Grain- and macro-scale kinematics for granular micromechanics based small deformation micromorphic continuum model

Macro-scale deformation of granular solids comprising large number of grains (>10⁶) are most efficiently described within the framework of continuum mechanics. It is notable, however that the micro-scale deformations in these materials are concentrated at the grain-boundaries or grain-contacts. Thus, the deformation energies in these systems must be modeled by considering the deformations concentrated in the neighborhood of the grain-boundaries or grain-contacts. To address this issue, grain-interactions has been widely described in the Hertzian sense by considering the relative movement of points on either side of a grain boundary or contact treated as an imperfect interface. This communication introduces the relevant kinematic variables given in the terms of the grain displacements, spins and size that can be used to estimate the relative movement of a grain boundary or contact. The macro-scale kinematic variables useful for continuum modeling are then identified with the grain-scale kinematic variables. The deformation energy density of the granular solid can thus be expressed both in terms of the grain-scale as well as the macro-scale kinematic variables providing the necessary pathway for micro-macro identification which can lead to non-classical micromorphic continuum models that incorporate grain-scale representation.     

Effects of technological delay on insulin and blood glucose in a physiological model

In this paper, a physiological model of invasive blood-glucose (BG) measurement is employed to consider the diabetic treatment by the external auxiliary system, i.e., artificial pancreas (AP). For such system, there are two time delays, i.e., technological and liver's physiological delay, where the former comes from external auxiliary system with the active pancreas inputting. The technological delay and the infection degree of patients are considered as two controlled parameters to regulate the BG level of patients. This two parameters can also lead to the non-resonant double Hopf bifurcations. The classification and unfolding for the non-resonant double Hopf bifurcation are performed in terms of non-linear dynamics. The results show that such controlled parameters are very important. They can determine the efficiency for the diabetic treatment. It implies whether the diabetic patients recover or are still tormented by the simple or complex glucose fluctuation. The results have also been promising applications on analyzing, predicting and optimizing the medical outcome, evaluating the medical risk and feasibility. The physiological meaning in this paper is that one is able to achieve the better medical outcomes for the different patients by controlling the technological delay qualitatively.     

On double reductions from symmetries and conservation laws

We present the theory of double reductions of PDEs with two independent variables that admit a Lie point symmetry and a conserved vector invariant under the symmetry. The theory is applied to a third order nonlinear partial differential equation which describes the filtration of a visco-elastic liquid with relaxation through a porous medium.     

Buchdahl limit for <Emphasis Type="Italic">d</Emphasis>-dimensional spherical solutions with a cosmological constant

We investigate the conditions for which a d-dimensional perfect fluid solution is in hydrostatic equilibrium with a cosmological constant. We find a generalization of Buchdahl inequality and obtain an upper bound for the degree of compactification. Using the Tolman–Oppenheimer–Volkoff equation to get a lower bound for the degree of compactification we analyse the regions where the solution is in hydrostatic equilibrium. We obtain the inner metric solution and the pressure for the constant fluid density model.     

Collisionally inhomogeneous Bose-Einstein condensates in double-well potentials

In this work, we consider quasi-one-dimensional Bose-Einstein condensates (BECs), with spatially varying collisional interactions, trapped in double-well potentials. In particular, we study a setup in which such a “collisionally inhomogeneous” BEC has the same (attractive-attractive or repulsive-repulsive) or different (attractive-repulsive) types of interparticle interactions. Our analysis is based on the continuation of the symmetric ground state and anti-symmetric first excited state of the non-interacting (linear) limit into their nonlinear counterparts. The collisional inhomogeneity produces a saddle-node bifurcation scenario between two additional solution branches; as the inhomogeneity becomes stronger, the turning point of the saddle-node tends to infinity and eventually only the two original branches remain, which is completely different from the standard double-well phenomenology. Finally, one of these branches changes its monotonicity as a function of the chemical potential, a feature especially prominent, when the sign of the nonlinearity changes between the two wells. Our theoretical predictions, are in excellent agreement with the numerical results.