On viscous film flows coating the interior of a tube: thin-film and long-wave models

2015 ◽  
Vol 772 ◽  
pp. 569-599 ◽  
Author(s):  
Roberto Camassa ◽  
H. Reed Ogrosky
Keyword(s):  
Thin Film ◽  
Numerical Solutions ◽  
Travelling Wave ◽  
Long Wave ◽  
Fluid Core ◽  
Primary Model ◽  
Long Time ◽  
Dynamical Regimes ◽  
Wave Models ◽  
Comparison Of The Results

A theoretical and numerical investigation of two classes of models for pressure-driven core–annular flow is presented. Both classes, referred to as ‘long-wave’ and ‘thin-film’ models, may be derived from a unified perspective using long-wave asymptotics, but are distinct from one another in the role played by the curved tube geometry with respect to the planar (limiting) case. Analytical and numerical techniques are used to show and quantify the significant differences between the behaviour of solutions to both model types. Temporal linear stability analysis of the constant solution is carried out first to pinpoint with closed-form mathematical expressions the different dynamical regimes associated with absolute or convective instabilities. Numerical simulations for the models are then performed and qualitative differences in the evolution of the free surface are explored. Mathematically, different levels of asymptotic accuracy are found to result in different regularizing properties affecting the long-time behaviour of generic numerical solutions. Travelling wave solutions are also studied, and qualitative differences in the topology of streamline patterns describing the flow of the film in a moving reference frame are discussed. These topological differences allow for further classification of the models. In particular, a transition from a regime in which waves trap a fluid core to one where waves travel faster than any parcel of the underlying fluid is documented for a variant of the primary model. In the corresponding thin-film model, no such transition is found to occur. The source of these differences is examined, and a comparison of the results with those of related models in the literature is given. A brief discussion of the merits of each class of models concludes this study.

scholarly journals On travelling wave solutions of a model of a liquid film flowing down a fibre

2021 ◽  
pp. 1-30
Author(s):  
HANGJIE JI ◽  
ROMAN TARANETS ◽  
MARINA CHUGUNOVA
Keyword(s):  
Thin Film ◽  
Weak Solutions ◽  
A Priori Estimates ◽  
Travelling Waves ◽  
A Priori ◽  
Travelling Wave ◽  
Model Parameters ◽  
Priori Estimates ◽  
Thin Film Model ◽  
Long Time

Abstract Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

scholarly journals Mathematical Modelling of the Effects of Mitotic Inhibitors on Avascular Tumour Growth

1999 ◽  
Vol 1 (4) ◽  
pp. 287-311 ◽  
Author(s):  
J. P. Ward ◽  
J. R. King
Keyword(s):  
Steady State ◽  
Tumour Growth ◽  
Numerical Solutions ◽  
Mitotic Rate ◽  
Travelling Wave ◽  
Living Cells ◽  
Cellular Material ◽  
Raw Material ◽  
Long Time ◽  
Mitotic Inhibitors

In this paper we build on the mathematical model of Ward and King (1998) to study the effects of high molecular mass mitotic inhibitors released at cell death. The model assumes a continuum of living cells which, depending on the concentration of a generic nutrient, generate movement (described by a velocity field) due to the changes in volumes caused by cell birth and death. The necrotic material is assumed to consist of two diffusible materials: 1) basic cellular material which is used by living cells as raw material for mitosis; 2) a generic non-utilisable material which may inhibit mitosis. Numerical solutions of the resulting system of partial differential equations show all the main features of tumour growth and heterogeneity. Material 2) is found to act in an inhibitive fashion in two ways: i) directly, by reducing the mitotic rate and ii) indirectly, by occupying space, thereby reducing the availability of the basic cellular material. For large time the solutions to the model tend either to a steady-state, reflecting growth saturation, or to a travelling wave, indicating continual linear growth. The steady-state and travelling wave limits of the model are derived and studied, the regions of existence of these two types of long-time solution being explored in parameter space using numerical methods.

scholarly journals On the stability of traveling wave solutions to thin-film and long-wave models for film flows inside a tube

2021 ◽  
Vol 415 ◽  
pp. 132750 ◽  
Author(s):  
Roberto Camassa ◽  
Jeremy L. Marzuola ◽  
H. Reed Ogrosky ◽  
Sterling Swygert
Keyword(s):  
Thin Film ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Wave Solutions ◽  
Long Wave ◽  
Wave Models ◽  
The Stability ◽  
Film Flows

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.

