Eugenia Corvera Poiré

Viscous fingering in non-Newtonian fluids

We study the viscous fingering or Saffman–Taylor instability in two different dilute or semi-dilute polymer solutions. The different solutions exhibit only one non-Newtonian property, in the sense that other non-Newtonian effects can be neglected. The viscosity of solutions of stiff polymers has a strong shear rate dependence. Relative to Newtonian fluids, narrower fingers are found for rigid polymers. For solutions of flexible polymers, elastic effects such as normal stresses are dominant, whereas the shear viscosity is almost constant. Wider fingers are found in this case. We characterize the non-Newtonian flow properties of these polymer solutions completely, allowing for separate and quantitative investigation of the influence of the two most common non-Newtonian properties on the Saffman–Taylor instability. The effects of the non-Newtonian flow properties on the instability can in all cases be understood quantitatively by redefining the control parameter of the instability.

Pushing a non-Newtonian fluid in a Hele-Shaw cell: From fingers to needles

We make a theoretical study of the finger behavior of a simple fluid displacing a non-Newtonian fluid confined in a Hele–Shaw cell. We study the Saffman–Taylor instability when the viscosity of the displaced fluid changes with shear. Our results predict a decrease of the finger width that goes to zero for large values of the velocity. An analytical treatment allows the predictions of the dynamics in radial growth.

Phase equilibria of confined liquid crystals

The differences between the phase diagram of the Gay-Berne potential confined by two identical walls versus the corresponding bulk phase diagram have been investigated. A wall-fluid interaction 9-3 Lennard-Jones potential was used. The study was performed in most cases by using the hybrid Monte Carlo method for the μVT ensemble. Several isotherms were analysed where vapour, liquid and smectic phases were observed. The smectic-isotropic coexistence region becomes wider, i.e. the isotropic coexistence line is shifted to lower densities but the smectic coexistence line remains nearly the same. The triple point temperature of the confined system is estimated to be in the vicinity of 0.45 versus 0.40 of the bulk system. For the isotherm at T* = 0.65 an orientational dependence was added to the 9-3 Lennard-Jones potential to model the wall-fluid interaction. For both kinds of walls, 9-3 LJ with and without orientational dependence, confinement was not found to stabilize a nematic phase as found by previous authors.

Flow and anastomosis in vascular networks.

We analyze the effect that the geometrical place of anastomosis in the circulatory tree has on blood flow. We introduce an idealized model that consists of a symmetric network for the arterial and venous vascular trees. We consider that the network contains a viscoelastic fluid with the rheological characteristics of blood, and analyze the network hydrodynamic response to a time-dependent periodic pressure gradient. This response is a measurement of the resistance to flow: the larger the response, the smaller the resistance to flow. We find that for networks whose vessels have the same radius and length, the outer the level of the branching tree in which anastomosis occurs, the larger the network response. Moreover, when anastomosis is incorporated in the form of bypasses that bridge vessels at different bifurcation levels, the further apart are the levels bridged by the bypass, the larger the response is. Furthermore, we apply the model to the available information for the dog circulatory system and find that the effect that anastomosis causes at different bifurcation levels is strongly determined by the structure of the underlying network without anastomosis. We rationalize our results by introducing two idealized models and approximated analytical expressions that allow us to argue that, to a large extent, the response of the network with anastomosis is determined locally. We have also considered the influence of the myogenic effect. This one has a large quantitative impact on the network response. However, the qualitative behavior of the network response with anastomosis is the same with or without consideration of the myogenic effect. That is, it depends on the structure that the underlying vessel network has in a small neighborhood around the place where anastomosis occurs. This implies that whenever there is an underlying tree-like network in an in vivo vasculature, our model is able to interpret the anastomotic effect.

