Cosserat interphase models for elasticity with application to the interphase bonding a spherical inclusion to an infinite matrix

Interphases are often modeled as interfaces with zero thickness using jump conditions that can be developed based on approximate shell or membrane models which are valid for specific limited ranges of the elastic material parameters. For a two-dimensional problem it has been shown (Rubin and Benveniste, 2004) that the Cosserat model of a finite thickness interphase is a unified model that is accurate over the full range of elastic parameters. In contrast, many other interphase models are valid for only limited ranges of the elastic parameters. In this paper, the accuracy of different Cosserat models of a finite thickness interphase that connects a spherical inclusion to an infinite matrix is examined. Specifically, four Cosserat interphase models are considered: a general shell (GS)$(\mathit{GS})$, a membrane-like shell (MS)$(\mathit{MS})$, a simple shell (SS)$(\mathit{SS})$ and a generalized membrane (GM)$(\mathit{GM})$. The models (GS)$(\mathit{GS})$ and (MS)$(\mathit{MS})$ both satisfy restrictions on the strain energy function of the interphase that ensure exact solutions for all homogeneous three-dimensional deformations, while the other models (SS)$(\mathit{SS})$ and (GM)$(\mathit{GM})$ do not satisfy these restrictions. The importance of these restrictions is examined for the three-dimensional inhomogeneous inclusion problem being considered. This is the first test of the accuracy of an elastic interphase model for a spherical interphase.     

Lateral migration of a capsule in a plane Poiseuille flow in a channel

Three-dimensional numerical simulation is presented on the motion of a deformable capsule undergoing large deformation in a plane Poiseuille flow in a channel at small inertia. The capsule is modeled as a liquid drop surrounded by an elastic membrane which follows neo-Hookean law. The numerical methodology is based on a mixed finite-difference/Fourier transform method for the flow solver and a front-tracking method for the deformable interface. The methodology can address large deformation of a capsule over a wide range of capsule-to-medium viscosity ratio. An extensive validation of the methodology is presented on capsule deformation in linear shear flow and compared with the boundary-element/integral simulations. Motion of a capsule in wall-bounded parabolic flow is simulated over an extended period of time to consider both transient and steady-state motion. Lateral migration of the capsule towards the centerline of the channel is observed. Results are presented over a range of capillary number, viscosity ratio, capsule-to-channel size ratio, and lateral location. After an initial transient phase during which the capsule deforms very quickly, the flow of the capsule is observed to be a quasi-steady process irrespective of capillary number (Ca)$(\mathit{Ca})$, capsule-to-channel size ratio (a/H)$(a/H)$, and viscosity ratio (λ)$(\lambda )$. Migration velocity and capsule deformation are observed to increase with increasing Ca$\mathit{Ca}$ and a/H$a/H$, but decrease with increasing λ$\lambda$, and increasing distance from the wall. Numerical results on the capsule migration are compared with the analytical results for liquid drops, and capsules with Hookean membrane which are valid in the limit of small deformation. Unlike the prediction for liquid drops, capsules are observed to migrate toward the centerline for 0.2?λ?5$0.2?\lambda ?5$ range considered here. The migration velocity is observed to depend linearly on (a/H)³$(a/H{)}^3$, in agreement with the small-deformation theory, but non-linearly on Ca$\mathit{Ca}$ and the distance from the wall, in violation of the theory. Using the present numerical results and the analytical results, we present a correlation that can reasonably predict migration velocity of a capsule for moderate values of a/H$a/H$ and Ca$\mathit{Ca}$.     

Wake response of a stationary finite-sized particle in a turbulent channel flow

Here we consider the effect of a finite-sized stationary particle in a channel flow of modest turbulence at _Reτ=178.12${\mathit{Re}}_{\tau }=178.12$. The size of particle is varied such that the particle Reynolds number ranges from about 40 to 450. The location of the particle is chosen to be either in the buffer layer (_yp+=17.81)$(y_{p+}=17.81)$ or at the channel center. Fully resolved direct numerical simulations of the turbulent channel flow around the particles is performed. Here the ambient turbulence intensity relative to the mean velocity seen by the particle is large (I=23.16%)$(I=23.16\%)$ in the buffer region, while it is substantially lower (I=4.09%)$(I=4.09\%)$ at the channel center. We present results on turbulence modulation due to the particle in terms of wake dynamics and vortex shedding.     

A Riemannian approach to strain measures in nonlinear elasticity

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n)$\mathrm{SO}(n)$ of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n)$\mathrm{GL}(n)$. Objectivity requires the Riemannian metric to be left-GL(n)$\mathrm{GL}(n)$-invariant, isotropy requires the Riemannian metric to be right-O(n)$\mathrm{O}(n)$-invariant. The latter two conditions are only satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n)$\mathrm{GL}(n)$. Surprisingly, the final result is basically independent of the chosen parameters.     

Chaining of weakly interacting particles suspended in viscoelastic fluids

An asymptotic theory based on multipole expansions is presented for multiparticle interactions in unbounded, weakly viscoelastic, creeping flows. The theory accounts for non-Newtonian sphere–sphere interactions that are of order O(De(a/R)²)$\text{O}(De{(a/R)}^2)$, where De is the Deborah number, a the sphere radius and R is the sphere–sphere separation. Analytic expressions are derived for the non-Newtonian correction to the multisphere mobility matrix for non-neutrally buoyant sedimenting spheres, and for neutrally buoyant spheres suspended in a shear flow. It is shown that these expressions give rise to particle chaining in simulations of interacting spherical particles.     

