The influence of capillary effects on the drainage of a viscous gravity current into a deep porous medium

2017 ◽  
Vol 817 ◽  
pp. 514-559 ◽  
Author(s):  
Ying Liu ◽  
Zhong Zheng ◽  
Howard A. Stone
Keyword(s):  
Porous Medium ◽  
Gravity Current ◽  
Early Time ◽  
Late Time ◽  
Time Period ◽  
Gravity Driven ◽  
Drainage Problem ◽  
Self Similar ◽  
The One ◽  
Similar Solutions

The drainage of a viscous gravity current into a deep porous medium driven by both the gravitational and capillary forces is considered in two steps. We first study the one-dimensional case where a layer of fluid drains vertically into an infinitely deep porous medium. We determine a transition from the capillary-driven regime to the gravity-driven regime as time proceeds. Second, we solve the coupled spreading and drainage problem. There are no self-similar solutions of the problem for the entire time period, so asymptotic analyses are developed for the height, depth and front location in both the early-time and the late-time periods. In addition, we present numerical results of the governing partial differential equations, which agree well with the self-similar solutions in the appropriate asymptotic limits.

A self-similar solution for expansion into a vacuum of a collisionless plasma bunch

1998 ◽  
Vol 59 (1) ◽  
pp. 83-90 ◽  
Author(s):  
A. V. BAITIN ◽  
K. M. KUZANYAN
Keyword(s):  
Electron Distribution Function ◽  
Collisionless Plasma ◽  
Electron Component ◽  
One Dimensional ◽  
Ultrarelativistic Electron ◽  
Plasma Bunch ◽  
Self Similar Solution ◽  
Self Similar ◽  
The One ◽  
Similar Solutions

The process of expansion into a vacuum of a collisionless plasma bunch with relativistic electron temperature is investigated for the one-dimensional case. Self-similar solutions for the evolution of the electron distribution function and ion acceleration are obtained, taking account of cooling of the electron component of plasma for the cases of non-relativistic and ultrarelativistic electron energies.

Converging gravity currents over a permeable substrate

2015 ◽  
Vol 778 ◽  
pp. 669-690 ◽  
Author(s):  
Zhong Zheng ◽  
Sangwoo Shin ◽  
Howard A. Stone
Keyword(s):  
Power Law ◽  
Gravity Current ◽  
Gravity Currents ◽  
Numerical Predictions ◽  
Front Position ◽  
Dependent Position ◽  
Self Similar ◽  
Vertical Leakage ◽  
Similar Solutions ◽  
Lock Gate

We study the propagation of viscous gravity currents along a thin permeable substrate where slow vertical drainage is allowed from the boundary. In particular, we report the effect of this vertical fluid drainage on the second-kind self-similar solutions for the shape of the fluid–fluid interface in three contexts: (i) viscous axisymmetric gravity currents converging towards the centre of a cylindrical container; (ii) viscous gravity currents moving towards the origin in a horizontal Hele-Shaw channel with a power-law varying gap thickness in the horizontal direction; and (iii) viscous gravity currents propagating towards the origin of a porous medium with horizontal permeability and porosity gradients in power-law forms. For each of these cases with vertical leakage, we identify a regime diagram that characterizes whether the front reaches the origin or not; in particular, when the front does not reach the origin, we calculate the final location of the front. We have also conducted laboratory experiments with a cylindrical lock gate to generate a converging viscous gravity current where vertical fluid drainage is allowed from various perforated horizontal substrates. The time-dependent position of the propagating front is captured from the experiments, and the front position is found to agree well with the theoretical and numerical predictions when surface tension effects can be neglected.

scholarly journals Cost Implications of Comorbidity for Autologous Stem Cell Transplantation in Elderly Patients with Multiple Myeloma Using SEER-Medicare

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Gunjan L. Shah ◽  
Aaron Winn ◽  
Pei-Jung Lin ◽  
Andreas Klein ◽  
Kellie A. Sprague ◽  
...  
Keyword(s):  
Older Patients ◽  
Early Period ◽  
Early Time ◽  
Cost Of Care ◽  
Late Time ◽  
Time Period ◽  
Comorbidity Burden ◽  
Impact On Survival ◽  
The Difference ◽  
Early Time Period

Comorbidity is more common in older patients and can increase the cost of care by increasing toxicity. Using the SEER-Medicare database from 2000 to 2007, we examined the costs and life-year benefit of Auto-HSCT for MM patients over the age of 65 by evaluating the difference over time relative to comorbidity burden. One hundred ten patients had an Auto-HSCT in the early time period (2000–2003) and 160 in the late time period (2004–2007). Patients were divided by a Charlson Comorbidity Index (CCI) of 0 or greater than 1 (CCI1+). Median overall survival was 53.5 months for the late time period patients compared to 40.3 months for the early time period patients (p=0.031). Median costs for CCI0 versus CCI1+ in the early period were, respectively, $70,900 versus $72,000 (100 d); $86,100 versus $98,300 (1 yr); and $139,200 versus $195,300 (3 yrs). Median costs for late period were, respectively, $58,400 versus $60,400 (100 d); $86,300 versus $77,700 (1 yr); and $124,400 versus $110,900 (3 yrs). Comorbidity had a significant impact on survival and cost among early time period patients but not among late time period patients. Therefore, older patients with some comorbidities can be considered for Auto-HSCT depending on clinical circumstances.

