Asymptotic methods are employed to analyse, interpret and improve on an often-used model for the dissolution of drug particles in a standardized flow-through dissolution cell, United States Pharmacopeia Apparatus 4 (USP4). The model is based on a simplified description for the mass and momentum conservation for an individual spherical drug particle that is free to move and dissolves in a solvent that is pumped vertically upwards into a cylindrical cell. Mathematically, the model consists of a nonlinear ordinary differential equation for the particle radius which is intricately coupled to an integrodifferential equation containing the Basset integral history term for the particle velocity; both equations evolve over time. We nondimensionalize the equations and derive novel asymptotic solutions for the two cases normally of practical interest: when the pumping is uniform in time and when it is pulsatile. Subsequent comparison with numerical results is found to show excellent agreement; moreover, whereas the asymptotic solutions are obtained instantaneously, the numerical solutions require many hours of calculation, as a consequence of the cumulative computational burden of the history term. The suitability of the often-cited Ranz-Marshall correlation for the mass transfer associated with particle dissolution is also investigated. Lastly, the relevance of the model as regards experimental dissolution data obtained from a USP4 cell is discussed.
The interaction of the edge and internal cracks on the axis of symmetry of the elastic wedge is considered. The wedge is in a state of plane deformation. The faces of the cracks are under constant pressure, and the edges of the wedge are free from stress. Using the Mellin integral transformation in the radial coordinate, an integral equation with a difference kernel on a system of finite and semi-infinite intervals is obtained. Using the generalized scheme of the Wiener–Hopf method, the integral equation of the problem is reduced to an infinite system of algebraic equations with exponentially decreasing coefficients. The distributions of normal stresses on the crack extension line and the normal displacements of the crack faces are obtained in the form of power series in the radial coordinate, the coefficients of which are expressed through the solution of an infinite system of algebraic equations. The stress intensity factors at the crack tips are found. It is shown that with a decrease in the distance between cracks, the stress intensity in the vicinity of the crack tips first decreases and then increases significantly, which sharply increases the probability of material failure in wedge-shaped bodies with cracks.
Rayleigh Bénard convection (RBC) represents a fundamental fluid mechanical phenomenon characterized by the simultaneous transport of heat and mass due to temperature and concentration gradients, respectively. This process generates buoyancy-driven instabilities that manifest as complex flow patterns in various natural and industrial systems. When combined with time controlled rotational effects and thermal boundary modulations, the convective dynamics become significantly more intricate due to the introduction of Coriolis forces and time-dependent forcing. The study of such systems has gained considerable attention due to their relevance in geophysical flows, industrial applications, and fundamental understanding of pattern formation in fluid systems. This study explores RBC in a Newtonian fluid between two infinite horizontal plates, with the lower plate heated and the upper plate cooled. The analysis considers the influence of system rotation and temperature modulation on the flow behavior. To examine the onset of instability, a weakly nonlinear stability approach is used, leading to the derivation of the Ginzburg–Landau amplitude equation, which describes how convective disturbances evolve near critical conditions. The efficiency of heat transfer was evaluated using the Nusselt number, providing a dimensionless measure of how convection enhances heat transport compared to pure conduction. The analysis revealed several critical insights into the system behavior. Graphical investigations demonstrated the distinct influences of key dimensionless parameters: the Prandtl number (Pr) controlling the ratio of momentum to thermal diffusivity, modulation amplitude ( δ _2 ) governing the strength of thermal forcing, and the modulation frequency ( ω _2 ) determining the temporal characteristics of boundary conditions. Most notably, the Taylor number (Ta), representing the ratio of centrifugal to viscous forces, exhibited a stabilizing effect on Rayleigh Bénard convection. This stabilization occurs through the suppression of convective instabilities via rotational constraints imposed by Coriolis forces. The findings contribute to the fundamental understanding of how rotational and thermal forcing mechanisms interact to influence on Rayleigh Bénard convection, with implications for both theoretical fluid mechanics and practical applications in rotating machinery and geophysical processes.
