Basic Steady State of Axisymmetric Dendritic Growth and Its Regular Perturbation Expansion

2017 ◽  
pp. 89-108
Author(s):  
Jian-Jun Xu
Keyword(s):  
Steady State ◽  
Dendritic Growth ◽  
Perturbation Expansion ◽  
Regular Perturbation

Dynamics of a non-spherical microcapsule with incompressible interface in shear flow

2011 ◽  
Vol 678 ◽  
pp. 221-247 ◽  
Author(s):  
P. M. VLAHOVSKA ◽  
Y.-N. YOUNG ◽  
G. DANKER ◽  
C. MISBAH
Keyword(s):  
Shear Rate ◽  
Shear Flow ◽  
Viscosity Ratio ◽  
Perturbation Expansion ◽  
Shear Plane ◽  
Creeping Flow ◽  
Regular Perturbation ◽  
Cell Dynamics ◽  
Initial Shape ◽  
Flow Equations

We study the motion and deformation of a liquid capsule enclosed by a surface-incompressible membrane as a model of red blood cell dynamics in shear flow. Considering a slightly ellipsoidal initial shape, an analytical solution to the creeping-flow equations is obtained as a regular perturbation expansion in the excess area. The analysis takes into account the membrane fluidity, area-incompressibility and resistance to bending. The theory captures the observed transition from tumbling to swinging as the shear rate increases and clarifies the effect of capsule deformability. Near the transition, intermittent behaviour (swinging periodically interrupted by a tumble) is found only if the capsule deforms in the shear plane and does not undergo stretching or compression along the vorticity direction; the intermittency disappears if deformation along the vorticity direction occurs, i.e. if the capsule ‘breathes’. We report the phase diagram of capsule motions as a function of viscosity ratio, non-sphericity and dimensionless shear rate.

On the Initiation of Necking Modes in Layered Plastic Solids

1987 ◽  
Vol 54 (1) ◽  
pp. 141-146 ◽  
Author(s):  
P. S. Steif
Keyword(s):  
Small Parameter ◽  
Finite Strain ◽  
Relative Rate ◽  
Perturbation Expansion ◽  
Nonlinear Field ◽  
Field Equations ◽  
Regular Perturbation ◽  
Layered Solid ◽  
Interface Imperfections ◽  
Nonlinear Field Equations

The deformation of a solid consisting of alternating material layers with interface undulations is studied analytically. We assess the relative rate of growth or decay of these interface imperfections with straining. The analysis is based on a regular perturbation expansion of the finite strain, nonlinear field equations, the small parameter being the amplitude of the undulations relative to the layer thickness. Attention is focused on the influence of material and geometric parameters on imperfection growth-rates. It is found that significant connections exist between the growth of imperfections and the tendency for undulatory bifurcation modes to emerge from a perfect, layered solid.

Microstructural theory and the rheology of concentrated colloidal suspensions

2012 ◽  
Vol 713 ◽  
pp. 420-452 ◽  
Author(s):  
Ehssan Nazockdast ◽  
Jeffrey F. Morris
Keyword(s):  
Normal Stress ◽  
Volume Fraction ◽  
Solid Volume Fraction ◽  
Perturbation Expansion ◽  
Colloidal Suspensions ◽  
Simple Shear Flow ◽  
Fluid Viscosity ◽  
Analytical Prediction ◽  
Regular Perturbation ◽  
Normal Stress Differences

