Microscopic Selection of Fluid Fingering Pattern

We study the issue of the selection of viscous fingering patterns in the limit of small surface tension. Through detailed simulations of anisotropic fingering, we demonstrate conclusively that no selection independent of the small-scale cutoff (macroscopic selection) occurs in this system. Rather, the small-scale cutoff completely controls the pattern, even on short time scales, in accord with the theory of microscopic solvability. We demonstrate that ordered patterns are dynamically selected only for not too small surface tensions. For extremely small surface tensions, the system exhibits chaotic behavior and no regular pattern is realized.Comment: 6 pages, 5 figure

Characteristic Angles in the Wetting of an Angular Region: Deposit Growth

As was shown in an earlier paper [1], solids dispersed in a drying drop migrate to the (pinned) contact line. This migration is caused by outward flows driven by the loss of the solvent due to evaporation and by geometrical constraint that the drop maintains an equilibrium surface shape with a fixed boundary. Here, in continuation of our earlier paper [2], we theoretically investigate the evaporation rate, the flow field and the rate of growth of the deposit patterns in a drop over an angular sector on a plane substrate. Asymptotic power laws near the vertex (as distance to the vertex goes to zero) are obtained. A hydrodynamic model of fluid flow near the singularity of the vertex is developed and the velocity field is obtained. The rate of the deposit growth near the contact line is found in two time regimes. The deposited mass falls off as a weak power Gamma of distance close to the vertex and as a stronger power Beta of distance further from the vertex. The power Gamma depends only slightly on the opening angle Alpha and stays between roughly -1/3 and 0. The power Beta varies from -1 to 0 as the opening angle increases from 0 to 180 degrees. At a given distance from the vertex, the deposited mass grows faster and faster with time, with the greatest increase in the growth rate occurring at the early stages of the drying process.Comment: v1: 36 pages, 21 figures, LaTeX; submitted to Physical Review E; v2: minor additions to Abstract and Introductio

Basal Signalling Through Death Receptor 5 and Caspase 3 Activates p38 Kinase To Regulate Serum Response Factor (SRF)-Mediated Myod Transcription

We have previously reported that stable expression of a dominant negative Death Receptor 5 (dnDR5) in skeletal myoblasts results in decreased basal caspase activity and decreased mRNA and protein expression of the muscle regulatory transcription factor MyoD in growth medium (GM), resulting in inhibited differentation when myoblasts are then cultured in differentiation media (DM). Further, this decreased level of MyoD mRNA was not a consequence of altered message stability, but rather correlated with decreased acetylation of histones in the distal regulatory region (DRR) of the MyoD extended promoter known to control MyoD transcription. As serum response factor (SRF) is the transcription factor known to be responsible for basal MyoD expression in GM, we compared the level of SRF binding to the non-canonical serum response element (SRE) within the DRR in parental and dnDR5 expressing myoblasts. Herein, we report that stable expression of dnDR5 resulted in decreased levels of serum response factor (SRF) binding to the CArG box in the SRE of the DRR. Total SRF expression levels were not affected, but phosphorylation indicative of SRF activation was impaired. This decreased SRF phosphorylation correlated with decreased phosphorylation-induced activation of p38 kinase. Moreover, the aforementioned signaling events affected by expression of dnDR5 could be appropriately recapitulated using either a pharmacological inhibitor of caspase 3 or p38 kinase. Thus, our results have established a signaling pathway from DR5 through caspases to p38 kinase activation, to SRF activation and the basal expression of MyoD

Distinct patterns of notochord mineralization in zebrafish coincide with the localization of Osteocalcin isoform 1 during early vertebral centra formation

In chondrichthyans, basal osteichthyans and tetrapods, vertebral bodies have cartilaginous anlagen that subsequently mineralize (chondrichthyans) or ossify (osteichthyans). Chondrocytes that form the vertebral centra derive from somites. In teleost fish, vertebral centrum formation starts in the absence of cartilage, through direct mineralization of the notochord sheath. In a second step, the notochord is surrounded by somite-derived intramembranous bone. In several small teleost species, including zebrafish (Danio rerio), even haemal and neural arches form directly as intramembranous bone and only modified caudalmost arches remain cartilaginous. This study compares initial patterns of mineralization in different regions of the vertebral column in zebrafish. We ask if the absence or presence of cartilaginous arches influences the pattern of notochord sheath mineralization. Results - To reveal which cells are involved in mineralization of the notochord sheath we identify proliferating cells, we trace mineralization on the histological level and we analyze cell ultrastructure by TEM. Moreover, we localize proteins and genes that are typically expressed by skeletogenic cells such as Collagen type II, Alkaline phosphatase (ALP) and Osteocalcin (Oc). Mineralization of abdominal and caudal vertebrae starts with a complete ring within the notochord sheath and prior to the formation of the bony arches. In contrast, notochord mineralization of caudal fin centra starts with a broad ventral mineral deposition, associated with the bases of the modified cartilaginous arches. Similar, arch-related, patterns of mineralization occur in teleosts that maintain cartilaginous arches throughout the spine.Throughout the entire vertebral column, we were able to co-localize ALP-positive signal with chordacentrum mineralization sites, as well as Collagen II and Oc protein accumulation in the mineralizing notochord sheath. In the caudal fin region, ALP and Oc signals were clearly produced both by the notochord epithelium and cells outside the notochord, the cartilaginous arches. Based on immunostaining, real time PCR and oc2:gfp transgenic fish, we identify Oc in the mineralizing notochord sheath as osteocalcin isoform 1 (Oc1). Conclusions - If notochord mineralization occurs prior to arch formation, mineralization of the notochord sheath is ring-shaped. If notochord mineralization occurs after cartilaginous arch formation, mineralization of the notochord sheath starts at the insertion point of the arches, with a basiventral origin. The presence of ALP and Oc1, not only in cells outside the notochord, but also in the notochord epithelium, suggests an active role of the notochord in the mineralization process. The same may apply to Col II-positive chondrocytes of the caudalmost haemal arches that show ALP activity and Oc1 accumulation, since these chondrocytes do not mineralize their own cartilage matrix. Even without cartilaginous preformed vertebral centra, the cartilaginous arches may have an inductive role in vertebral centrum formation, possibly contributing to the distinct mineralization patterns of zebrafish vertebral column and caudal fin vertebral fusion.Peer Reviewe