A scaling analysis for point–particle approaches to turbulent multiphase flows

Simple dimensional arguments are used in establishing three different regimes of particle time scale, where explicit expression for particle Reynolds number and Stokes number are obtained as a function of nondimensional particle size (d/η)$(d/\eta )$ and density ratio. From a comparative analysis of the different computational approaches available for turbulent multiphase flows it is argued that the point–particle approach is uniquely suited to address turbulent multiphase flows where the Stokes number, defined as the ratio of particle time scale to Kolmogorov time scale (_τp/_τk)$({\tau }_p/{\tau }_k)$, is greater than 1. The Stokes number estimate has been used to establish parameter range where point–particle approach is ideally suited. The point–particle approach can be extended to handle “finite-sized” particles whose diameter approach that of the smallest resolved eddies. However, new challenges arise in the implementation of Lagrangian–Eulerian coupling between the particles and the carrier phase. An approach where the inter-phase momentum and energy coupling can be separated into a deterministic and a stochastic contribution has been suggested.     

A twofold strength and toughness criterion for the onset of free-edge shear delamination in angle-ply laminates

Free edge delamination in composite structures results from very localised stress fields which induce a stress concentration promoting the nucleation of an interfacial crack. To predict such a delamination onset at the free edge of a $(\pm \theta {)}_s$ laminate in traction, use is made of a strength and toughness criterion which combines a stress condition with an energy analysis. A generalised plane strain model allows to determine the stress distribution near the free edge and the energy released by the nucleation of an interfacial crack. The results show that this approach can predict the delamination onset for $((\pm 10{)}_s\text{,}(\pm 20{)}_s)$ laminates provided the interfacial fracture energy and interlaminar shear strength are known. These characteristic values can be identified with the help of traction tests performed on samples with different thicknesses.     

Scale-by-scale energy budget in a turbulent boundary layer over a rough wall

Hot-wire velocity measurements are carried out in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing (w) of two consecutive rods and the rod height (k). The pressure distribution around the roughness element is used to accurately measure the mean friction velocity ($U_{\tau }$) and the error in the origin. It is found that $U_{\tau }$ remained practically constant in the streamwise direction suggesting that the boundary layer over this surface is evolving in a self-similar manner. This is further corroborated by the similarity observed at all scales of motion, in the region $0.2⩽y/\delta ⩽0.6$, as reflected in the constancy of Reynolds number ($R_{\lambda }$) based on Taylor’s microscale and the collapse of Kolmogorov normalized velocity spectra at all wavenumbers.A scale-by-scale budget for the second-order structure function $〈{(\delta u)}^2〉$ ($\delta u=u(x+r)-u(x)$, where u is the fluctuating streamwise velocity component and r is the longitudinal separation) is carried out to investigate the energy distribution amongst different scales in the boundary layer. It is found that while the small scales are controlled by the viscosity, intermediate scales over which the transfer of energy (or $〈{(\delta u)}^3〉$) is important are affected by mechanisms induced by the large-scale inhomogeneities in the flow, such as production, advection and turbulent diffusion. For example, there are non-negligible contributions from the large-scale inhomogeneity to the budget at scales of the order of $\lambda$, the Taylor microscale, in the region of the boundary layer extending from $y/\delta =0.2$ to 0.6 ($\delta$ is the boundary layer thickness).     

A new constitutive equation for elastoviscoplastic fluid flows

From thermodynamic theory, a new three-dimensional model for elastoviscoplastic fluid flows is presented. It extends both the Bingham viscoplastic and the Oldroyd viscoelastic models. Fundamental flows are studied: simple shear flow, uniaxial elongation and large amplitude oscillatory shear. The complex moduli (G^′,G^″)$(G^{\prime }\text{,}{G}^{″})$ are found to be in qualitative agreement with experimental data for materials that present microscopic network structures and large scale rearrangements. Various fluids of practical interest, such as liquid foams, droplet emulsions or blood, present such elastoviscoplastic behavior: at low stress, the material behaves as a viscoelastic solid, whereas at stresses above a yield stress, the material behaves as a fluid.     

Flow-induced vibration of a square cylinder without and with interference

This paper presents the results of an investigation on the interference effects of a rigid square cylinder on the transverse vibrations of a spring-mounted square cylinder (test cylinder) exposed to a uniform flow. The interference effects were studied for the tandem, side-by-side and staggered arrangements. Experiments have been carried out for various relative dimensions of the test cylinder and the interfering cylinder; the tests for the staggered arrangements were conducted at several tandem distances between the two. The results indicate that there is a critical combination of relative dimensions and spacing that gives rise to maximum amplitude of vibration. Among the cases studied, tandem arrangement with $L/B=1.25$ and $b/B=0.5$ gives rise to maximum amplitude of vibration with $(a/B{)}_{\max }=0.57$. A tentative explanation is offered for the observed features based on flow-visualization studies conducted as a part of the experimental investigation.