Ahead of Publication / Just Accepted

To date, natural and synthetic polymer-based microcapsules have been used extensively in various dermato-cosmetic applications, with an emphasis on the targeted delivery of active ingredients, including therapeutic and aesthetic interventions. Although numerous polymer candidates have been comprehensively investigated, polyurethane based microcapsules have received comparatively minor attention, despite possessing a multitude of intrinsic benefits. However, in recent years, although there has been an upsurge of studies involving polyurethane, predominantly as a capsule wall or shell component, towards tangible dermato-cosmetic applications, these are only intermittently documented. In the current review, we target this lacuna, explore, and collate only the most contemporary trends and advances (2017-to date) in the field. In addition, despite the significance, and pertaining to the acute deficiency of rheological studies targeting polyurethane-based microcapsules in dermato-cosmetic applications, we critically examine and lay a comprehensive interpretation, based on the current state-of-the-art, inevitability for systematic inquiries, and identification of several target domains that need urgent attention. Finally, we deliberate on the challenges and the impending projections from a diverse outlook. We focus on a steady and more sustainable path forward via incorporation of green raw materials, cumulative domain-optimized and customer-focused applications, and a significantly improved understanding of the microcapsule mechanical behavior via implementation of novel rheological characterization procedures.

The Prandtl mixing length plane turbulent boundary flow is governed by the nonlinear ordinary differential equation (ODE): d u + d y + + κ 2 y + 2 d u + d y + 2 = 1 $\frac{\mathrm{d}{u}^{+}}{\mathrm{d}{y}^{+}}+{\kappa }^{2}{{y}^{+}}^{2}{\left(\frac{\mathrm{d}{u}^{+}}{\mathrm{d}{y}^{+}}\right)}^{2}=1$ with boundary condition y + = 0 : u + = 0. Although van Driest first derived the closed-form solution for this ODE in 1956 (E. R. van Driest, “On turbulent flow near a wall,” J. Aeronaut. Sci. , vol. 23, no. 11, pp. 1007–1011, 1956), but the lack of detailed solution process led to its long-term neglect in subsequent fluid mechanics textbooks and research. As a result, the problem of finding the solution has remained an open question in the community. In this work, we independently rederive the solution and provide a complete, step-by-step solution process. Furthermore, we explicitly clarify the validity conditions of the solution (i.e., smooth wall, zero pressure gradient, fully developed turbulent flow, incompressible flow) and propose a modified analytical extension that conforms well to direct numerical simulation (DNS) results. The proposed modification incorporating van Driest-type damping into the solution framework is physically motivated, and this extension is semi-empirical, thus should be interpreted as a model enhancement/refinement rather than an exact mathematical solution. The present work not only supplements the missing solution details of van Driest’s solution but also improves its applicability.

Sulfolane is a groundwater-miscible pollutant that is widely reported to exhibit negligible sorption onto minerals. Yet sub-monolayer uptake can evade detection in conventional batch tests while still affecting plume evolution over transport distances. Here we combine synchrotron soft X-ray spectroscopy, electronic-structure calculations, and transport modeling to test whether “undetectable” sorption can matter at the aquifer scale. Using O K-edge X-ray absorption spectroscopy (XAS) in a liquid flow cell with silicon nitride windows, we observe a sulfolane signal after flushing with water, consistent with slow desorption or irreversible retention on experimental timescales despite negligible sorption inferred from batch measurements. Complementary electronic-structure calculations support the plausibility of sub-monolayer sulfolane association on hydrated motifs representative of calcite and iron hydroxides, for which laboratory studies have reported no detectable uptake. We quantify transport implications of low-capacity interactions using two frameworks: (i) a capacity-limited storage model that represents finite interfacial capacity and (ii) a kinetic sink model that treats uptake as rate-limited removal rather than instantaneous equilibrium partitioning. Across soils spanning specific surface areas and site densities, even sub-monolayer surface excess can attenuate dissolved concentrations over kilometer-scale distances under plausible parameter combinations. Under sustained releases, the solid phase can accumulate inventories comparable to the dissolved mass while progressively saturating capacity along the flow path. This distributed surface reservoir is poised to desorb under changing conditions (e.g., dilution or geochemical shifts), sustaining long-term residual aqueous concentrations even when equilibrium sorption appears negligible. Although focused on sulfolane, the same mechanism may apply to other mobile groundwater contaminants.

