reinterpreted to reveal deep physical mechanisms. It is shown that variable viscosity introduces a strong asymmetry between momentum and thermal diffusion layers, while variable thermal conductivity fundamentally alters how heat propagates away from the surface, especially in radiative environments. Second grade elasticity is found to either stabilize or destabilize the flow depending on whether elastic memory reinforces or resists surface stretching. Magnetic fields suppress velocity while enhancing thermal energy retention, a trend that becomes even more pronounced in viscoelastic fluids due to the additional elastic stresses.
Fractional Order Dynamical Systems, Differential Transform Techniques, and Fuzzy Extensions for Modeling Nonlinear and Chaotic Phenomena
Fractional order differential equations have emerged as one of the most powerful mathematical tools for describing memory dependent, nonlocal, and hereditary phenomena in physical, biological, engineering, and social systems. Unlike classical integer order models, fractional calculus provides a mathematically rigorous and physically meaningful way to incorporate long range temporal dependence and complex internal structure into dynamic models. This theoretical advantage becomes particularly important in the analysis of nonlinear and chaotic systems, where classical approaches often fail to explain observed behaviors such as long term correlations, anomalous diffusion, and multi scale instability. At the same time, modern analytical and semi analytical methods such as the differential transform method, homotopy perturbation, and Adomian decomposition have revolutionized the ability of researchers to approximate and interpret solutions of highly nonlinear and fractional models without relying on purely numerical discretization. In parallel, the integration of fuzzy theory into differential equations has opened new directions for modeling uncertainty, imprecision, and vagueness in real world systems, especially when precise initial conditions or parameters are unavailable. This article develops a comprehensive theoretical and methodological synthesis of fractional differential equations, differential transform techniques, fuzzy extensions, and chaotic dynamical systems. Drawing exclusively on foundational and contemporary research in fractional calculus, chaos theory, numerical analysis, and fuzzy differential equations, the study explores how fractional order models fundamentally alter the qualitative behavior of classical systems such as the Lorenz and Rossler models, how differential transform methods enable efficient solution generation for linear, nonlinear, and fractional systems, and how generalized differentiability allows fuzzy valued functions to be meaningfully embedded in fractional dynamic frameworks. The article elaborates the mathematical philosophy behind Caputo type derivatives, the generation of fractional semi dynamical systems, and the dimension theory of attractors, while also examining how numerical multi step and differential transform schemes preserve stability and accuracy in nonlocal models. The results of this synthesis demonstrate that fractional order modeling not only generalizes classical differential equations but also reveals hidden dynamical regimes such as hyperchaos, bistability, and hidden attractors that are invisible in integer order frameworks. Furthermore, the differential transform method and its multi step and generalized variants provide a unifying computational language that bridges ordinary, partial, integral, and fuzzy differential equations. By integrating these perspectives, this article contributes a unified theoretical platform for understanding and modeling complex systems characterized by nonlinearity, memory, uncertainty, and chaos.
Differential Transform Based Analytical Frameworks for Nonlinear Dynamical Systems in Physical and Biological Modeling
The increasing complexity of nonlinear dynamical systems in modern science has generated an urgent need for analytical and semi analytical techniques capable of producing reliable and interpretable solutions without relying exclusively on heavy numerical simulations. Physical wave propagation, epidemiological processes, and population dynamics are all governed by nonlinear differential equations whose solutions determine prediction, stability, and control. Within this scientific context, the differential transform method has emerged as one of the most powerful series based analytical techniques for solving both linear and nonlinear differential equations across multiple domains. The present article develops a comprehensive, unified, and original analytical framework that integrates classical differential transform theory with its modern multi step, projected, adaptive, and hybrid extensions as reported in the literature. Drawing strictly from the foundational and applied studies of Kangalgil and Ayaz, Ravi Kanth and Aruna, Hassan, Ayaz, Jang, Yildirim, Gokdogan, Odibat, and their collaborators, this study presents a fully synthesized theoretical structure that explains why and how differential transform based methods outperform traditional perturbation, decomposition, and purely numerical approaches for nonlinear models. The article does not merely summarize previous findings but instead reconstructs the conceptual logic that underlies them, tracing the epistemological roots of transform based series representations and explaining how they generate stability, convergence, and accuracy in nonlinear contexts. A particular emphasis is placed on how differential transform techniques enable the extraction of physically meaningful behaviors from nonlinear partial differential equations such as the Korteweg de Vries and modified Korteweg de Vries equations, which describe solitary wave propagation in fluid and plasma physics. Through detailed theoretical analysis, the article demonstrates how differential transform solutions preserve wave coherence, soliton structure, and nonlinear interactions in a way that conventional discretization schemes often fail to capture. These theoretical insights are directly connected to the work of Kangalgil and Ayaz, who showed that differential transform based solutions reproduce solitary wave profiles with remarkable precision when compared with exact analytical solutions.
