Narrow Results

In this paper we propose a model for the evolution of a tumor spheroid assuming a structure in which the central necrotic region contains an inner liquid core surrounded by dead cells that keep some mechanical integrity. This partition is a consequence of assuming that a finite delay is required for the degradation of dead cells into liquid. The phenomenological assumption of constant local volume fraction of cells is also made. The above structure is coupled with a mechanical two-phase model that views the cell component as a Bingham-like fluid and the extracellular liquid as an inviscid fluid. By imposing the continuity of the normal stress throughout the whole spheroid, we can describe the spheroid evolution and characterize the possible steady state. Depending on the values of mechanical parameters, the model predicts either an evolution toward the steady state or an unbounded growth. An existence and uniqueness result has been proved under suitable assumptions, along with some qualitative properties of the solution.

We are concerned with a system of partial differential equations (PDEs) describing internal flows of homogeneous incompressible fluids of Bingham type in which the value of activation (the so-called yield) stress depends on the internal pore pressure governed by an advection–diffusion equation. After providing the physical background of the considered model, paying attention to the assumptions involved in its derivation, we focus on the PDE analysis of the initial and boundary value problems. We give several equivalent descriptions for the considered class of fluids of Bingham type. In particular, we exploit the possibility to write such a response as an implicit tensorial constitutive equation, involving the pore pressure, the deviatoric part of the Cauchy stress and the velocity gradient. Interestingly, this tensorial response can be characterized by two scalar constraints. We employ a similar approach to treat stick-slip boundary conditions. Within such a setting we prove long-time and large-data existence of weak solutions to the evolutionary problem in three dimensions.

In this paper, we study the effects of heat transfer on the peristaltic magneto-hydrodynamic (MHD) flow of a Bingham fluid through a porous medium in a channel. Long wavelength approximation (that is, the wavelength of the peristaltic wave is large in comparison with the radius of the channel) and low Reynolds number are used to linearize the governing equations. The velocity field for the model of interest is solved by Adomian decomposition method. The expressions for pressure rise, flow rate and frictional force are obtained. The effect of magnetic field, Darcy number, yield stress, amplitude ratio and the temperature on the axial pressure gradient, pumping characteristics and frictional force are discussed through graphs.