Bioprinting enables fabrication of three-dimensional, patient-specific, tissue-like structures, for which printing resolution and printability are key determinants of construct fidelity. Achieving consistent control of printed filament dimensions remains challenging due to its strong dependence on interacting process parameters, while existing print-and-test approaches are often time-consuming, expensive, and specific to particular materials and experimental set-ups. This study proposes a generalised, physics-based model to predict printed filament diameters for moderately shear-thinning materials under pneumatic extrusion-based bioprinting. By adopting an arc length coordinate system and exploiting the slender geometry of the extruded filament, asymptotic techniques are employed to reduce the incompressible Navier–Stokes equations to a closed system describing filament radius and centre-line orientation. Non-Newtonian behaviour characteristic of bioinks is captured by modelling the material as a power-law fluid. Applied prior to printing, the model enables identification of process parameter combinations corresponding to a bioink’s window of printability, reducing reliance on print-and-test methodologies. Preliminary validation using pneumatically extruded Nivea Crème demonstrates good predictive capability. This work highlights the value of theoretical modelling for supporting optimisation-driven design in extrusion-based bioprinting.
Orthotropic–piezoelectric materials are broadly used in advanced electromechanical devices due to their strong coupling and directional stiffness. The present study examines the propagation characteristics of shear-horizontal (SH) waves in a cylindrical core–shell system incorporating an imperfect interface. Three configurations are examined: Model-1 (FGOPE core–FGOPE shell), Model-2 (FGPE core–FGOPE shell), and Model-3 (FGPE core–FGPE shell), where FGOPE and FGPE denote functionally graded orthotropic and transversely isotropic piezoelectric materials, respectively. Two interfacial conditions are considered: a coupled electromechanical spring-type interface and a piezoelectric membrane-type interface. The governing equations in cylindrical coordinates reduce to Bessel-type differential equations, and exact solutions for displacement and electric potential are expressed in Bessel functions of the first and second kinds. By enforcing the boundary and interface conditions, closed-form dispersion relations are derived for both interface models for all three configurations. Numerical results illustrate the effects of interfacial imperfection parameters, functional gradedness, anisotropy, and the core and shell radii on phase velocity. The membrane-type interface generally yields lower phase velocities than the coupled spring-type interface. The proposed formulation offers a unified analytical framework for assessing interfacial and material effects in functionally graded piezoelectric cylindrical waveguides and provides guidelines for the design of smart sensors, transducers, and wave-based electromechanical devices.
The nonlinear oscillations of a micron-sized air bubble in water driven by a spatially uniform, temporally periodic, single-frequency electric field are considered. Using a model that accounts for shape mode interactions to second order, thermal damping of the interior gas, viscous damping of the liquid and weak compressibility, the resultant volume mode oscillations and shape deformation are studied in detail. For a range of driving frequencies and electric field strengths, after an initial transition phase, the bubble is shown to assume a sustained, finite amplitude, oscillating ellipsoidal shape dominated by the prolate/oblate mode ( k=2 shape mode) which oscillates at twice the driving frequency. Both the volume mode and k=2 shape mode are directly excited by the electric field while higher-order even shape modes are excited through nonlinear shape mode interactions consistent with previous work. The dynamical behaviour of the k=2 shape mode is shown to depend on the difference between the shape mode’s natural frequency and twice the driving frequency and on whether the bubble is driven below, at, or above the resonance of the k=2 shape mode. In all considered cases, the volume mode oscillations are shown to be prohibitively small, even at volume resonance, to induce parametric instability growth, leaving the directly excited k=2 shape mode to dominate the resultant bubble dynamics.
The stability and rupture dynamics of a gravity-driven thin liquid film coating a vertical cylinder are analyzed, incorporating van der Waals forces, thermocapillary stresses, and wall slip. A nonlinear evolution equation is derived under the assumption of small film thickness relative to cylinder radius. Linear stability analysis shows that van der Waals attraction destabilizes the film, with wall slip amplifying disturbance growth and shifting the most unstable wavenumber. Weakly nonlinear analysis reveals both supercritical and subcritical regimes, underscoring the interplay among slip length, thermocapillarity, and intermolecular forces. Numerical simulations validate these predictions, demonstrating rupture dynamics governed by the disturbance spectrum. Spatiotemporal analysis identifies conditions for absolute instability driven by van der Waals forces, with thresholds that depend on the slip length. Finally, using self-similar scaling, a power law is proposed between minimum film thickness and time during film thinning close to rupture. These findings provide a unified framework for thin-film instabilities on curved substrates, with relevance to coating flows and microfluidic applications.
