On the evolution of thermally driven shallow cavity flows

1994 ◽  
Vol 259 ◽  
pp. 107-124 ◽  
Author(s):  
P. G. Daniels ◽  
P. Wang
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Lateral Wall ◽  
Cavity Flows ◽  
Thermally Driven ◽  
Order Of Magnitude ◽  
Final Steady State ◽  
Shallow Cavity

The temporal evolution of thermally driven flow in a shallow laterally heated cavity is studied for the nonlinear regime where the Rayleigh number R based on cavity height is of the same order of magnitude as the aspect ratio L (length/height). The horizontal surfaces of the cavity are assumed to be thermally insulating. For a certain class of initial conditions the evolution is found to occur over two non-dimensional timescales, of order one and of order L2. Analytical solutions for the motion throughout most of the cavity are found for each of these timescales and numerical solutions are obtained for the nonlinear time-dependent motion in end regions near each lateral wall. This provides a complete picture of the evolution of the steady-state flow in the cavity for cases where instability in the form of multicellular convection does not occur. The final steady state evolves on a dimensional timescale proportional to l2/κ, where l is the length of the cavity, κ is the thermal diffusivity of the fluid and the constant of proportionality depends on the ratio R/L.

Thermally driven shallow cavity flows in porous media: the intermediate régime

1986 ◽  
Vol 406 (1831) ◽  
pp. 263-285 ◽  
Keyword(s):  
Heat Transfer ◽  
Porous Media ◽  
Numerical Solutions ◽  
Cavity Flows ◽  
Horizontal Temperature Gradient ◽  
Intermediate Regime ◽  
Flows In Porous Media ◽  
Thermally Driven ◽  
Shallow Cavities ◽  
Shallow Cavity

Steady cavity flows in porous media, driven by a horizontal temperature gradient, are examined in the intermediate limit when the Rayleigh number is comparable with the cavity aspect ratio, L (≫ 1). In this régime the flow departs from the single-cell Hadley structure, valid only for very shallow cavities, and separate circulations can develop at each end of the cavity. The existence of these closed cells is consistent with numerical solutions of the full cavity problem. Suitable heat-transfer laws are developed for conducting and for adiabatic horizontal boundaries.

scholarly journals Modelling the Effect of Cell Shedding on Avascular Tumour Growth

2000 ◽  
Vol 2 (3) ◽  
pp. 155-174 ◽  
Author(s):  
J. P. Ward ◽  
J. R. King
Keyword(s):  
Steady State ◽  
Tumour Growth ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Travelling Waves ◽  
Tumour Size ◽  
Travelling Wave ◽  
Live Cells ◽  
Long Time ◽  
Cell Shedding

Earlier mathematical models of the authors which describe avascular tumour growth are extended to incorporate the process of cell shedding, a feature known to affect the growth of multicell spheroids. A continuum of live cells is assumed within which, depending on the concentration of a generic nutrient, movement (described by a velocity field) occurs due to volume changes caused by cell birth and death. The necrotic material is assumed to contain a mixture of basic cellular material (assumed necessary for creating new cells) and a non-utilisable material which may inhibit mitosis. The rate of cell shedding is taken to be proportional to the mitotic rate, with constant of proportionality θ. Numerical solutions of the resulting system of partial differential equations indicate that, depending on θ and the initial conditions, the solution may either tend to the trivial state in finite time (by which we mean complete death of the tumour), or to one of two non-trivial states, namely a steady-state (indicating growth saturation) or a travelling wave (indicating continual linear growth). These long time outcomes are explored by deriving the travelling wave and steady-state limits of the model. Numerical solutions demonstrate that there are two branches of solutions, which we have termed the ′Major′ and ′Minor′ branches, consisting of both travelling waves and steady-states. The behaviour of the solutions along each branch is discussed, with those of the Major branch expected to be stable. Beyond some critical θ,where the Major and Minor branches merge, the spheroid ultimately vanishes whatever the initial tumour size due to the effects of cell shedding being too strong for it to survive. The regions of existence of the two long time outcomes are investigated in parameter space, cell shedding being shown to expand significantly the parameter ranges within which growth saturation occurs.