A New Class of Nonsingular Exact Solutions for Laplacian Pattern Formation

We present a new class of exact solutions for the so-called {\it Laplacian Growth Equation} describing the zero-surface-tension limit of a variety of 2D pattern formation problems. Contrary to common belief, we prove that these solutions are free of finite-time singularities (cusps) for quite general initial conditions and may well describe real fingering instabilities. At long times the interface consists of N separated moving Saffman-Taylor fingers, with ``stagnation points'' in between, in agreement with numerous observations. This evolution resembles the N-soliton solution of classical integrable PDE's.Comment: LaTeX, uuencoded postscript file

Gravity-driven instability in a spherical Hele-Shaw cell

A pair of concentric spheres separated by a small gap form a spherical Hele-Shaw cell. In this cell an interfacial instability arises when two immiscible fluids flow. We derive the equation of motion for the interface perturbation amplitudes, including both pressure and gravity drivings, using a mode coupling approach. Linear stability analysis shows that mode growth rates depend upon interface perimeter and gravitational force. Mode coupling analysis reveals the formation of fingering structures presenting a tendency toward finger tip-sharpening.Comment: 13 pages, 4 ps figures, RevTex, to appear in Physical Review

Multidimensional Pattern Formation Has an Infinite Number of Constants of Motion

Extending our previous work on 2D growth for the Laplace equation we study here {\it multidimensional} growth for {\it arbitrary elliptic} equations, describing inhomogeneous and anisotropic pattern formations processes. We find that these nonlinear processes are governed by an infinite number of conservation laws. Moreover, in many cases {\it all dynamics of the interface can be reduced to the linear time--dependence of only one ``moment" $M_0$} which corresponds to the changing volume while {\it all higher moments, $M_l$, are constant in time. These moments have a purely geometrical nature}, and thus carry information about the moving shape. These conserved quantities (eqs.~(7) and (8) of this article) are interpreted as coefficients of the multipole expansion of the Newtonian potential created by the mass uniformly occupying the domain enclosing the moving interface. Thus the question of how to recover the moving shape using these conserved quantities is reduced to the classical inverse potential problem of reconstructing the shape of a body from its exterior gravitational potential. Our results also suggest the possibility of controlling a moving interface by appropriate varying the location and strength of sources and sinks.Comment: CYCLER Paper 93feb00

Mode-coupling approach to non-Newtonian Hele-Shaw flow

The Saffman-Taylor viscous fingering problem is investigated for the displacement of a non-Newtonian fluid by a Newtonian one in a radial Hele-Shaw cell. We execute a mode-coupling approach to the problem and examine the morphology of the fluid-fluid interface in the weak shear limit. A differential equation describing the early nonlinear evolution of the interface modes is derived in detail. Owing to vorticity arising from our modified Darcy's law, we introduce a vector potential for the velocity in contrast to the conventional scalar potential. Our analytical results address how mode-coupling dynamics relates to tip-splitting and side branching in both shear thinning and shear thickening cases. The development of non-Newtonian interfacial patterns in rectangular Hele-Shaw cells is also analyzed.Comment: 14 pages, 5 ps figures, Revtex4, accepted for publication in Phys. Rev.

Rotating Hele-Shaw cells with ferrofluids

We investigate the flow of two immiscible, viscous fluids in a rotating Hele-Shaw cell, when one of the fluids is a ferrofluid and an external magnetic field is applied. The interplay between centrifugal and magnetic forces in determining the instability of the fluid-fluid interface is analyzed. The linear stability analysis of the problem shows that a non-uniform, azimuthal magnetic field, applied tangential to the cell, tends to stabilize the interface. We verify that maximum growth rate selection of initial patterns is influenced by the applied field, which tends to decrease the number of interface ripples. We contrast these results with the situation in which a uniform magnetic field is applied normally to the plane defined by the rotating Hele-Shaw cell.Comment: 12 pages, 3 ps figures, RevTe