The dynamic response of underground cavities in layered sites to plane SH waves is a fundamental research topic in underground seismic engineering. Using the wave function expansion method combined with the plane wave spectrum superposition technique, this paper establishes an analytical model for a cylindrical cavity embedded in the substratum. The problem is reduced to a system of infinite linear equations and solved numerically using series truncation and complex damping. Accuracy verification and comparison with classical solutions confirm the reliability of the method. Parametric analysis demonstrates that the shear modulus ratio exerts a remarkable influence on the displacement distribution around the cavity, particularly within its low-value range. Overburden thickness and incidence angle jointly determine the displacement evolution law, and soft and hard overburdens exhibiting mirror-image patterns. Variations in cavity depth trigger periodic evolution of displacement responses. Additionally, the dimensionless frequency has a strong controlling effect on the dynamic behavior of the cavity, and the overburden layer produces an obvious amplification effect on the cavity response under high-frequency excitation. The findings provide a theoretical basis for the seismic design of underground structures in layered sites.
A numerical investigation is conducted on mixed convection of viscoplastic nanofluids, modeled as a Bingham plastic, within a ventilated enclosure incorporating relative slip velocity between the base fluid and nanoparticles. The enclosure features heated walls, with cold fluid entering through an inlet and exiting through an outlet on opposite vertical walls. The governing equations are solved using a control volume approach based on the two-phase nanofluid model to assess thermal performance and the influence of yield stress on flow behavior. Heat transfer characteristics are analyzed through the average Nusselt number, entropy generation, cup mixing temperature, and pressure drop. Slip effects induced by Brownian diffusion and thermophoresis enhance heat transfer compared to the homogeneous model. The impacts of Reynolds number, Richardson number, nanoparticles volume fraction, particle diameter, and Joule heating are examined. Results indicate that nanoparticles addition improves heat transfer more significantly than the accompanying rise in entropy generation and pressure drop, while the Bingham yield stress diminishes heat transfer but enhances thermal mixing. These findings provide valuable insights for optimizing the design and performance of viscoplastic nanofluids-based thermal systems.
Large-scale energy storage is increasingly required to support offshore renewable generation, motivating interest in floating mechanical storage systems. Here we study the large-scale storage of kinetic energy via a floating offshore flywheel. We develop a simple mathematical model for the flywheel dynamics subject to hydrodynamic drag, bounded applied torque, and time-varying electricity prices, with optional wind-driven torque from surface-mounted sails. The operation of the system is formulated as an optimal control problem that maximises net energy revenue over a periodic operating cycle. The resulting optimal control problem is studied using a combination of asymptotic analysis and numerical optimisation. We show that across a wide range of physically relevant parameters, optimal operation reduces to a regime of near-constant angular velocity. In this regime, charging and discharging are governed by a simple switching rule based on a critical electricity price that can be determined directly from price data. Wind-driven torque enhances performance but does not alter the basic switching control structure. Case studies using idealised price profiles and German electricity market data demonstrate how the resulting switching-based control strategy operates under realistic conditions.
This paper presents a comprehensive spectral analysis of the Dirichlet-to-Neumann operator associated with the Laplace equation in thin spherical shells in ℝ^3 . Using spherical harmonic expansions and separation of variables, we derive an explicit spectral representation that diagonalizes the operator with respect to the spherical harmonic basis. We establish sharp continuity bounds in fractional Sobolev spaces on the sphere, proving explicit norm estimates in ℒ(H^1/2(𝕊^2) , H^-1/2(𝕊^2)) . For the thin shell regime δ≪ R , we develop rigorously justified first- and third-order asymptotic expansions with precise error estimates in operator norm after a frequency cut-off. The spectral properties of the operator are fully characterized, including self-adjointness, dissipativity, and high-frequency asymptotics. Our results provide both theoretical insights into boundary operators in singular geometries and practical approximation formulas for computational applications in domain decomposition methods and thin-layer modeling.