Forcing fluids to interact at high speeds creates a multitude of instabilities. Prior studies considered the consequences of these instabilities for core-annular flow (CAF) or atomization systems. However, the existence of spontaneous axis-switching swirling (SASS) modalities in CAF have never been explored. We computationally demonstrate this new behavior for a CAF of steam surrounded by a cold non-Newtonian fuel. Wave pile-up created 3-D fuel surface disturbances which eventually became thin slender lobes with radial modal deformation. Those slender lobes were subject to further instabilities and resulted in oscillatory tangential velocities that were a large fraction of the core steam velocity and orthogonal to the axially flowing base fuel wave. Since these tangential velocities were not externally imposed by atomizer geometry or boundary conditions, it was concluded that they are self-generating. Frequency analysis revealed that SASS tended to operate at the base wave frequency. We additionally explored numerical uncertainty with our previously validated computational tool and found that SASS behavior can be achieved on a coarse, limited mesh at relatively low simulation times.

The formulation of constitutive relations is central to turbulence modeling, yet their evolution has always been constrained by requirements of physical consistency, mathematical well-posedness and numerical stability. This review integrates these constraints into a unified framework, tracing how they have shaped the trajectory of turbulence closures from classical formulations to contemporary data-driven approaches. Fundamental principles including conservation laws, realizability, invariance, dimensional homogeneity, memory effects and asymptotic consistency delineate the admissible space of models while simultaneously guiding their refinement. Historical progress, from the Boussinesq approximation through nonlinear eddy-viscosity models to explicit algebraic stress closures, demonstrates the progressive incorporation of such constraints as corrective mechanisms that enhance robustness and predictive fidelity. In parallel, recent advances in physics-informed and machine learning-based models confirm that these same constraints remain indispensable for ensuring generalizability, physical admissibility and solver compatibility. By framing turbulence modeling through the lens of constraints, this review highlights their dual role as both limitations and design principles while underscoring their continuing relevance in shaping next-generation hybrid frameworks that unite physical rigor with data-driven adaptability.

This paper develops a comprehensive mathematical framework for modeling the coupled hydroelastic dynamics of sea-ice floes of arbitrary shape and non-uniform thickness under linear ocean wave forcing, with particular emphasis on oblique wave incidence. We simultaneously incorporate all six rigid-body motions (heave, surge, sway, roll, pitch, and yaw) and the complete spectrum of flexural deformation modes within a unified Green’s function formulation. The water flow is modeled using potential theory governed by Laplace’s equation, while the floe obeys the generalized Kirchhoff-Love plate equation with spatially varying flexural rigidity. We formulate the coupled fluid-structure interaction problem through kinematic velocity-matching conditions and dynamic pressure-continuity conditions at the ice-water interface. The submerged floe geometry is represented as a non-axisymmetric, piece-wise smooth orientable surface parameterized by a piece-wise differentiable draft function, accommodating realistic irregular features including thickness variations, keels, and submerged ridge structures. The elastic eigenproblem with free-edge boundary conditions yields a complete orthogonal basis of deformation modes, accounting for added-mass effects through modified natural frequencies. By decomposing the velocity potential into partial potentials associated with incident waves, scattered waves, rigid motions, and elastic modes, we reduce the problem to a system of Fredholm integral equations of the second kind for surface density functions on all boundary segments. We derive the complete 6 × 6 mass, added-mass, and hydrostatic stiffness matrices for floes of arbitrary geometry, enabling analysis of fully coupled dynamics. A numerical implementation is developed and validated against multiple floe geometries, including elliptical disk-cone configurations and complex asymmetric shapes with variable thickness and ice keels. Results demonstrate that oblique wave incidence excites all six rigid degrees of freedom, with significant energy exchange between rigid motions and flexural modes near resonance conditions. The Response Amplitude Operators (RAOs) reveal geometry-dependent resonance phenomena and strong coupling between modes for non-symmetric configurations. Energy budget analysis quantifies the distribution among heave, surge, sway, roll, pitch, and elastic deformation modes across the frequency spectrum.

Based on a cocycle structure, we identify a new derivation of the Boltzmann distribution for finite energy-level systems from the maximal entropy principle (MEP). Our approach does not rely on the method of the Lagrange multiplier, and it provides a more transparent way to understand the dependence on the energy levels of the temperature T = 1/ β for the equilibrium distribution. Finally, we make two curious observations associated with our derivations.

