Ben Nadler

We use the principles of non-equilibrium thermodynamics to rigorously formulate the transport equations for granular systems consisting of oriented particles. The state variables are taken to be the density, velocity, thermal temperature, granular temperature (particles agitation) and the orientation tensor. The evolution of the state variables is governed by the associated balance laws in terms of fluxes. The contributions of the granular agitation energy and orientation to entropy are introduced into the Gibbs equation. The balance of entropy is used to identify the entropy production as the product of thermodynamics forces and fluxes. Using classical linear non-equilibrium thermodynamics the fluxes are considered to be linear in the thermodynamic forces. The Onsager–Casimir reciprocal relations and the representation theorem of isotropic tensors provide further restrictions that simplify the formulation. The non-negative entropy production requirement is satisfied by restricting the matrix of phenomenological coefficients to be positive semidefinite. Similarly the boundary conditions are constructed. The transport coefficients are then determined by comparison with available results from the granular kinetic theory of spherical particles and other available results for oriented particles. It is shown that not only these results are well captured, but also the formulation provides a framework for further generalization. The significant contribution of this work is the rigorous formulation of a physically admissible generalization to granular gases of oriented particles which reveals the role of the orientation in the transport equations and identifies couplings that might otherwise be omitted.

We join the theories that describe the orientation, treated as a tensor, of liquid crystals and the agitation of inelastic grains to obtain a mathematical model of non-spherical particles contained in a quasi-2D square box and driven into dissipative collisions through the vibration of two of the four flat walls, in the absence of gravity and mean flow. The particle agitation induces spatial inhomogeneities in the density and the isotropic–nematic transition to take place somewhere inside the box, if the particle shape is sufficiently far from spherical. We show quantitative agreement between the theory and discrete numerical simulations of ellipsoids of different length-to-diameter ratio. We need to fit two dimensionless parameters that were not previously available or determined in different configurations. These parameters, of order unity and weakly dependent on the shape of the particles, are indicative of the resistance to alignment distortion associated with entropic elasticity.

We generalize kinetic theory of inelastic spheres to uniaxial, nonspherical grains by including the orientational tensor as a state variable. The theory has three phenomenological parameters, with one parameter to account for the dependency of the stresses on the orientation, which is exactly one for frictionless cylinders, and the other two parameters to predict the orientational tensor. We model the competition between the alignment induced by shearing and the misalignment due to collisions in the evolution law for the orientational tensor. The theory can predict a significant reduction in the viscosity in response to alignment measured in discrete simulations of homogeneous shear flows of prolate and oblate frictionless cylinders.

Cancer cell migration, as the principal element of tumor invasion, involves different cellular mechanisms. Various modes of cell migration including single and collective motions contribute to the invasion patterns. The competition between adhesive cell–cell and cell–matrix forces is a key factor that determines such patterns. In this paper, we study a distinct shell-type mode of tumor invasion observed in brain and breast tumors. In this mode, cells at the outer layer of the tumor collectively move away from the core and form a shell-type shape. Both the core and the shell sustain a sharp interface between cells and the surrounding matrix. To model the preserved interface, we adopted a Cahn–Hilliard-type free energy relation with the contribution of the interfacial stress. This nonconvex form of free energy allows for cells to remain together and preserve the tumor core via adhesive cell–cell forces while separating the core from the surrounding matrix across a continuous sharp interface. In addition, the motion of the shell was modeled using the chemotactic migration of cells in response to the gradient of nutrients. The associated fluxes of cells were implemented in a general form of balance law. A non-Michaelis–Menten kinetics model was adopted for the proliferation rate of cells. The flux of nutrients was also modeled using a simple diffusion equation. The comparison between the model predictions and experimental observations indicates the ability of the model to manifest the salient features of the invasion pattern.