We develop uniquely solvable second kind boundary integral equations systems for solving the transmission problem associated with linear elastic waves propagation in two-dimensional isotropic and homogeneous media. With the help of the Günter derivative, the transmission boundary conditions are rewritten in such a way the physical system can be reduced to a compact perturbation of an invertible zero-order operator—a lower off-diagonal perturbation of identity—without regularization procedure. Inspired by the Müller formulation developed for the electromagnetic case, hypersingular and strong singularities of the boundary integral operator kernels are cancelled out by considering particular combinations of the Calderón projections on the contact boundary curve. Irregular frequencies do not occur with this method. Existence and uniqueness can be established on either a single function space or on a mixed regularity solution space.
A mathematical model is derived for the dynamics of a cylinder, or wheel, rolling over a thin viscous film. The model combines the Reynolds lubrication equation for the fluid with an equation of motion for the wheel. Two asymptotic limits are studied in detail to interrogate the dynamics of levitation: an infinitely wide wheel and a relatively narrow one. In both cases, the front and back of the fluid-filled gap are either straight or nearly so. To bridge the gap between these two asymptotic limits, wheels of finite width are considered, introducing a further simplifying approximation: although the front and back are no longer expected to remain straight for a finite width, the footprint of the fluid-filled gap is still taken to be rectangular, with boundary conditions imposed at the front and back in a wheel-averaged sense. The Reynolds equation can then be solved by separation of variables. For wider wheels, with a large amount of incoming flux or a relatively heavy loading of the wheel, the system is prone to flooding by back flow with fluid unable to pass underneath. Otherwise steady planing states are achieved. Both lift-off and touch-down are explored for a wheel rolling over a film of finite length. Theoretical predictions are compared with a set of experimental data.
Steady, fully developed Poiseuille flow of an incompressible Newtonian fluid in a circular-segment duct is analysed. The problem is formulated in bipolar coordinates and the axial-velocity field is reduced to a Laplace problem for a harmonic correction. A Fourier cosine transform with respect to the unbounded coordinate yields an elementary boundary-value problem in the remaining variable, which is solved in transform space and inverted by residue calculus. This procedure leads to an explicit series representation for the axial velocity. The volumetric flow rate and the associated Poiseuille number are then obtained in a form suitable for computation, and closed-form expressions are derived for several opening angles. In addition, a 6/6 Padé approximant for the Poiseuille number as a function of the central angle is constructed, which reproduces the analytical results with high accuracy over the full range of circular-segment geometries.
We present a theoretical framework for estimating the added mass of a partially submerged spherical buoy undergoing heave oscillations. Asymptotic analyses are conducted in the limiting cases of low and high oscillation frequencies, where the classical potential flow theory applies. For each frequency regime, we consider two canonical geometries that approximate the wetted surface of the buoy: (i) a small perturbation about a reference flat disk, relevant when the immersion depth is small compared to the buoy radius (immersion ratio ε _d≪ 1 ), and (ii) a small perturbation about a reference hemisphere, appropriate when the immersion depth is comparable to the radius. Perturbation expansions for the added mass are derived in each case. By smoothly bridging the results from these asymptotic limits, we construct a composite approximation for the added mass across a wide range of immersion ratios. This theoretical prediction is validated against direct numerical simulations, showing excellent agreement: the error remains below 1.1 ε _d≤ 1 , below 6 1 < ε _d≤ 2 , and below 25 2 < ε _d≤ 3.75 . Our results offer a tractable and yet physically grounded model for added mass evaluation, applicable to the analysis and optimization of systems involving floating oscillating bodies, such as two-body wave energy converters.
We develop a mathematical model to predict the most effective design of porous tissue-engineering scaffolds and analyze the choice of values for input parameters such as channel radius, shear stress, nutrient concentration, and nutrient flow pressure, while assuming a constant inlet flux of suspended nutrients. Our model utilizes a branching structure in which scaffold pores bifurcate at each layer junction, thereby allowing for investigation of the input parameters at a pore level. By assuming all branching structures to be equivalent, we assume homogeneity along two axes, meaning the biological scaffold’s geometry is effectively reduced to one dimension. We employ established fluid dynamics equations such as Darcy’s law, the continuity equation, and the advection–diffusion-reaction equation to model the evolution of parameters such as pore radius, shear stress, pressure, fluid velocity, and nutrient concentration. Further simplification of these equations is achieved via nondimensionalization and asymptotic analysis based on the small aspect ratio of the scaffold. Our results establish quantitative relationships among the input parameters to provide a foundation for effectual construction of engineered tissue in the shortest time possible. Further, we devise a trade-off analysis for differing scaffold geometries by comparing ratios of decreasing layer thickness and decreasing initial pore radius in successive downstream layers; these ratios guide predictions for candidates for ideal scaffold geometries according to priorities of time, cost, or total tissue volume. Particularly, we find that minimal values of both these ratios together yield larger volumes of new tissue, while jointly large ratios produce far less tissue but in much less time. Mixed values (e.g., small layer thickness ratio and large initial radius ratio) and intermediate values (in the middle range for both ratios) of these ratios produce moderate tissue growth, in a moderate amount of time, and yield less waste of material used to construct the scaffolds.
