The Penna model for biological ageing on a lattice: spatial consequences of child-care

We introduce a square lattice into the Penna bit-string model for biological ageing and study the evolution of the spatial distribution of the population considering different strategies of child-care. Two of the strategies are related to the movements of a whole family on the lattice: in one case the mother cannot move if she has any child younger than a given age, and in the other case if she moves, she brings these young children with her. A stronger condition has also been added to the second case, considering that young children die with a higher probability if their mothers die, this probability decreasing with age. We show that a highly non uniform occupation can be obtained when child-care is considered, even for an uniform initial occupation per site. We also compare the standard survival rate of the model with that obtained when the spacial lattice is considered (without any kind of child-care).Comment: 8 pages, 6 Postscript figure

High performance microfluidic rectifiers for viscoelastic fluid flow

The flow of Newtonian and non-Newtonian fluids within microfluidic rectifiers with a hyperbolic shape was investigated to assess the effect of the bounding walls on the diodicity of the microfluidic device and achieve high flow anisotropy. Three microchannels were used, with different depths and the same geometrical configuration, which creates a strong extensional flow and generates high anisotropic flow resistance between the two flow directions. The Newtonian fluid, de-ionized water, was used as a reference fluid. The viscoelastic fluid used was an aqueous solution of polyethylene oxide (0.1% w/w) with high molecular weight. The flow patterns were visualized using streak photography and the velocity field was investigated using micro-particle image velocimetry. Moreover, pressure drop measurements were performed in order to compare the diodicity achieved in the microfluidic rectifiers. For the Newtonian fluid flow, the experimental results are compared with numerical predictions obtained using a finite-volume method and good agreement was found between both approaches. For the viscoelastic fluid, significant anisotropic flow resistance can be achieved. The effect of the bounding walls was analysed and found to be qualitatively similar for all microchannels. Nevertheless, in quantitative terms, the diodicity is enhanced when the wall effect is reduced, i.e. when the channels are deeper. A maximum diodicity above six was found for the deeper channel, a value well beyond those previously reported

Laminar flow in three-dimensional square-square expansions

In this work we investigate the three-dimensional laminar flow of Newtonian and viscoelastic fluids through square–square expansions. The experimental results obtained in this simple geometry provide useful data for benchmarking purposes in complex three-dimensional flows. Visualizations of the flow patterns were performed using streak photography, the velocity field of the flow was measured in detail using particle image velocimetry and additionally, pressure drop measurements were carried out. The Newtonian fluid flow was investigated for the expansion ratios of 1:2.4, 1:4 and 1:8 and the experimental results were compared with numerical predictions. For all expansion ratios studied, a corner vortex is observed downstream of the expansion and an increase of the flow inertia leads to an enhancement of the vortex size. Good agreement is found between experimental and numerical results. The flow of the two non-Newtonian fluids was investigated experimentally for expansion ratios of 1:2.4, 1:4, 1:8 and 1:12, and compared with numerical simulations using the Oldroyd-B, FENE-MCR and sPTT constitutive equations. For both the Boger and shear-thinning viscoelastic fluids, a corner vortex appears downstream of the expansion, which decreases in size and strength when the elasticity of the flow is increased. For all fluids and expansion ratios studied, the recirculations that are formed downstream of the square–square expansion exhibit a three-dimensional structure evidenced by a helical flow, which is also predicted in the numerical simulations

Effect of the contraction ratio upon viscoelastic fluid flow in three-dimensional square-square contractions

In this work we investigate the laminar flow through square–square sudden contractions with various contraction ratios (CR¼2.4, 4,8and12), using a Newtonian fluid and a shear-thinning viscoelastic fluid. Visualizations of the flow patterns were carried out using streakline photography and detailed velocity field measurements were performed using particle image velocimetry. The experimental results are compared with numerical predictions obtained using a finite-volume method. For the Newtonian fluid, a corner vortex is found upstream of the contraction and increasing flow inertia leads to a reduction of the vortex size. Good agreement is observed between experiments and numerical simulations. For the shear-thinning fluid flow a corner vortex is also observed upstream of the contraction independently of the contraction ratio. Increasing the elasticity of the flow, while still maintaining low inertia flow conditions, leads to a strong increase of the vortex size, until an elastic instability sets in and the flow becomes time-dependent at DeE200, 300, 70 and 450 for CR¼2.4, 4, 8 and 12, respectively. At low contraction ratios, viscoelasticity brings out an anomalous divergent flow upstream of the contraction. For both fluids studied the flow presents a complex three-dimensional helical vortex structure which is well predicted by numerical simulations. However, for the viscoelastic fluid flow the maximum Deborah number achieved in the numerical simulations is about one order of magnitude lower than the critical Deborah number for the onset of the elastic instability found in the experiments

Sharp mixed norm spherical restriction

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{f\sigma}\big\|_{L^q_{{\rm rad}}L^2_{{\rm ang}}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}\sigma)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions.Comment: 21 page

Three-dimensional flow of Newtonian and Boger fluids in square-square contractions

The flow of a Newtonian fluid and a Boger fluid through sudden square–square contractions was investigated experimentally aiming to characterize the flow and provide quantitative data for benchmarking in a complex three-dimensional flow. Visualizations of the flow patterns were undertaken using streakline photography, detailed velocity field measurementswere conducted using particle image velocimetry (PIV) and pressure drop measurements were performed in various geometries with different contraction ratios. For the Newtonian fluid, the experimental results are compared with numerical simulations performed using a finite volume method, and excellent agreement is found for the range of Reynolds number tested (Re2 ≤23). For the viscoelastic case, recirculations are still present upstream of the contraction but we also observe other complex flow patterns that are dependent on contraction ratio (CR) and Deborah number (De2) for the range of conditions studied: CR = 2.4, 4, 8, 12 and De2 ≤150. For low contraction ratios strong divergent flow is observed upstream of the contraction, whereas for high contraction ratios there is no upstream divergent flow, except in the vicinity of the re-entrant corner where a localized a typical divergent flow is observed. For all contraction ratios studied, at sufficiently high Deborah numbers, strong elastic vortex enhancement upstream of the contraction is observed, which leads to the onset of a periodic complex flow at higher flow rates. The vortices observed under steady flow are not closed, and fluid elasticity was found to modify the flow direction within the recirculations as compared to that found for Newtonian fluids. The entry pressure drop, quantified using a Couette correction, was found to increase with the Deborah number for the higher contraction ratios