Nonlinear dispersion hyperbolic models of continuous media

2007 ◽  
Vol 5 ◽  
pp. 273-278
Author(s):  
V.Yu Liapidevskii
Keyword(s):  
Boussinesq Equations ◽  
Nonlinear Dispersion ◽  
Order Of Accuracy ◽  
Flow Models ◽  
Wave Approximation ◽  
Long Wave ◽  
Primary Model ◽  
Nonequilibrium Flows ◽  
Internal Inertia ◽  
Long Wave Approximation

Nonequilibrium flows of an inhomogeneous liquid in channels and pipes are considered in the long-wave approximation. Nonlinear dispersion hyperbolic flow models are derived allowing taking into account the influence of internal inertia during the relative motion of phases upon the structure of nonlinear wave fronts. The asymptotic derivation of dispersion hyperbolic models is shown on the example of classical Boussinesq equations. It is shown that the hyperbolic approximation of the equations has the same order of accuracy as the primary model.

A high-order spectral method for the study of nonlinear gravity waves

1987 ◽  
Vol 184 ◽  
pp. 267-288 ◽  
Author(s):  
Douglas G. Dommermuth ◽  
Dick K. P. Yue
Keyword(s):  
Gravity Waves ◽  
Mode Coupling ◽  
Arbitrary Order ◽  
Travelling Wave ◽  
Computational Effort ◽  
Zakharov Equation ◽  
Long Time ◽  
Nonlinear Free Surface ◽  
High Order Spectral Method ◽  
Nonlinear Gravity

We develop a robust numerical method for modelling nonlinear gravity waves which is based on the Zakharov equation/mode-coupling idea but is generalized to include interactions up to an arbitrary order M in wave steepness. A large number (N = O(1000)) of free wave modes are typically used whose amplitude evolutions are determined through a pseudospectral treatment of the nonlinear free-surface conditions. The computational effort is directly proportional to N and M, and the convergence with N and M is exponentially fast for waves up to approximately 80% of Stokes limiting steepness (ka ∼ 0.35). The efficiency and accuracy of the method is demonstrated by comparisons to fully nonlinear semi-Lagrangian computations (Vinje & Brevig 1981); calculations of long-time evolution of wavetrains using the modified (fourth-order) Zakharov equations (Stiassnie & Shemer 1987); and experimental measurements of a travelling wave packet (Su 1982). As a final example of the usefulness of the method, we consider the nonlinear interactions between two colliding wave envelopes of different carrier frequencies.

scholarly journals Gravity currents and internal waves in a stratified fluid

2008 ◽  
Vol 616 ◽  
pp. 327-356 ◽  
Author(s):  
BRIAN L. WHITE ◽  
KARL R. HELFRICH
Keyword(s):  
Internal Waves ◽  
Wave Speed ◽  
Numerical Solutions ◽  
Gravity Current ◽  
Gravity Currents ◽  
Stratified Fluid ◽  
Front Speed ◽  
Long Wave ◽  
Conjugate State ◽  
Energy Conserving

A steady theory is presented for gravity currents propagating with constant speed into a stratified fluid with a general density profile. Solution curves for front speed versus height have an energy-conserving upper bound (the conjugate state) and a lower bound marked by the onset of upstream influence. The conjugate state is the largest-amplitude nonlinear internal wave supported by the ambient stratification, and in the limit of weak stratification approaches Benjamin's energy-conserving gravity current solution. When the front speed becomes critical with respect to linear long waves generated above the current, steady solutions cannot be calculated, implying upstream influence. For non-uniform stratification, the critical long-wave speed exceeds the ambient long-wave speed, and the critical-Froude-number condition appropriate for uniform stratification must be generalized. The theoretical results demonstrate a clear connection between internal waves and gravity currents. The steady theory is also compared with non-hydrostatic numerical solutions of the full lock release initial-value problem. Some solutions resemble classic gravity currents with no upstream disturbance, but others show long internal waves propagating ahead of the gravity current. Wave generation generally occurs when the stratification and current speed are such that the steady gravity current theory fails. Thus the steady theory is consistent with the occurrence of either wave-generating or steady gravity solutions to the dam-break problem. When the available potential energy of the dam is large enough, the numerical simulations approach the energy-conserving conjugate state. Existing laboratory experiments for intrusions and gravity currents produced by full-depth lock exchange flows over a range of stratification profiles show excellent agreement with the conjugate state solutions.

