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Boundary element methods provide powerful techniques for the analysis of problems involving coupled multi-physical response. This paper presents the integral equation formulation for the size-dependent thermoelastic response of solids under steady-state conditions in three dimensions. The formulation is based upon consistent couple stress theory, which features a skew-symmetric couple-stress pseudo-tensor. For general anisotropic thermoelastic material, there is not only thermal strain deformation, but also thermal mean curvature deformation. Interestingly, in this size-dependent multi-physics model, the thermal governing equation is independent of the deformation. However, the mechanical governing equations depend on the temperature field. First, thermal and mechanical weak forms and reciprocal theorems are developed for this theory. Then, an integral equation formulation for three-dimensional size-dependent thermoelastic isotropic materials is derived, along with the corresponding singular infinite-space fundamental solutions or kernel functions. For isotropic materials, there is no thermal mean curvature deformation, and the thermoelastic effect is solely the result of thermal strain deformation. As a result, the size-dependent behavior is specified entirely by a single characteristic length scale parameter l, while the thermal coupling is defined in terms of the thermal expansion coefficient α, as in the classical theory of steady-state isotropic thermoelasticity.

The surface shear viscosity of an insoluble surfactant monolayer often depends strongly on its surface pressure. Here, we show that a particle moving within a bounded monolayer breaks the kinematic reversibility of low-Reynolds-number flows. The Lorentz reciprocal theorem allows such irreversibilities to be computed without solving the full nonlinear equations, giving the leading-order contribution of surface pressure-dependent surface viscosity. In particular, we show that a disc translating or rotating near an interfacial boundary experiences a force in the direction perpendicular to that boundary. In unbounded monolayers, coupled modes of motion can also lead to non-intuitive trajectories, which we illustrate using an interfacial analogue of the Magnus effect. This perturbative approach can be extended to more complex geometries, and to two-dimensional suspensions more generally.

In the study of fluid dynamics and transport phenomena, key quantities of interest are often the force and torque on objects and total rate of heat/mass transfer from them. Conventionally, these integrated quantities are determined by first solving the governing equations for the detailed distribution of the field variables (i.e. velocity, pressure, temperature, concentration, etc.) and then integrating the variables or their derivatives on the surface of the objects. On the other hand, the divergence form of the conservation equations opens the door for establishing integral identities that can be used for directly calculating the integrated quantities without requiring the detailed knowledge of the distribution of the primary variables. This shortcut approach constitutes the idea of the reciprocal theorem, whose closest relative is Green’s second identity, which readers may recall from studies of partial differential equations. Despite its importance and practicality, the theorem may not be so familiar to many in the research community. Ironically, some believe that the extreme simplicity and generality of the theorem are responsible for suppressing its application! In this Perspectives piece, we provide a pedagogical introduction to the concept and application of the reciprocal theorem, with the hope of facilitating its use. Specifically, a brief history on the development of the theorem is given as a background, followed by the discussion of the main ideas in the context of elementary boundary-value problems. After that, we demonstrate how the reciprocal theorem can be utilized to solve fundamental problems in low-Reynolds-number hydrodynamics, aerodynamics, acoustics and heat/mass transfer, including convection. Throughout the article, we strive to make the materials accessible to early career researchers while keeping it interesting for more experienced scientists and engineers.

Since its development, Stokesian dynamics has been a leading approach for the dynamic simulation of suspensions of particles at arbitrary concentrations with full hydrodynamic interactions. Although developed originally for the simulation of passive particle suspensions, the Stokesian dynamics framework is equally well suited to the analysis and dynamic simulation of suspensions of active particles, as we elucidate here. We show how the reciprocal theorem can be used to formulate the exact dynamics for a suspension of arbitrary active particles, and then show how the Stokesian dynamics method provides a rigorous way to approximate and compute the dynamics of dense active suspensions where many-body hydrodynamic interactions are important.

We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $\\Delta p$. We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ($De$), we provide analytical expressions for the velocity, stress and the $q$–$\\Delta p$ relation in the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit the reciprocal theorem derived by Boyko $\\&$ Stone (Phys. Rev. Fluids, vol. 6, 2021, L081301) to obtain the $q$–$\\Delta p$ relation at the next order, $O(De^3)$, using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$. We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain.