Effects of a third rank permeability term in chiral solids are studied. Fluid flow through such materials acquires vorticity upon emergence from the material. Materials of interest include chiral surface lattices such as the gyroid, chiral rib lattices, and granular materials comprised of sugar crystals, quartz sand, wheat or beans. A characteristic length scale is associated with the chirality. The length scale can be obtained by several methods. Contacts with nonlocal permeability, elasticity and piezoelectricity are explored.

A discrete binomial random-walk description of molecular collisions is used to quantify the variance of coarse-grained velocity fields arising solely from collision-induced momentum exchange. Closed-form expressions for the growth of velocity variance as functions of coarse-graining scale and time are derived and shown to imply a power-law decay of variance with averaging scale. Particle-based ensemble simulations validate the predicted scaling and temporal behaviour; surrogate ensemble tests demonstrate that phase/temporal coherence is required for the observed integrated transfer diagnostics. The analysis is intentionally restricted to collision-generated fluctuations in quiescent fluids and does not model cascade dynamics; implications for possible amplification under inertial dynamics are discussed cautiously. All data and the minimal verification instructions required to reproduce the summary tables are embedded in Appendix.

Coherent Rayleigh-Brillouin scattering (CRBS) holds great promise for the characterization of gas properties and the investigation of gas kinetic processes. The CRBS spectrum exhibits a strong dependence on the Knudsen number (Kn), revealing its inherently multiscale nature. In the unified gas-kinetic scheme (UGKS), collisions are intrinsically coupled with free transport during flux construction, endowing the method with distinct multiscale capabilities. Specifically, the UGKS reduces to a Boltzmann solver when the relaxation time τ ≥Δ t , and to the gas-kinetic scheme (GKS)-a Navier–Stokes solver-when τ ≪Δ t , thereby accommodating flow regimes without constraints on the molecular mean free path or collision time. In this study, the UGKS is extended to simulate CRBS phenomena, with the governing equation formulated based on the Bhatnagar–Gross–Krook (BGK)-Shakhov model. Detailed derivations are provided. To account for the additional perturbation source term, a second-order accurate numerical algorithm is developed using the Strang splitting method within the UGKS framework. The proposed model is validated against argon CRBS experiments, demonstrating excellent agreement. Building on this validated framework, the impact of incident signal intensity on CRBS spectra across a range of Knudsen numbers is systematically examined, accompanied by an in-depth analysis of the underlying physical mechanisms. This work broadens the applicability of computational fluid dynamics (CFD)-based CRBS simulations and provides a reliable numerical foundation for exploring high-intensity, multiscale gas-kinetic phenomena in future research.

One of the most intriguing experiments in polymer physics, is the superposition of small shearing oscillations upon steady shear flow. This experiment allows us to explore how the steady shear flow affects the complex viscosity, and the complex components of the first and second normal stress differences. The novelty of this paper is its exploration of the how the direction of this superposition matters to the complex components of the first and second normal stress differences in orthogonal superposition. Whereas, these complex components of the normal stress differences have been measured in parallel superposition, these have not been measured in orthogonal. Previously, we derived material functions from rotarance theory for the (previously measured) normal stress responses to the parallel superposition of small-amplitude oscillatory shear flow upon steady shear flow [ Phys. Fluids , 37 , 033133 (2025)]. The novelty of this work is that we arrive at analytical expressions for the (not previously measured) orthogonal superposition counterparts. We choose rotarance theory for our exploration, for its unique diversity of macromolecular structure. By rotarance theory , we mean accounting for the elasticity of polymeric liquids by considering, and by only considering, the physics of macromolecular orientation. We find that both the first and second normal stress difference responses in orthogonal superposition differ importantly from their parallel superposition counterparts.