Obstructions in Vascular Networks: Relation Between Network Morphology and Blood Supply

We relate vascular network structure to hemodynamics after vessel obstructions. We consider tree-like networks with a viscoelastic fluid with the rheological characteristics of blood. We analyze the network hemodynamic response, which is a function of the frequencies involved in the driving, and a measurement of the resistance to flow. This response function allows the study of the hemodynamics of the system, without the knowledge of a particular pressure gradient. We find analytical expressions for the network response, which explicitly show the roles played by the network structure, the degree of obstruction, and the geometrical place in which obstructions occur. Notably, we find that the sequence of resistances of the network without occlusions strongly determines the tendencies that the response function has with the anatomical place where obstructions are located. We identify anatomical sites in a network that are critical for its overall capacity to supply blood to a tissue after obstructions. We demonstrate that relatively small obstructions in such critical sites are able to cause a much larger decrease on flow than larger obstructions placed in non-critical sites. Our results indicate that, to a large extent, the response of the network is determined locally. That is, it depends on the structure that the vasculature has around the place where occlusions are found. This result is manifest in a network that follows Murray’s law, which is in reasonable agreement with several mammalian vasculatures. For this one, occlusions in early generation vessels have a radically different effect than occlusions in late generation vessels occluding the same percentage of area available to flow. This locality implies that whenever there is a tissue irrigated by a tree-like in vivo vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.

When do redundant fluidic networks outperform non-redundant ones?

Redundancy constitutes a fundamental and intrinsic aspect of healthy vasculatures. Built-in redundancy might also be a desirable feature in man-made microfluidic devices. We show that redundant and non-redundant networks, built to have identical resistances to flow when unobstructed, allow for very different flows when they are occluded; redundant ones??densely occluded at a certain bifurcation level??allowing for larger flows than non-redundant ones??obstructed above relatively small thresholds. We also show that redundancy protects vessels against the large shear-rate gradients that occlusions would cause if it were not present. Our study allows one to quantify a network tolerance against blockage, provides guidance in the tailoring of microfluidic devices, and offers novel insights into why nature has selected intrinsic redundancy over thicker vessels to assure blood supply at key places of the organisms.

Obstructions in Vascular Networks. Critical vs Non-critical Topological Sites for Blood Supply

We relate vascular network structure to hemodynamics after vessel obstructions. We consider tree-like networks with a viscoelastic fluid with the rheological characteristics of blood. We analyze the network hemodynamic response, which is a function of the frequencies involved in the driving, and a measurement of the resistance to flow. This response function allows the study of the hemodynamics of the system, without the knowledge of a particular pressure gradient. We find analytical expressions for the network response, that explicitly show the roles played by the network structure, the degree of obstruction, and the geometrical place in which obstructions occur. Notably, we find that the sequence of resistances of the network without occlusions, strongly determines the tendencies that the response function has with the anatomical place where obstructions are located. We identify anatomical sites in a network that are critical for its overall capacity to supply blood to a tissue after obstructions. We demonstrate that relatively small obstructions in such critical sites are able to cause a much larger decrease on flow than larger obstructions placed in non-critical sites. Our results indicate that, to a large extent, the response of the network is determined locally. That is, it depends on the structure that the vasculature has around the place where occlusions are found. This result is manifest in a network that follows Murray’s law, which is in reasonable agreement with several mammalian vasculatures. For this one, occlusions in early generation vessels have a radically different effect than occlusions in late generation vessels occluding the same percentage of area available to flow. This locality implies that whenever there is a tissue irrigated by a tree-like in-vivo vasculature, our model is able to interpret how important obstructions are for the irrigation of such tissue.

Correction: Tumor Angiogenesis and Vascular Patterning: A Mathematical Model

The following information is missing from the Funding section: This work was also supported by Fundos FEDER through Programa Operacional Factores de Competitividade COMPETE through the project with reference number FCOMP-01-0124-FEDER-015708.

Morphological instability at the early stages of heteroepitaxial growth on vicinal surfaces

We study the strain-induced morphological instability at the submonolayer coverage stage of heteroepitaxial growth on a vicinal substrate with regularly spaced steps. We study the regime in which diffusion along the film edge is the dominant mechanism of transport of matter. We perform a linear stability analysis and determine for which conditions of coverage a flat front is unstable and for which conditions it is stable. We discuss the effect of step energy. Our results predict that when the thin film covers less than one-half of the terraces the flat front is unstable. For very small coverage, the front will spontaneously break into a regular array of islands. We obtain expressions for the aspect ratio, the size and the spacing of the islands forming this array. This opens the possibility of inducing the spontaneous formation of an array of two-dimensional quantum structures with the desired size and spacing by controlling the cutting angle of the vicinal surface and the fraction of the surface covered by the material.