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or with each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point-singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head–tail reversal of the swimmer, in the near field, microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of approximately two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.

The purpose of this article is to improve the accuracy and expand the force analysis of the lens by using the theory of plates and shells and reciprocal theorem. The first one concerns the deflection expression of a circular lens of equal thickness supported at several uniform points. The second calculates the deflection of the lens under its own weight, and then starting from known Zernike coefficient, PV value and RMS value in optics, the first 37 Zernike coefficients are obtained. The third compares the Zernike coefficients and lens center deflections of partic-ular cases. The derivation process based on plates and shells theory and reciprocal theorem is given in details. As validation, the lens deflec-tion expression is applied to calculate 4 points support, the predicted re-sults illustrate great agreements with the finite element simulation solution. Finally, the proposed method has shown great potential for engi-neering analysis and may also bring inspirations for optical field, as well as other plate and shell theory system.

Active particles driven by chemical reactions are the subject of intense research to date due to their rich physics, being intrinsically far from equilibrium, and their multiple technological applications. Recent attention in this field is now shifting towards exploring the fascinating dynamics of active and passive mixtures. Here we realize active colloidal rafts, composed of a single catalytic particle encircled by several shells of passive microspheres, and assembled via light-activated chemophoresis. We show that the cluster propulsion mechanism transits from diffusiophoretic to diffusioosmotic as the number of colloidal shells increases. Using the Lorentz reciprocal theorem, we demonstrate that in large clusters self-propulsion emerges by considering the hydrodynamic flow via the diffusioosmotic response of the substrate. The dynamics in our active colloidal rafts are governed by the interplay between phoretic and osmotic effects. Thus, our work highlights their importance in understanding the rich physics of active catalytic systems. Diffusiophoresis and diffusioosmosis govern the dynamics of catalytically active colloids. Here, the authors show that the competition between these effects can control the propulsion direction of active colloidal rafts composed of a central apolar particle and several shells of passive ones.

In this work, we study active particles with prescribed surface velocities in non-Newtonian fluids. We employ the reciprocal theorem to obtain the velocity of an active spherical particle with an arbitrary axisymmetric slip velocity in an otherwise quiescent second-order fluid. We then determine how the motion of a diffusiophoretic Janus particle is affected by complex fluid rheology, namely viscoelasticity and shear-thinning viscosity, compared to a Newtonian fluid, assuming a fixed slip velocity. We find that a Janus particle may go faster or slower in a viscoelastic fluid, but is always slower in a shear-thinning fluid as compared to a Newtonian fluid.

Despite significant advances in the field of man-made micro- and nanomotors, it remains a challenge to precisely control their motion in bounded environments. Here, we present a theoretical analysis of a thermally activated micromotor near a plane wall under the action of a background linear temperature field. The coupling between the autonomous and field-directed motions has been resolved using a combined analytical–numerical framework comprising general solutions in bispherical coordinates and the reciprocal theorem for creeping flows. Results reveal giant augmentation in swimming speeds, the controlling parameter zones for positive and negative thermotaxes and the flexibility of steering perpendicular to the field gradient for an isolated micromotor. Boundary-instigated thermo-fluidic modulations at different levels of confinements and preferential orientations cause directional switching of both the vertical translation and rotation parallel to the wall, thereby drastically altering the phase portraits of the swimmer dynamics. Contrasting trajectory characteristics, e.g. escape, attraction, are partitioned by unstable separatrices in the phase portraits, while competitive repulsion (attraction) after attraction (repulsion) characteristics emerge for different relative field strengths $\\mathcal {S}$ and gradient orientations $\\theta _T$. Below $\\mathcal {S}=0.25$, highly counter-intuitive trajectories result when the micromotor is initially launched from an overlapping escape zone. Moreover, external-field-assisted microswimming can uniquely tune the directionality of wall-parallel translation, broadening the scope of dynamic regulation of self-propulsion. Thus, providing insights into a precisely controlled fuel-free actuation of micromotors near a physical obstacle, the present study stands as a step toward addressing the increasing demand for successful implementation of micromotors in futuristic clinical and environmental applications.