This work develops a unified, dimensionless framework for comparing reacting systems that are geometrically similar but differ in size, particle size distribution (PSD), and internal pore structure of the solid reactant, with emphasis on hydrometallurgical heap leaching. For two heaps operated under dynamically similar hydrodynamic conditions, the dimensionless residence time distribution (RTD) of the liquid is identical. However, differences in PSD and internal porosity break microscopic similarity and can strongly alter heap-scale conversion. Using the shrinking-core model (SCM), we show how a PSD induces a distribution of particle-scale Damköhler numbers that controls the heap-averaged conversion. We derive explicit change-of-variable expressions mapping a PSD in particle diameter to Damköhler-number distributions for (i) external film (convection) control, (ii) intraparticle diffusion control, and (iii) mixed control approximated by additive resistances. We then couple SCM kinetics to dual-porosity hydrology (mobile/immobile liquid domains) and identify the additional dimensionless groups governing interporosity exchange. Two numerical examples illustrate the approach: (1) transforming a lognormal PSD into external- and diffusion-controlled Damköhler distributions, and (2) a toy column/heap calculation comparing conversion trajectories for different PSDs under film versus diffusion control. Finally, we outline a practical workflow for column-test interpretation, combining SCM diagnostics with tracer-derived RTD and exchange parameters. The framework explains the heightened sensitivity of diffusion-controlled systems to PSD tails and dual porosity and provides a compact set of dimensionless groups for robust scale-up from columns to industrial heaps.

Fragmentation population-balance equations describe how particle-size distributions evolve under breakage and daughter fragment redistribution. By log-size we mean the logarithmic internal coordinate ξ = ln( x / x 0 ), where x is a size-like variable taken proportional to a conserved mass. From a standard self-similar fragmentation class we derive an exact conservative transport equation in ξ for the normalized mass fraction : a state-dependent pure-jump master equation (nonlocal internal-coordinate mass transfer). We also give an explicit Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) factorization whose diagonal sector reproduces this master equation, used here as an optional structure-preserving operator representation and constrained parameterization for inverse modeling. In a controlled small-jump regime, the nonlocal jump transport reduces to a drift–diffusion (Fokker–Planck) operator in log-size space. Under detailed-balance conditions this operator admits the standard symmetrization to a self-adjoint Schrödinger-type spectral problem, motivating compact parametric hypothesis classes for distribution shapes. We then present two inverse routes: (i) time-resolved parametric fitting of transport/spectral parameters, and (ii) a regularized steady inversion that reconstructs an effective potential from a measured steady PSD. We include numerical benchmarks: forward simulation of the jump transport model (continuous-time Markov chain discretization) and its drift–diffusion reduction, quantitative discrepancy metrics, and inverse parameter recovery on an Airy half-line synthetic benchmark under multiplicative noise.

This paper investigates the behavior of a fluid characterized by a viscosity simultaneously depending on pressure and shear rate within a Hele-Shaw cell featuring a sharp corner geometry. The study extends previous analyses conducted on purely pressure-dependent (piezo-viscous) and yield-stress fluids, providing a new perspective on confined complex flows. Motivated by practical applications related to designing biomedical devices and flows of relevance to bio-medicine area, thin film technologies, injection molding – to name only a few – the flow configuration considered here can highlight essential features of complex fluid behavior in narrow-gap geometries around a sharp edge. Starting from the governing equations for an incompressible generalized Newtonian fluid and employing an appropriate rheological model, we derive the modified flow equations adapted to the Hele-Shaw flow. A particular solution is obtained near the corner region. Numerical simulations complement the theoretical results, illustrating the influence of the rheological parameters on the flow behavior.

We study coupled mass transport in a tumor–microenvironment regime with two motile densities ( S , R ) and non-motile state switching ( P , A ). The populations diffuse and undergo chemotactic drift; ( P , A ) follow pointwise ODE switching. A decoupled inhibitory field D satisfies a damped Neumann heat equation, giving maximum-principle bounds and exponential decay. Together with the pointwise invariant P + A , these identities yield global existence, positivity, and long-time reduction to limiting ( S , R ) kinetics with a unique globally attracting coexistence state. Neumann eigenmode reduction gives closed dispersion relations. The base ( S , R ) reaction–diffusion (RD) block remains stable for all nonconstant modes for any d S , d R > 0, excluding classical Turing destabilization. Chemotaxis is posed via a diffusive cue c , since ∇ A is undefined for non-diffusive A . In one-way damped coupling, the linearized mode matrix is block triangular and leaves the ( S , R ) spectrum unchanged. Two-way coupling adds a feedback rank-one mobility correction, induces effective cross-diffusion, and admits mode growth. We give explicit trace/determinant criteria for unstable Laplacian modes and the resulting instability thresholds.