Molecular Dynamics Study of Cohesionless Granular Materials: Size Segregation by Shaking

1993 ◽  
Vol 07 (09n10) ◽  
pp. 1865-1872 ◽  
Author(s):  
Toshiya OHTSUKI ◽  
Yoshikazu TAKEMOTO ◽  
Tatsuo HATA ◽  
Shigeki KAWAI ◽  
Akihisa HAYASHI
Keyword(s):  
Molecular Dynamics ◽  
Steady State ◽  
Specific Gravity ◽  
Pressure Gradient ◽  
Granular Materials ◽  
Buoyancy Force ◽  
Temporal Evolution ◽  
Particle Radius ◽  
Size Segregation ◽  
Increasing Function

The Molecular Dynamics technique is used to investigate size segregation by shaking in cohesionless granular materials. Temporal evolution of the height h of the tagged particle with different size and mass is measured for various values of the particle radius and specific gravity. It becomes evident that h approaches the steady state value h∞ independent of initial positions. There exists a threshold of the specific gravity of the particle. Below the threshold, h∞ is an increasing function of the particle size, whereas above it, h∞ decreases with increasing the particle radius. The relaxation time τ towards the steady state is calculated and its dependence on the particle radius and specific gravity is clarified. The pressure gradient of pure systems is also measured and turned out to be almost constant. This suggests that the buoyancy force due to the pressure gradient is not responsible to h∞.

scholarly journals The Process of Stellar Tidal Disruption by Supermassive Black Holes

2021 ◽  
Vol 217 (3) ◽  
Author(s):  
E. M. Rossi ◽  
N. C. Stone ◽  
J. A. P. Law-Smith ◽  
M. Macleod ◽  
G. Lodato ◽  
...  
Keyword(s):  
Black Holes ◽  
Initial Conditions ◽  
Numerical Models ◽  
Binary Star ◽  
Tidal Disruption ◽  
Massive Black Hole ◽  
Accretion Flows ◽  
Simple Order ◽  
Order Of Magnitude ◽  
A Chain

AbstractTidal disruption events (TDEs) are among the brightest transients in the optical, ultraviolet, and X-ray sky. These flares are set into motion when a star is torn apart by the tidal field of a massive black hole, triggering a chain of events which is – so far – incompletely understood. However, the disruption process has been studied extensively for almost half a century, and unlike the later stages of a TDE, our understanding of the disruption itself is reasonably well converged. In this Chapter, we review both analytical and numerical models for stellar tidal disruption. Starting with relatively simple, order-of-magnitude physics, we review models of increasing sophistication, the semi-analytic “affine formalism,” hydrodynamic simulations of the disruption of polytropic stars, and the most recent hydrodynamic results concerning the disruption of realistic stellar models. Our review surveys the immediate aftermath of disruption in both typical and more unusual TDEs, exploring how the fate of the tidal debris changes if one considers non-main sequence stars, deeply penetrating tidal encounters, binary star systems, and sub-parabolic orbits. The stellar tidal disruption process provides the initial conditions needed to model the formation of accretion flows around quiescent massive black holes, and in some cases may also lead to directly observable emission, for example via shock breakout, gravitational waves or runaway nuclear fusion in deeply plunging TDEs.

Steady-State Dynamic Response of a Limited Slip System

1968 ◽  
Vol 35 (2) ◽  
pp. 322-326 ◽  
Author(s):  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Dynamic Response ◽  
Slip System ◽  
External Load ◽  
Initial Conditions ◽  
Viscous Damping ◽  
Steady State Response ◽  
Response Curves ◽  
State Response ◽  
System Nonlinearity

The steady-state response of a system constrained by a limited slip joint and excited by a trigonometrically varying external load is discussed. It is shown that the system may possess such features as disconnected response curves and jumps in response depending on the strength of the system nonlinearity, the level of excitation, the amount of viscous damping, and the initial conditions of the system.

An Improved Assembling Algorithm in Boundary Elements With Galerkin Weighting Applied to Three-Dimensional Stokes Flows

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Sofia Sarraf ◽  
Ezequiel López ◽  
Laura Battaglia ◽  
Gustavo Ríos Rodríguez ◽  
Jorge D'Elía
Keyword(s):  
Boundary Element Method ◽  
Linear System ◽  
Boundary Element ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Computational Time ◽  
Creeping Flows ◽  
Order Of Magnitude ◽  
Element Method

In the boundary element method (BEM), the Galerkin weighting technique allows to obtain numerical solutions of a boundary integral equation (BIE), giving the Galerkin boundary element method (GBEM). In three-dimensional (3D) spatial domains, the nested double surface integration of GBEM leads to a significantly larger computational time for assembling the linear system than with the standard collocation method. In practice, the computational time is roughly an order of magnitude larger, thus limiting the use of GBEM in 3D engineering problems. The standard approach for reducing the computational time of the linear system assembling is to skip integrations whenever possible. In this work, a modified assembling algorithm for the element matrices in GBEM is proposed for solving integral kernels that depend on the exterior unit normal. This algorithm is based on kernels symmetries at the element level and not on the flow nor in the mesh. It is applied to a BIE that models external creeping flows around 3D closed bodies using second-order kernels, and it is implemented using OpenMP. For these BIEs, the modified algorithm is on average 32% faster than the original one.

