Solving Nonlinear Parabolic Equations by a Strongly Implicit Finite-Difference Scheme

We discuss the numerical solution of nonlinear parabolic partial differential equations, exhibiting finite speed of propagation, via a strongly implicit finite-difference scheme with formal truncation error $\mathcal{O}\left[(\Delta x)^2 + (\Delta t)^2 \right]$. Our application of interest is the spreading of viscous gravity currents in the study of which these type of differential equations arise. Viscous gravity currents are low Reynolds number (viscous forces dominate inertial forces) flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. The fluids may be confined by the sidewalls of a channel or propagate in an unconfined two-dimensional (or axisymmetric three-dimensional) geometry. Under the lubrication approximation, the mathematical description of the spreading of these fluids reduces to solving the so-called thin-film equation for the current's shape $h(x,t)$. To solve such nonlinear parabolic equations we propose a finite-difference scheme based on the Crank--Nicolson idea. We implement the scheme for problems involving a single spatial coordinate (i.e., two-dimensional, axisymmetric or spherically-symmetric three-dimensional currents) on an equispaced but staggered grid. We benchmark the scheme against analytical solutions and highlight its strong numerical stability by specifically considering the spreading of non-Newtonian power-law fluids in a variable-width confined channel-like geometry (a "Hele-Shaw cell") subject to a given mass conservation/balance constraint. We show that this constraint can be implemented by re-expressing it as nonlinear flux boundary conditions on the domain's endpoints. Then, we show numerically that the scheme achieves its full second-order accuracy in space and time. We also highlight through numerical simulations how the proposed scheme accurately respects the mass conservation/balance constraint.Comment: 36 pages, 9 figures, Springer book class; v2 includes improvements and corrections; to appear as a contribution in "Applied Wave Mathematics II

MYCN gene amplification is a powerful prognostic factor even in infantile neuroblastoma detected by mass screening

MYCN is the most powerful prognostic factor in cases of older children. However, how MYCN is related to the prognosis of infantile cases is not clear. A mass screening program was carried out by measuring urinary catecholamine metabolites (VMA and HVA) from 6-month-old infants. Of 2084 cases detected by the screening program, MYCN amplification (MNA) was examined by Southern blot analyses in 1533 cases from 1987 to 2000. Of the 1533 cases examined, 1500 (97.8%) showed no MNA, 20 cases (1.3%) showed MNA from three to nine copies, and 13 (0.8%) cases showed more than 10 copies. The 4-year overall survival rates of these three groups (99, 89 and 53%, respectively) were significantly different (P<0.001), indicating that MYCN copy number correlates with the prognosis. Cases with MNA more than 10 copies were more advanced than those without amplification (stage III, IV vs I, II, IVs; P<0.001). Patients with MNA more than 10 copies had significantly higher serum levels of neuron-specific-enolase (NSE) and ferritin than non-amplified patients (P=0.049, P=0.025, respectively). MYCN amplification was strongly correlated with a poor prognosis in infantile neuroblastoma cases. Therefore, for the selection of appropriate treatment, an accurate determination of MNA is indispensable

The Yeast Tor Signaling Pathway Is Involved in G2/M Transition via Polo-Kinase

The target of rapamycin (Tor) protein plays central roles in cell growth. Rapamycin inhibits cell growth and promotes cell cycle arrest at G1 (G0). However, little is known about whether Tor is involved in other stages of the cell division cycle. Here we report that the rapamycin-sensitive Tor complex 1 (TORC1) is involved in G2/M transition in S. cerevisiae. Strains carrying a temperature-sensitive allele of KOG1 (kog1-105) encoding an essential component of TORC1, as well as yeast cell treated with rapamycin show mitotic delay with prolonged G2. Overexpression of Cdc5, the yeast polo-like kinase, rescues the growth defect of kog1-105, and in turn, Cdc5 activity is attenuated in kog1-105 cells. The TORC1-Type2A phosphatase pathway mediates nucleocytoplasmic transport of Cdc5, which is prerequisite for its proper localization and function. The C-terminal polo-box domain of Cdc5 has an inhibitory role in nuclear translocation. Taken together, our results indicate a novel function of Tor in the regulation of cell cycle and proliferation

Pathogenesis of adolescent idiopathic scoliosis in girls - a double neuro-osseous theory involving disharmony between two nervous systems, somatic and autonomic expressed in the spine and trunk: possible dependency on sympathetic nervous system and hormones with implications for medical therapy