In this paper, I further continue an investigation on Beltrami Flows began in 2015 with A. Sorin and amply revised and developed in 2022 with M. Trigiante. Instead of a compact 3-torus T 3 = R 3 / Λ ${T}^{3}={\mathbb{R}}^{3}/{\Lambda}$ where Λ is a crystallographic lattice, as done in previous work, here I considered flows confined in a cylinder with identified opposite bases. In this topology I considered axial symmetric flows and found a complete basis of axial symmetric harmonic 1-forms that, for each energy level, decomposes into six components: two Beltrami, two antiBeltrami and two closed forms. These objects, that are written in terms of trigonometric and Bessel functions, constitute a function basis for an L 2 space of axial symmetric flows. I have presented a general scheme for the search of axial symmetric solutions of Navier Stokes equation by reducing the latter to a hierachy of quadratic relations on the development coefficients of the flow in the above described functional basis. It is proposed that the coefficients can be determined by means of a Physics Informed like Neural Network optimization recursive algorithm. Indeed the present paper provides the theoretical foundations for such an algorithmic construction that is planned for a future publication.

This study investigates effect of externally imposed axial-flow in Taylor-Couette flow (TCF) with smooth rotating inner-cylinder and stationary outer-cylinder for radius ratio = 0.5, and Re = 60 to 650. Previous studies by Razzak et al. [M. A. Razzak, B. C. Khoo, and K. B. Lua, “Numerical study on wide gap Taylor Couette flow with flow transition,” Phys. Fluids , vol. 31, no. 11, 2019, M. A. Razzak, B. C. Khoo, and K. B. Lua, “Numerical study of Taylor Couette flow with longitudinal corrugated surface,” Phys. Fluids , vol. 32, no. 5, 2020, Drag reduction study of naturally occurring oscillating axial flow induced by helical corrugated surface in Taylor–Couette flow,” Phys. Fluids , vol. 35, no. 8, p. 083608, 2023] showed that macroscale corrugated surfaces in TCF passively generate axial secondary flows and reduce drag without external power, but their complex design limits practical use in turbomachinery. To overcome this, present study applies external axial-flow without altering cylinder geometry to evaluate its effect on torque and power. Unlike earlier periodic-boundary studies, parabolic inlet velocity and outflow condition are imposed at the cylinder ends. Three axial-flow types are examined steady (non-zero), oscillating with non-zero mean, and oscillating with zero-mean and results are compared with baseline TCF(no axial-flow) from Razzak et al. [M. A. Razzak, B. C. Khoo, and K. B. Lua, “Numerical study on wide gap Taylor Couette flow with flow transition,” Phys. Fluids , vol. 31, no. 11, 2019] for same radius and aspect ratios. Based on results, Taylor-vortices are found to be stretched along axial direction under the application of axial-flows. This was found to suppress the strength of jet impingement at the inner wall resulting in torque and power reduction for Re ≤ 450 and Re ≥ 540. In contrast, within range 450 < Re < 540, stronger naturally induced axial-flows in baseline case achieve better suppression, leading to lower torque than the externally applied axial-flow cases. Among the three axial-flow types, steady axial-flow yields the greatest vortex stretching and jet impingement suppression, resulting in the highest torque and power savings. Adding oscillation with non-zero mean slightly weakens these effects, while zero-mean oscillation offers the least suppression and torque reduction. Overall, stronger imposed axial-flow can further enhance drag reduction, offering a geometry-independent control for TCF.

Eyring’s absolute rate theory relates fluid flow with the activation energy necessary for a molecule to go from one equilibrium position to another. Developed nearly a century ago, it remains a powerful approach for understanding transport processes in liquids. This work presents a revision of Eyring’s theory by replacing the vaporization-based parts of the theory with a residual approach that conceptualizes transport as local (re)movement of molecules within the system rather than removal from it. In this new approach, the energy barrier corresponds to the difference between a molecule with and without intermolecular interactions, effectively treating it as the residual property related to an ideal gas reference state. Furthermore, we explore the physical connections between Eyring’s absolute rate theory and Rosenfeld excess entropy scaling, revealing that both approaches describe complementary aspects of the same transport phenomena. It is tempting to link both theories. The activation parameters in Eyring’s theory, particularly the energy of activation, are shown to relate to the residual entropy used in entropy scaling. This provides a pathway to establish entropy scaling on a more rigorous physical foundation while offering deeper insights into the molecular mechanisms governing viscous flow. In comparison to experimental data, the revised theory demonstrated significant predictive power for viscosity across a wide range of thermodynamic conditions and species, including associating liquids. Moreover, parameters are transferable to other properties, such as self-diffusion.