Adaptive Multi Step Differential Transform Analysis of Fractional Order Chaotic and Stiff Dynamical Systems
Fractional order dynamical systems have emerged as one of the most powerful mathematical frameworks for understanding complex physical, biological, and engineering phenomena that exhibit memory, hereditary behavior, and long term dependency. Classical integer order models often fail to capture the full richness of such processes, particularly when chaos, stiffness, or multiscale dynamics are involved. The fractional order Rossler system, in particular, represents one of the most widely studied prototypes for fractional chaotic dynamics because it combines nonlinear feedback, sensitive dependence on initial conditions, and nonlocal temporal effects. However, solving such systems remains a fundamental challenge because fractional derivatives introduce nonlocality, while chaos amplifies numerical errors, and stiffness creates extreme sensitivity to step size. This article develops a comprehensive analytic and numerical investigation of fractional order chaotic and stiff systems using the differential transform method and its multi step and adaptive extensions. Building strictly upon the theoretical and computational foundations established by Zhang and Zhou, Freihat and Momani, Odibat and collaborators, Alomari, Yildirim, Gokdogan, and other contributors, the study examines how generalized and multi step differential transform techniques overcome the core limitations of classical solvers such as Runge Kutta and standard MATLAB routines when applied to chaotic fractional dynamics. The methodology section presents a detailed conceptual explanation of how the differential transform method converts differential equations into recursive algebraic structures that preserve nonlinearity and fractional memory while maintaining computational efficiency. The results demonstrate that multi step and adaptive differential transform strategies achieve superior stability, accuracy, and long time convergence when compared to conventional numerical approaches, particularly in stiff and chaotic regimes. The discussion elaborates on the theoretical implications of these findings for chaos theory, fractional calculus, and numerical analysis, and highlights how the interaction between memory effects and adaptive step control leads to qualitatively different dynamical representations. Limitations of the approach are critically analyzed, including convergence radius, truncation sensitivity, and computational cost. Finally, the article outlines future research directions for hybrid fractional chaos solvers and real world applications in physics, engineering, and biological modeling.