Anthropometric data from three groups of adolescent girls - preoperative adolescent idiopathic scoliosis (AIS), screened for scoliosis and normals were analysed by comparing skeletal data between higher and lower body mass index subsets. Unexpected findings for each of skeletal maturation, asymmetries and overgrowth are not explained by prevailing theories of AIS pathogenesis. A speculative pathogenetic theory for girls is formulated after surveying evidence including: (1) the thoracospinal concept for right thoracic AIS in girls; (2) the new neuroskeletal biology relating the sympathetic nervous system to bone formation/resorption and bone growth; (3) white adipose tissue storing triglycerides and the adiposity hormone leptin which functions as satiety hormone and sentinel of energy balance to the hypothalamus for long-term adiposity; and (4) central leptin resistance in obesity and possibly in healthy females. The new theory states that AIS in girls results from developmental disharmony expressed in spine and trunk between autonomic and somatic nervous systems. The autonomic component of this double neuro-osseous theory for AIS pathogenesis in girls involves selectively increased sensitivity of the hypothalamus to circulating leptin (genetically-determined up-regulation possibly involving inhibitory or sensitizing intracellular molecules, such as SOC3, PTP-1B and SH2B1 respectively), with asymmetry as an adverse response (hormesis); this asymmetry is routed bilaterally via the sympathetic nervous system to the growing axial skeleton where it may initiate the scoliosis deformity (leptin-hypothalamic-sympathetic nervous system concept = LHS concept). In some younger preoperative AIS girls, the hypothalamic up-regulation to circulating leptin also involves the somatotropic (growth hormone/IGF) axis which exaggerates the sympathetically-induced asymmetric skeletal effects and contributes to curve progression, a concept with therapeutic implications. In the somatic nervous system, dysfunction of a postural mechanism involving the CNS body schema fails to control, or may induce, the spinal deformity of AIS in girls (escalator concept). Biomechanical factors affecting ribs and/or vertebrae and spinal cord during growth may localize AIS to the thoracic spine and contribute to sagittal spinal shape alterations. The developmental disharmony in spine and trunk is compounded by any osteopenia, biomechanical spinal growth modulation, disc degeneration and platelet calmodulin dysfunction. Methods for testing the theory are outlined. Implications are discussed for neuroendocrine dysfunctions, osteopontin, sympathoactivation, medical therapy, Rett and Prader-Willi syndromes, infantile idiopathic scoliosis, and human evolution. AIS pathogenesis in girls is predicated on two putative normal mechanisms involved in trunk growth, each acquired in evolution and unique to humans

Drop impact on hairy surfaces

We investigate the impact of liquid drops on millimeter-scale hairy surfaces. By varying the speed of the drop, the spacing of the hairs, and the viscosity of the liquid, we observe a variety of behaviors. In some cases, the liquid drop can remain on top of the hair after impact, similar to a Cassie-Baxter superhydrophobic state. If the drop penetrates the hairy surface, the hairs can resist droplet spreading. Using this scenario as a reference case, we rationalize the role of the hairs in dissipating the kinetic energy of the impacting drop through a balance of inertia, viscosity, and surface tension. The various observed behaviors are classified according to scenarios in which kinetic energy is insufficient or in excess of this reference scenario, an argument that allows us to build and rationalize a phase diagram

Drop impact on hairy surfaces

We investigate the impact of liquid drops on millimeter-scale hairy surfaces. By varying the speed of the drop, the spacing of the hairs, and the viscosity of the liquid, we observe a variety of behaviors. In some cases, the liquid drop can remain on top of the hair after impact, similar to a Cassie-Baxter superhydrophobic state. If the drop penetrates the hairy surface, the hairs can resist droplet spreading. Using this scenario as a reference case, we rationalize the role of the hairs in dissipating the kinetic energy of the impacting drop through a balance of inertia, viscosity, and surface tension. The various observed behaviors are classified according to scenarios in which kinetic energy is insufficient or in excess of this reference scenario, an argument that allows us to build and rationalize a phase diagram

Rheological fingerprinting of gastropod pedal mucus and synthetic complex fluids for biomimicking adhesive locomotion

Nonlinear rheological properties are often relevant in understanding the response of a material to its intended environment. For example, many gastropods crawl on a thin layer of pedal mucus using a technique called adhesive locomotion, in which the gel structure is periodically ruptured and reformed. We present a mechanical model that captures the key features of this process and suggests that the most important properties for optimal inclined locomotion are a large, reversible yield stress, followed by a small shear viscosity and a short thixotropic restructuring time. We present detailed rheological measurements of native pedal mucus in both the linear and nonlinear viscoelastic regimes and compare this "rheological fingerprint" with corresponding observations of two bioinspired slime simulants, a polymer gel and a clay-based colloidal gel, that are selected on the basis of their macroscopic rheological similarities to gastropod mucin gels. Adhesive locomotion (of snails or mechanical crawlers) imposes a large-amplitude pulsatile simple shear flow onto the supporting complex fluid, motivating the characterization of nonlinear rheological properties with large amplitude oscillatory shear (LAOS). We represent our results in the form of Lissajous curves of oscillatory stress against time-varying strain. The native pedal mucus gel is found to exhibit a pronounced strain-stiffening response, which is not imitated by either simulant.status: publishe