We use chemical reaction-diffusion waves in narrow channels to model electron drift in wires. By varying the initial conditions of an excitable Belousov–Zhabotinsky (BZ) medium, we achieve careful, quantitative control of BZ wave speeds in the range of electron drift speeds in conductors, a few millimeters per minute. We compare the speeds of the easily observable BZ waves and their computer simulations with theoretical electron drift speeds to explore the effects of wire radius, electric current, and material composition. Such BZ waves are compelling visual analogues of electron drift.

This paper discusses the structure of turbulence, categorizing it into elementary structures and composite structures, while also outlining the discovery processes of these structures. It strives to present the different viewpoints regarding their formation, with a focus on describing their dynamic roles in the process of turbulence formation. This paper does not critique past interpretations, as researchers at the time had limited and possibly one-sided evidence. The author also presents an own understanding of the physical processes and dynamic relationships involved in the formation of these structures, offering reasonable explanations for these processes.

In fully developed homogeneous and isotropic turbulence, the Lagrangian and Eulerian descriptions of motion, although formally equivalent, become statistically decoupled. In this work, by invoking Liouville theorem, we show that the joint probability density function (PDF) of the Eulerian and Lagrangian fields, evolving from arbitrary initial conditions, relaxes exponentially toward a factorized form given by the product of the corresponding marginal PDFs. This relaxation is governed by a genuine spectral gap of the Liouville operator, whose magnitude is primarily set by the bifurcation rate of the velocity-gradient dynamics, whereas the contribution of Lyapunov exponents is shown to be significantly smaller. As a consequence, Eulerian–Lagrangian correlations decay rapidly, and if the joint PDF is initially factorized, its factorized structure is preserved at all subsequent times, with each marginal evolving independently under the corresponding dynamics. We further show that the formal equivalence between the two descriptions implies the invariance of the relative kinetic energy between arbitrarily chosen points. When combined with the asymmetric statistics of instantaneous finite-scale Lyapunov exponents in incompressible turbulence, this property provides a quantitative interpretation of particle-pair separation and of the turbulent energy cascade. Finally, these results naturally lead to nondiffusive closure relations for the von Kármán–Howarth and Corrsin equations, which coincide with those previously proposed by the author, thereby providing an independent theoretical validation of those closures.

Emulsions are useful for formulating compartmentalized structures with tunable characteristics, making their dried state a potential stimuli-responsive edible substrate for four-dimensional (4D) printing applications. Yet, establishing component, microstructure, and response relationships is crucial for controlled transformations under stimulus exposure, thereby enabling the technology for edible applications. The oil phase incorporation requires particular attention, due to its hydrophobicity and its impact on the response behavior, which can be determined by the swelling rate during water immersion tests. This study focused on flaxseed oil-in-water (O/W) emulsion-based edible films, stabilized by gelatin and lecithin. The dried films based on two oil-to-water (O:W) ratios (5:95 and 10:90 v/v), were submerged in distilled water (pH: 4, 7, and 10). The film-forming emulsions were yield stress fluids exhibiting solid-like properties in the linear viscoelastic region. Network strength and connectivity of emulsions were significantly different based on complex modulus as a function of frequency ( p < 0.05). The films were characterized via thickness (0.19–0.66 mm), weight (0.41–1.18 g), moisture content, opacity, and color. The film matrix properties affected the response behavior of the samples depending on the swelling media. The film survivability depended on the swelling medium pH ( p < 0.05). All samples remained intact overnight at pH 4 and 7. Only sample 5O95W-10 swelled up to 60 min at pH 10, which can be attributed to longer gel setting time and the presence of more protein and emulsifier allowing better network connectivity. Swelling rates of the films were similar at pH 4 and pH 7 ( p > 0.05), and their governing mechanism was less Fickian diffusion. Understanding the effects of the oil phase and the pH of the swelling medium on the response behavior provides insights for tailoring these films as novel substrates for 4D printing and enables controlled transformations.