A Unified Theoretical and Computational Framework for Nonlinear Telegraph Reaction Diffusion Dynamics and Stochastic Control Using Differential Transformation Methods
Nonlinear reaction diffusion and telegraph type equations have long served as foundational models for understanding wave propagation in excitable media, biological tissues, electrical transmission systems, and stochastic control processes. From the early conceptualization of bistable transmission lines and neuristor dynamics to modern developments in density dependent diffusion and random differential operational calculus, a continuous theoretical lineage connects electrical engineering, mathematical biology, and applied probability. This article develops an integrated, publication ready theoretical and computational framework that synthesizes the Nagumo type reaction diffusion and telegraph models with deterministic and stochastic differential transformation methodologies. Drawing exclusively from the provided references, the study constructs a unified interpretation of how nonlinear wave propagation, stability, and control can be modeled, approximated, and interpreted in systems subject to both deterministic and random influences. The paper begins by tracing the historical evolution of the Nagumo and neuristor models from early transmission line theories to biologically inspired nerve impulse representations. Building on Scott and Nagumo and later Buratti and Lindgren, the conceptualization of bistable dynamics and waveforms is reframed as a general theory of excitable systems that can be extended into reaction diffusion and telegraph frameworks. The work of Pickard is incorporated to emphasize how physiological conduction in medullated and unmedullated fibers can be described by nonlinear wave equations, while Ahmed and Abdusalam are used to link these ideas to telegraph reaction diffusion and coupled map lattices in biological systems. The analytical core of the article draws heavily on Abdusalam, Van Gorder, Vajravelu, Pedersen, Mansour, and others who developed analytic, variational, and numerical perspectives on density dependent and nonlinear Nagumo diffusion equations. Their results are interpreted not merely as isolated solution techniques but as part of a broader epistemological shift toward viewing wave speed, stability, and pattern formation as emergent properties of nonlinear transport under feedback controlled reaction mechanisms. A second major theoretical pillar is the differential transformation method and its variants, including projected, improved, and random differential transformations as developed by Zhou, Bert and Zeng, Arikoglu and Ozkol, Jang, Ebaid, and Villafuerte and collaborators. These methods are presented as a coherent computational philosophy that transforms complex nonlinear and stochastic differential equations into tractable recursive structures, allowing both deterministic and random effects to be studied in a unified manner.
A Multiphysics Framework for Magnetohydrodynamic and Rheological Effects in Cilia Driven Mucociliary Transport within Constrained Channels
Mucociliary transport is a fundamental physiological process that ensures the clearance of inhaled pathogens, particulates, and excess secretions from the respiratory tract. It is driven by the coordinated beating of cilia embedded in a viscoelastic mucus layer that overlies a watery periciliary fluid. Over the past several decades, mathematical and biomechanical models have been developed to elucidate the fluid mechanical principles governing this transport mechanism, incorporating increasingly sophisticated descriptions of mucus rheology, ciliary kinematics, and geometrical constraints. In parallel, advances in magnetohydrodynamics and non Newtonian fluid theory have opened new avenues for understanding how external fields and complex material responses can modulate microscale transport phenomena. The present study synthesizes and extends these lines of inquiry by developing a unified theoretical framework for cilia driven flow in channels that accounts simultaneously for rheological complexity and magnetohydrodynamic effects. Drawing strictly on established results from the literature, including the foundational work on mucociliary transport, viscoelastic fluid mechanics, fractional and second grade models, and magnetically influenced channel flows, this article constructs a detailed conceptual model of how metachronal ciliary waves interact with electrically conducting mucus to generate transport under physiological and pathological conditions. Particular attention is paid to the role of epithelial health, mucus viscoelasticity, and external magnetic fields in altering flow resistance, transport efficiency, and mixing. By interpreting the classical and contemporary studies within a common theoretical language, the article highlights previously underexplored synergies between biological transport and applied magnetofluid dynamics. The results provide a descriptive yet rigorous account of how modifications in cilia density, beat frequency, mucus rheology, and magnetic field strength collectively shape the velocity profiles, shear distributions, and clearance rates within airway like channels. This integrative perspective not only deepens our understanding of normal mucociliary function but also offers insight into disease states such as severe asthma, chronic infection, and impaired epithelial function, where deviations in any of these parameters can lead to compromised clearance and tissue damage.
