We analyze how nonlocal stresslet interactions generate modal instability and finite-time nonmodal amplification in a particle-resolved zero-Reynolds-number active suspension. Three configuration-dependent operators are compared side by side: the orientation-projected pairwise interaction matrix W ( q ) , its activity-weighted counterpart L _p( q ,A)=A W ( q ) , and the full tangent Jacobian J ( q )=D_ q F ( q ) of the deterministic drift, constructed by centered finite differences at snapshots on an H-weighted state space. Within the sampled parameter range, once the direct activity scaling is divided out, the projected operator provides no incremental information beyond the imposed activity ramp. By contrast, the full tangent operator contains spatial derivatives, orientational derivatives, and translation–orientation coupling that generate substantial finite-time amplification in addition to modal growth: the pseudospectral abscissa a_ε =10^-1( J _H) exceeds the spectral abscissa a( J ) by 0.1–0.2, the eigenvector condition number κ (V) is of order 10^2 – 10^3 , and the normalized nonmodal gain R_nm(T=3) reaches ≈ 4.5 at the largest sampled activity. A logarithmic norm bound obtained through Grönwall’s inequality, ‖ (t_0+T,t_0)‖ _2^2≤exp[2∫ _t_0^t_0+Tω ( J _H(s)) ds] , transfers the frozen-time numerical abscissa into a rigorous upper bound on the trajectory propagator. The sampled regime is modally unstable within the parameter range studied; a stable-nonnormal regime a<0<ω is not exhibited by the simulations, and a weakly nonlinear normal form analysis is retained only as a formal outlook. The main limitation of the present study is that all numerical results concern a denominator-regularized minimum-image model rather than a fully smooth periodic Ewald Stokes formulation.
Capillary rise is a fundamental transport mechanism in natural and engineered porous systems, where inertial effects govern the early-time dynamics despite the small length scales involved. Recent analytical models incorporate an added mass term to remove the classical singularity in capillary rise dynamics, commonly using a hemispherical control volume and an imposed discontinuous velocity profile at the tube entrance. In this study, we develop an analytical potential-flow framework for estimating the added mass coefficient beneath the capillary entrance, avoiding the hemispherical velocity discontinuity present in classical reduced-order approaches. A generalized coaxial cylindrical geometry is first considered, from which three limiting cases are analyzed: (i) a cylindrical tube immersed in an infinitely deep reservoir, (ii) a coaxial cylindrical configuration, and (iii) a finite-depth reservoir. Closed-form analytical expressions and numerical evaluations of the added mass coefficient are presented, yielding a coefficient for the classical configuration that is lower than the widely used hemispherical-cap estimate. The results provide a mathematically consistent inertial closure parameter for reduced-order capillary rise models and complex geometrical configurations.
This work presents a discrete mechanical formulation for the nonlinear static bending analysis of stepped three-directional functionally graded material (3D-FGM) beams under different loading conditions. The beam is modeled by transverse displacement degrees of freedom connected through rotational springs for bending rigidity and longitudinal springs for the stretching effect induced by large deflections. The material properties vary continuously in the axial, width, and thickness directions, while the stepped geometry is described by piecewise constant cross-sectional dimensions. The nonlinear equilibrium equations are derived from the stationarity of the total potential energy. To reduce the computational effort, an adapted second reduced formulation is employed, where the dominant generalized displacement is treated explicitly, and the remaining correction components are obtained from a reduced linear system. The nonlinear deflection field is then reconstructed using the eigenvector basis of the linear stiffness matrix. Parametric investigations are carried out to assess the effects of step ratio, step location, gradation indices, boundary conditions, loading type, and load intensity on the nonlinear load–deflection response. The results show that the proposed discrete approach efficiently captures the coupled effects of material gradation, geometric discontinuities, and large-deflection nonlinearity.
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.