Among different hallmarks of cancer, understanding biomechanics of tumor growth and remodeling benefits the most from the theoretical framework of continuum mechanics. Tumor remodeling initiates when cancer cells seek new homeostasis in response to the microenvironmental stimuli. Cells within a growing tumor are capable to remodel their inter- and intra-connections and become more mobile to achieve a new homeostasis. This mobility enables the tumor to undergo large deformation. In this work, we studied the remodeling of homogeneous tumors, at their early stage of growth, in the context of continuum mechanics. We developed an evolution law for the remodeling-associated deformation which correlates the remodeling to a characteristic tensor of external stimuli. The asymmetric remodeling and the induced mechanical stresses were analyzed for different types of biochemical distributions. To experimentally investigate the model, we studied the remodeling of human glioblastoma (hGB) tumoroids in response to the gradient of nutrients. Using a tumoroid-on-a-chip platform, the degree of remodeling was estimated for the ellipsoidal tumoroids over time. It was observed that higher gradient of nutrients induces higher degree of ellipticity suggesting that the gradient of nutrient is a characteristic property of nutrient distribution that derives the remodeling. We also showed that remodeling gives rise to heterogeneity in cell distribution forming circumferentially aligned cells within the tumors. Compared to the existing studies on tumor growth, our work provides a biomechanical module that relates the remodeling to biochemical stimuli, and allows for large deformation. It also includes experimental component, a necessary but challenging step, that connects the theory and reality to evaluate the practicability of the model.

Oscillatory shear flows of axisymmetric grains exhibit amplitude-dependent rheological responses, which is related to the evolution of the microstructure. In this work, it is shown that the highly ordered configuration of grains at steady-state shear flow undergoes microstructural rearrangement when subjected to shear oscillations. This rearrangement may lead to reduced ordering configurations which give rise to macroscale shear hardening, which can result in shear jamming if the applied shear traction is below the critical shear resistance. On the other hand, it was observed that applying oscillatory shear to the primary condensed shear flow enhances flowability due to microstructure rearrangement. In this study, we investigate the amplitude-dependent rheological responses of axisymmetric grains subjected oscillatory shear flows. First, we look into the evolution of grains alignment subjected to a range of oscillation amplitudes, where we show that the lower oscillation amplitudes have the potential to change the orientation from the ordered steady state to a completely disordered (isotropic) orientation. Next, we study the dependency of the shear flow resistance on the microstructure configuration, and show that the strain hardening and potential jamming have strong dependency on the oscillations amplitude. We also show that, in the case of jamming, the shear strain and the corresponding number of oscillation cycles depend not only on the grains aspect ratio but also on the oscillation amplitude.

The rheological response of oriented axisymmetric grains has additional degrees of complexity associated with their microstructure orientation. These additional kinematic degrees of freedom that give rise to complex transient macroscale rheological responses are not well understood. In this Letter, we study the rheology of axisymmetric grains subjected to transient flow. We identify strong coupling between the microstructure rearrangement and strain hardening which, under certain conditions, can yield jamming. We identify the critical conditions corresponding to jamming and the dependency on the shape of the grains. It is shown that this is a particular form of jamming that is directional in nature, since unjamming occurs if the shear direction is reversed.

In this work, we studied the stability of radially symmetric growth in tumor spheroids using a reaction-diffusion model. In this model, nutrient concentration and internal pressure are local variables that implicitly relate the proliferation of cells to the growth of the tumor. The analytical solution of the governing model was presented in an orthonormal spherical harmonic basis. It was shown that the radially symmetric steady-state solution to the growth of tumor spheroids, under symmetric growth conditions, was unstable with respect to small asymmetric perturbations. Such perturbations excited the asymmetric modes of growth, which could grow in time and change the spherical configuration of the tumor. The number of such modes and their rates of growth depended on parameters such as surface tension, external energy and the rate of nutrient consumption. This analysis indicated that the spherical configuration of tumor spheroids, even under experimentally controlled symmetric growth conditions, were naturally unstable. This was confirmed by a comparison between the shapes of in vitro human glioblastoma (hGB) spheroids and the configuration of the first few asymmetric modes predicted by the model.