AbstractA theory for the analytical prediction of microstructure of concentrated Brownian suspensions of spheres in simple-shear flow is developed. The computed microstructure is used in a prediction of the suspension rheology. A near-hard-sphere suspension is studied for solid volume fraction $\phi \leq 0. 55$ and Péclet number $Pe= 6\lrm{\pi} \eta \dot {\gamma } {a}^{3} / {k}_{b} T\leq 100$; $a$ is the particle radius, $\eta $ is the suspending Newtonian fluid viscosity, $\dot {\gamma } $ is the shear rate, ${k}_{b} $ is the Boltzmann constant and $T$ is absolute temperature. The method developed determines the steady pair distribution function $g(\mathbi{r})$, where $\mathbi{r}$ is the pair separation vector, from a solution of the Smoluchowski equation (SE) reduced to pair level. To account for the influence of the surrounding bath of particles on the interaction of a pair, an integro-differential form of the pair SE is developed; the integral portion represents the forces due to the bath which drive the pair interaction. Hydrodynamic interactions are accounted for in a pairwise fashion, based on the dominant influence of pair lubrication interactions for concentrated suspensions. The SE is modified to include the influence of shear-induced relative diffusion, and this is found to be crucial for success of the theory; a simple model based on understanding of the shear-induced self-diffusivity is used for this property. The computation of the microstructure is split into two parts, one specific to near-equilibrium ($Pe\ll 1$), where a regular perturbation expansion of $g$ in $Pe$ is applied, and a general-$Pe$ solution of the full SE. The predicted microstructure at low $Pe$ agrees with prior theory for dilute conditions, and becomes increasingly distorted from the equilibrium isotropic state as $\phi $ increases at fixed $Pe\lt 1$. Normal stress differences are predicted and the zero-shear viscosity predicted agrees with simulation results obtained using a Green–Kubo formulation (Foss & Brady, J. Fluid Mech., vol. 407, 2000, pp. 167–200). At $Pe\geq O(1)$, the influence of convection results in a progressively more anisotropic microstructure, with the contact values increasing with $Pe$ to yield a boundary layer and a wake. Agreement of the predicted microstructure with observations from simulations is generally good and discrepancies are clearly noted. The predicted rheology captures shear thinning and shear thickening as well as normal stress differences in good agreement with simulation; quantitative agreement is best at large $\phi $.

CRYSTAL GROWTH AND THE THERMODYNAMICS OF IRREVERSIBLE PROCESSES

1959 ◽  
Vol 37 (6) ◽  
pp. 739-754 ◽  
Author(s):  
J. S. Kirkaldy
Keyword(s):  
Free Energy ◽  
Steady State ◽  
Entropy Production ◽  
Dendritic Growth ◽  
Transport Processes ◽  
Irreversible Processes ◽  
Interface Morphology ◽  
Crystal Face ◽  
Thermodynamics Of Irreversible Processes ◽  
Alloy Crystal

The principle of minimum rate of entropy production is applied to steady-state transport processes in the neighborhood of an alloy crystal face growing into its melt. The procedure gives a satisfactory rationale of observed interface morphology. It is noted that segregation, which occurs in cellular or dendritic growth of alloys, is a direct manifestation of the system's attempt to minimize entropy production by conserving free energy. The general problems of growth of pure and impure single crystals from the melt and vapor are discussed.

scholarly journals Analysis of Brinkman-Forchheimer extended Darcy's model in a fluid saturated anisotropic porous channel

2012 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Timir Karmakar ◽  
Meraj Alam ◽  
G. P. Raja Sekhar
Keyword(s):  
Porous Medium ◽  
Approximate Solution ◽  
Existence And Uniqueness ◽  
Perturbation Expansion ◽  
Positive Semidefinite ◽  
Darcy Number ◽  
Momentum Equation ◽  
Matched Asymptotic Expansion ◽  
Regular Perturbation ◽  
Uniqueness Results

<p style='text-indent:20px;'>We present asymptotic analysis of Couette flow through a channel packed with porous medium. We assume that the porous medium is anisotropic and the permeability varies along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We developed existence and uniqueness results corresponding to the anisotropic Brinkman-Forchheimer extended Darcy's equation in case of fully developed flow using the Browder-Minty theorem. Complemented with the existence and uniqueness analysis, we present an asymptotic solution by taking Darcy number as the perturbed parameter. For a high Darcy number, the corresponding problem is dealt with regular perturbation expansion. For low Darcy number, the problem of interest is a singular perturbation. We use matched asymptotic expansion to treat this case. More generally, we obtained an approximate solution for the nonlinear problem, which is uniformly valid irrespective of the porous medium parameter values. The analysis presented serves a dual purpose by providing the existence and uniqueness of the anisotropic nonlinear Brinkman-Forchheimer extended Darcy's equation and provide an approximate solution that shows good agreement with the numerical solution.</p>

Sign in / Sign up

Export Citation Format

Share Document