The location of the Touch Down Point (TDP) of a slack mooring line, closely linked to its residual resistance, requires accurate modeling. We introduce an analytical formulation of seabed contact dynamics as a rigorous boundary condition for the system. Building on this, a modal analysis is developed to fully characterize the natural oscillation modes of a slack mooring line without assumptions on tension perturbations. The framework captures the dynamics of the moving boundary at the seabed and establishes a direct link between the maximum oscillation amplitudes and the onset of shock formation at the contact point. A complementary formulation describes, in static or quasi-static regimes, the relationship between fairlead displacement and seabed contact point motion, providing a solid basis for the study of seabed interaction mechanics. Finally, a numerical investigation contrasts the proposed exact seabed boundary condition with a widely used empirical approximation. The comparison is carried out against two exact benchmarks: (i) natural frequencies from modal analysis and (ii) analytically derived seabed contact point displacement under steady fairlead tension. Results confirm that the new boundary condition improves predictive accuracy without added computational cost. These findings advance the understanding of slack-line dynamics and support future analytical and numerical studies.
This study presents a novel strain-based rectangular plate finite element, denoted as SBRPFE20, developed for the accurate analysis of thin plate bending problems. The formulation is established within the framework of Kirchhoff plate theory and incorporates a coupled strain-based membrane–bending approach to effectively capture the interaction between bending and in-plane responses, particularly in functionally graded materials. Unlike existing strain-based plate elements, the proposed formulation is specifically developed to accurately model functionally graded material (FGM) plates, accounting for the continuous variation of material properties through the thickness. The proposed element employs four corner nodes with essential kinematic degrees of freedom and is designed to achieve high accuracy while maintaining computational efficiency. The performance of the SBRPFE20 element is systematically assessed through a comprehensive set of numerical benchmarks, including static bending and free vibration analyses of isotropic and functionally graded plates under various loading conditions, boundary configurations, and geometrical complexities. The numerical results demonstrate that the proposed element exhibits rapid and smooth convergence, excellent numerical stability, and superior accuracy compared with well-established finite elements available in the literature. The formulation demonstrates a locking-free behavior, particularly in the thin plate limit, and does not exhibit spurious stiffness effects, as confirmed through convergence studies and mesh distortion analyses. Owing to its robustness, efficiency, and accuracy, the SBRPFE20 element provides a reliable and effective tool for the static and dynamic analysis of thin isotropic and functionally graded plates.
Silicon carbide is a widely used material due to its unique combination of physical and chemical properties. However, existing Acheson furnaces, which cause direct CO_2 emissions, are expensive and energy inefficient. Rotary kilns are a promising alternative, but their use for silicon carbide production is still under development. In this article, we present a mathematical model for the consumption of quartz and carbon, and the formation of silicon carbide in a rotary kiln. We focus on the interplay between reaction kinetics, solid and gas transport, and thermal effects. Assuming radial mixing within the bed, we derive a simplified one-dimensional model that captures the dominant physics of the system. The model tracks the evolution of quartz, carbon, and silicon carbide, as well as gas-phase species and temperature, down the length of the kiln. We nondimensionalise the model and identify key parameter groupings—relating supplied heat to kiln fill level, initial carbon particle size, and the relative speed of the two chemical reactions we consider. Then, we examine the model’s behaviour via asymptotic analysis, before presenting numerical simulations. Our analysis shows that there is only one dimensionless parameter group that strongly influences reactor performance, with silicon carbide yield increasing monotonically with the value of this parameter. These findings offer broad-stroke insights into design principles and parameter operating regimes that favour efficient silicon carbide reactors.