Thermal stress vs. thermal transpiration: A competition in thermally driven cavity flows

2015 ◽  
Vol 27 (11) ◽  
pp. 112001 ◽  
Author(s):  
Alireza Mohammadzadeh ◽  
Anirudh Singh Rana ◽  
Henning Struchtrup
Keyword(s):  
Thermal Stress ◽  
Cavity Flows ◽  
Driven Cavity ◽  
Thermal Transpiration ◽  
Thermally Driven

scholarly journals Temporal Evolution of Immunity Distributions in a Population with Waning and Boosting

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
M. V. Barbarossa ◽  
M. Polner ◽  
G. Röst
Keyword(s):  
Mathematical Model ◽  
Steady State ◽  
Temporal Evolution ◽  
Disease Outbreaks ◽  
Population Level ◽  
Coupled System ◽  
Disease Outbreak ◽  
Periodic Disease ◽  
Evolution Of Immunity

We investigate the temporal evolution of the distribution of immunities in a population, which is determined by various epidemiological, immunological, and demographical phenomena: after a disease outbreak, recovered individuals constitute a large immune population; however, their immunity is waning in the long term and they may become susceptible again. Meanwhile, their immunity can be boosted by repeated exposure to the pathogen, which is linked to the density of infected individuals present in the population. This prolongs the length of their immunity. We consider a mathematical model formulated as a coupled system of ordinary and partial differential equations that connects all these processes and systematically compare a number of boosting assumptions proposed in the literature, showing that different boosting mechanisms lead to very different stationary distributions of the immunity at the endemic steady state. In the situation of periodic disease outbreaks, the waveforms of immunity distributions are studied and visualized. Our results show that there is a possibility to infer the boosting mechanism from the population level immune dynamics.

Oscillatory flow past a circular cylinder in a rotating frame

1997 ◽  
Vol 345 ◽  
pp. 101-131
Author(s):  
M. D. KUNKA ◽  
M. R. FOSTER
Keyword(s):  
Circular Cylinder ◽  
Steady Flow ◽  
Stagnation Point ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Temporal Integration ◽  
Oncoming Flow ◽  
Flow Past ◽  
Rear Stagnation Point

Because of the importance of oscillatory components in the oncoming flow at certain oceanic topographic features, we investigate the oscillatory flow past a circular cylinder in an homogeneous rotating fluid. When the oncoming flow is non-reversing, and for relatively low-frequency oscillations, the modifications to the equivalent steady flow arise principally in the ‘quarter layer’ on the surface of the cylinder. An incipient-separation criterion is found as a limitation on the magnitude of the Rossby number, as in the steady-flow case. We present exact solutions for a number of asymptotic cases, at both large frequency and small nonlinearity. We also report numerical solutions of the nonlinear quarter-layer equation for a range of parameters, obtained by a temporal integration. Near the rear stagnation point of the cylinder, we find a generalized velocity ‘plateau’ similar to that of the steady-flow problem, in which all harmonics of the free-stream oscillation may be present. Further, we determine that, for certain initial conditions, the boundary-layer flow develops a finite-time singularity in the neighbourhood of the rear stagnation point.

Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions

2010 ◽  
Vol 65 (11) ◽  
pp. 935-949 ◽  
Author(s):  
Mehdi Dehghan ◽  
Jalil Manafian ◽  
Abbas Saadatmandi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Analysis Method ◽  
Partial Differential ◽  
Integer Order ◽  
Homotopy Analysis

In this paper, the homotopy analysis method is applied to solve linear fractional problems. Based on this method, a scheme is developed to obtain approximation solution of fractional wave, Burgers, Korteweg-de Vries (KdV), KdV-Burgers, and Klein-Gordon equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The fractional derivatives are described in the Caputo sense. So the homotopy analysis method for partial differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique.

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