scholarly journals The Excess Secular Change in the Obliquity of the Ecliptic and Its Relation to the Internal Motion of the Earth

1972 ◽  
Vol 48 ◽  
pp. 192-195 ◽  
Author(s):  
Chuichi Kakuta ◽  
Shinko Aoki
Keyword(s):  
Time Variation ◽  
Rotational Speed ◽  
Internal Motion ◽  
Secular Change ◽  
Electrically Conducting ◽  
The Core ◽  
The Earth ◽  
Fluid Core ◽  
Electrical Conductivities ◽  
Long Time

The previous result (Aoki, 1969) on the explanation of the excess secular change in the obliquity of the ecliptic frictional couplings in the rigid constituents, the mantle and the core, is extended by using a model of an elastic and electrically conducting mantle and a hydromagnetic core. The secular change of the obliquity of the ecliptic referred to the mantle is found to be 1/3.2 times of the observed value, if the electrical conductivities of the fluid core and the mantle are assumed to be 3·10−6 emu and 3·10−9 emu respectively. A large secular deceleration of the Earth's rotational speed obtained in the previous result is proved to be strongly reduced because of weak excitation of the perturbing potential for a long time variation.

DESICCATION–REHYDRATION STRESS REVEALED BY SUGAR-METABOLITE-RESERVE MODEL

2021 ◽  
pp. 1-19
Author(s):  
JULIANA M. BERBERT ◽  
KAREN A. OLIVEIRA ◽  
RAFAELA F. MARTIN ◽  
DANILO C. CENTENO
Keyword(s):  
Water Supply ◽  
Climate Changes ◽  
Compartmental Model ◽  
Numerical Solutions ◽  
Mathematical Formulation ◽  
Photosynthetic Organisms ◽  
Crop Field ◽  
Long Time ◽  
External Conditions ◽  
New Perspective

We focus on the evaluation of photosynthetic organisms. Some species and tissues can endure periods of the dry season because they rely on a robust dynamics of metabolites. The metabolic dynamics are complex and challenging to address because it involves several steps, usually with hundreds of metabolites. The metabolites densities vary among species and tissues and respond to external conditions, such as an environmental stimulus like water supply. Understanding these responses, particularly the desiccation–rehydration processes, are important both economically and evolutionarily, especially in the presence of climate change. Therefore, we propose a new way to analyze the dynamics of metabolites with a compartmental model which explores the metabolites densities’ dependence on water explicitly. We use a mathematical formulation to model the dynamics among three essential metabolites classes: sugar ([Formula: see text]), active metabolite ([Formula: see text]), and reserve accumulation ([Formula: see text]). Through stability analysis and numerical solutions, we characterize regions on the phase space, defined by the transition rates between the classes [Formula: see text] to [Formula: see text] and [Formula: see text] to [Formula: see text], where the system diverges or approaches zero. We show that different species and tissues respond distinctly to desiccation processes, being more or less resilient according to the transitions rate between the compartments of the model. Furthermore, the effects of water supply fluctuation, due to the desiccation–rehydration processes, show that unless the organism has a robust reservoir metabolism, the system cannot support itself for a long time. Many results corroborate experimental observations, and others provide a new perspective on the studies of metabolic dynamics, such as the significance of the reservoir metabolism. We understand that knowing the organism’s response to abiotic changes, particularly that of the water supply, may improve our management of the use of these organisms, for example, in the crop field during climate changes.

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