An approximate analytical solution for the laminar boundary layer on a flat plate governed by the Blasius equation is presented. Although the numerical solution of the Blasius equation is well established, no closed-form analytical solution has yet been found. In this study, the approximate analytical solutions are developed by extending Lin's (1999) exponential formulation of the dimensionless streamwise velocity to include quadratic and cubic exponents. Both proposed forms satisfy the boundary conditions at the wall and the free-stream, and the corresponding analytical expressions for the dimensionless stream function, shear stress, and cross-stream velocity are derived. Comparison with the numerical Blasius solution shows that the quadratic form provides a reasonable approximation, while the cubic form exhibits excellent agreement over the entire range of the similarity variable η which is defined as η = y U ∞ / ν x $\eta =y\sqrt{{U}_{\infty }/\left(\nu x\right)}$ , where x and y are the wall-parallel and wall-normal coordinates respectively, U ∞ is the free-stream velocity and ν is the kinematic viscosity. The maximum residual of the cubic solution is on the order of 10 −3 at the wall, indicating high accuracy. The proposed cubic analytical solution thus provides a compact and explicit expression that closely represents the Blasius solution, offering a practical alternative for engineering applications requiring analytical evaluation of boundary-layer quantities.

The rheological behavior of polymer melts under high pressure is a critical factor in many industrial processes like injection molding and extrusion, yet it is often inadequately characterized. At operating pressures that can exceed 100 MPa, viscosity can increase by orders of magnitude, making atmospheric-pressure data insufficient for accurate process simulation. This pressure-induced viscosity increase is highly dependent on molecular architectures of the materials. This study aims to deconstruct the influence of specific structural features such as short-chain branching (SCB), long-chain branching (LCB), and polydispersity on the pressure sensitivity of the viscosity of polyethylene. Utilizing a high-pressure sliding plate rheometer (HPSPR) to ensure accurate measurements under uniform shear and pressure, we characterized four distinct polyethylene melts. All samples, regardless of their structure, exhibited piezorheologically simple behavior, allowing the application of time–pressure superposition over the entire shear rate range. A key finding is that the long-chain branched sample, known from the literature to be thermorheologically complex, was found to be piezorheologically simple. This dichotomy is explained by the different physical mechanisms of temperature and pressure. The pressure sensitivity of the viscosity, quantified by the pressure–viscosity coefficient, was found to be strongly dependent on molecular branching. Both SCB and LCB significantly increase the pressure sensitivity while polydispersity had a negligible effect. These results demonstrate that molecular branches are the dominant structural parameter controlling the rheological response of polyethylene to pressure, providing crucial insights for the development of more accurate predictive models for high-pressure polymer processing.

We develop a geometric framework for irreversible transport phenomena in which macroscopic evolution equations arise from the combined structure of a thermodynamic state metric and an Onsager-based dissipation metric. The construction begins by defining a pseudo-Riemannian manifold from the Hessian of an appropriate thermodynamic potential. When the enthalpy is used and written in variables ( S , P ), the resulting metric possesses a Lorentzian-type signature: entropy acts as a time-like coordinate, while pressure forms a spatial-like coordinate associated with mechanical response. Local irreversible dynamics are incorporated through the inverse Onsager matrix, which defines a positive-definite dissipation metric on the space of fluxes and gradients. A thermodynamic action integrating these two geometric layers yields geodesic evolution equations. For a Newtonian fluid with constant viscosity, the resulting Euler–Lagrange equations reproduce the incompressible Navier–Stokes equations without requiring an externally imposed constitutive closure. Within this framework, turbulence scaling emerges from competition between inertial curvature and dissipation metric stiffness. The Kolmogorov length scale appears as a minimum geometric resolution length where these contributions balance, providing a geometric interpretation of energy cascade termination and dissipation onset. Finite-time singularities in the classical PDE formulation correspond to curvature divergences in the transport geometry; however, the thermodynamic proper time diverges in such limits, suggesting that blow-up is dynamically suppressed in single-phase continua. Breakdown occurs only when the thermodynamic metric itself becomes degenerate, such as at cavitation or critical points where compressibility or heat capacity diverge, corresponding to a change in manifold topology. Although derived explicitly for fluid flow, the framework is general: by choosing different thermodynamic potentials and Onsager matrices, the same geometric formulation applies to heat conduction, diffusion, electrochemical transport, and other irreversible processes. The results provide a unified thermodynamic geometric basis for transport equations, clarify the conditions under which continuum models remain valid, and offer a new interpretation of turbulence scaling and phase-transition-induced breakdown.