A Comprehensive Analytical Framework for Solving Fuzzy and Singular Differential Systems Using Differential Transform and Homotopy Based Semi Analytical Methods
The rapid expansion of real world systems characterized by uncertainty, vagueness, and incomplete information has driven increasing attention toward fuzzy differential equations as a powerful modeling framework. In engineering, biological sciences, economics, and fluid mechanics, classical deterministic differential equations frequently fail to capture the imprecision inherent in experimental data, environmental variability, or subjective expert judgment. Fuzzy differential equations provide a mathematically rigorous way to incorporate this uncertainty, yet their solution is often significantly more complex than their crisp counterparts. This difficulty has motivated the development of advanced semi analytical and numerical techniques that can handle both nonlinearity and fuzziness in an efficient and stable manner. Among these, the differential transform method, the multi step differential transform method, and homotopy based analytical approaches have emerged as especially promising tools. This study presents a comprehensive theoretical and methodological synthesis of differential transform based techniques and homotopy analysis in the context of fuzzy differential equations and singularly perturbed problems. Drawing strictly from established literature, this article integrates foundational work on fuzzy differential equations, classical differential transform theory, multi step extensions, adaptive step size strategies, and homotopy analysis. It explains how these approaches can be systematically adapted to fuzzy systems through appropriate definitions of differentiability, fuzzy arithmetic, and iterative approximation structures. The analysis demonstrates that differential transform based methods provide highly structured series representations of fuzzy solutions that are computationally efficient and conceptually transparent. When enhanced by multi step and adaptive step size strategies, these methods overcome convergence limitations and enable accurate treatment of stiff, nonlinear, and boundary layer dominated systems. Homotopy analysis further extends this capability by allowing flexible control over convergence through auxiliary parameters and embedding techniques. By synthesizing these frameworks, this article establishes a coherent analytical foundation for solving a wide class of fuzzy ordinary and partial differential equations, including hybrid fuzzy systems and singularly perturbed models. The discussion highlights theoretical implications for fuzzy calculus, computational efficiency, and modeling fidelity, while also identifying limitations and directions for future research.
Electromagnetohydrodynamic and Reactive Transport Phenomena in Second Grade Fluids over Moving and Stretching Boundaries in Porous Media under Oscillatory and Unsteady Conditions
The study of non Newtonian fluids has grown into one of the most important branches of modern fluid mechanics due to its strong relevance in polymer processing, biomedical engineering, geophysical flows, and advanced manufacturing systems. Among the many non Newtonian models, second grade fluids occupy a particularly important position because they are able to represent elastic effects, normal stress differences, and memory behavior while still allowing analytical and semi analytical treatment. When second grade fluids interact with magnetic fields, porous substrates, chemical reactions, and moving or oscillating boundaries, the resulting transport phenomena become highly nonlinear, coupled, and extremely sensitive to both physical and mathematical modeling assumptions. The references forming the basis of this research collectively address magnetohydrodynamic flows, Stokes problems, Couette flows, stretching and shrinking surfaces, porous media, and homotopy based analytical techniques. However, these studies remain scattered across different physical settings and mathematical approaches, often addressing isolated cases rather than offering an integrated theoretical perspective. The present research develops a unified theoretical framework for understanding unsteady and oscillatory flows of second grade electrically conducting fluids in porous media over moving, stretching, shrinking, and oscillating boundaries with heat and mass transfer and chemical reactions. The framework is built strictly on the ideas, methods, and physical assumptions presented in the provided references, especially those of Cortell, Nazar, Tan and Masuoka, Hayat, Asghar, Yao, Liao, Abbasbandy, Nadeem, Erdogan, Vajravelu, Zeng, and Siddiqui. A comprehensive description of how magnetic fields, porosity, elastic effects, thermal boundary conditions, and surface kinematics interact to shape velocity, temperature, and concentration fields is provided in purely descriptive form, without mathematical expressions. The homotopy analysis method plays a central role in this work because it allows the construction of convergent series solutions for highly nonlinear boundary layer and Stokes type problems. Drawing from Liao and Abbasbandy, the present study elaborates on how this method reveals the multiplicity of solutions, sensitivity to boundary conditions, and stability features of second grade fluid flows. By synthesizing results from oscillatory wall problems, stretching and shrinking sheet flows, and chemically reactive magnetohydrodynamic transport in porous media, the paper shows that these flows exhibit strong memory dependent behavior, enhanced or suppressed transport depending on magnetic and porous effects, and complex transient dynamics under unsteady forcing.