Self-similar solutions of the second kind for the modified porous medium equation

1994 ◽  
Vol 5 (3) ◽  
pp. 391-403 ◽  
Author(s):  
Josephus Hulshof ◽  
Juan Luis Vazquez
Keyword(s):  
Porous Medium ◽  
Porous Medium Equation ◽  
Self Similarity ◽  
Compactly Supported ◽  
Medium Equation ◽  
Self Similar ◽  
Image Position ◽  
Similar Solutions

We construct compactly supported self-similar solutions of the modified porous medium equation (MPME)They have the formwhere the similarity exponents α and β depend on ε, m and the dimension N. This corresponds to what is known in the literature as anomalous exponents or self-similarity of the second kind, a not completely understood phenomenon. This paper performs a detailed study of the properties of the anomalous exponents of the MPME.

Self-similar viscous gravity currents: phase-plane formalism

1990 ◽  
Vol 210 ◽  
pp. 155-182 ◽  
Author(s):  
Julio Gratton ◽  
Fernando Minotti
Keyword(s):  
Scaling Laws ◽  
Phase Plane ◽  
Gravity Current ◽  
Classical Problem ◽  
Gravity Currents ◽  
Steady Flows ◽  
Two Phase ◽  
Integral Curves ◽  
Self Similar ◽  
Similar Solutions

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

Gravity currents over fractured substrates in a porous medium

2007 ◽  
Vol 584 ◽  
pp. 415-431 ◽  
Author(s):  
DAVID PRITCHARD
Keyword(s):  
Porous Medium ◽  
Steady State ◽  
Gravity Current ◽  
Gravity Currents ◽  
Limited Extent ◽  
Form Part ◽  
Long Time ◽  
Self Similar ◽  
Steady State Solutions ◽  
A Current

We consider the behaviour of a gravity current in a porous medium when the horizontal surface along which it spreads is punctuated either by narrow fractures or by permeable regions of limited extent. We derive steady-state solutions for the current, and show that these form part of a long-time asymptotic description which may also include a self-similar ‘leakage current’ propagating beyond the fractured region with a length proportional to t1/2. We discuss the conditions under which a current can be completely trapped by a permeable region or a series of fractures.

scholarly journals The problem of melting ice in a porous medium saturated with ice and gas

2019 ◽  
Vol 14 (1) ◽  
pp. 59-62
Author(s):  
M.N. Zapivakhina ◽  
D.A Umerov
Keyword(s):  
Porous Medium ◽  
Warm Water ◽  
Water Ice ◽  
Higher Temperature ◽  
Self Similar ◽  
Porous Soil ◽  
Temperature And Pressure ◽  
Lower Temperature ◽  
Similar Solutions ◽  
Constructed Self

The problem of ice formation in a dry, cold, porous medium saturated with ice and gas (air) when pumping warm water is considered in a flat one-dimensional self-similar formulation. The task was considered in volume area. During the injection of warm water from the beginning deep into the reservoir, it spread in a volume region that will divide the reservoir into 3 zones. The first zone was filled with water, the second zone was filled with ice and water, and the third zone was filled with ice and gas. To describe the process of heat and mass transfer, the following hypotheses were used: the temperature of the saturated substance (water, ice or gas) is equal to the temperature of the porous medium; ice and skeleton still; water, ice and skeleton of the reservoir are incompressible; skeletal porosity is constant. On the basis of constructed self-similar solutions, a numerical analysis was performed illustrating the effect of the initial parameters of a dry porous medium saturated with ice and gas, as well as the temperature of the injected water on the temperature and pressure distribution in the porous medium. It has been established that an increase in the temperature of the injected water does not lead to a significant increase in the area of ice decomposition. It is also established that if the pressure of the injected water is increased, this will not lead to a large increase in the area of ice decomposition. However, based on the results obtained, it can be seen that the speed of movement of the melting boundary increases, in particular, as the pressure increases by <i>p<sub>e</sub></i>=0.05 MPa, the intermediate region increases by one and a half times. It was found that it is economically more profitable to pump water with a lower temperature, because water with a higher temperature slightly increases the freezing area of the porous soil.

scholarly journals Early Time Modifications to the Buoyancy-Drag Model for Richtmyer–Meshkov Mixing

2020 ◽  
Vol 142 (12) ◽  
Author(s):  
David L. Youngs ◽  
Ben Thornber
Keyword(s):  
Initial Conditions ◽  
Mean Field ◽  
Early Time ◽  
Growth Exponent ◽  
Late Time ◽  
Time Behavior ◽  
Mixing Zone ◽  
Drag Model ◽  
Model Based ◽  
Self Similar

Abstract The Buoyancy-Drag model is a simple model, based on ordinary differential equations, for estimating the growth in the width of a turbulent mixing zone at an interface between fluids of different densities due to Richtmyer–Meshkov and Rayleigh–Taylor instabilities. The model is calibrated to give the required self-similar behavior for mixing in simple situations. However, the early stages of the mixing process are very dependent on the initial conditions and modifications to the Buoyancy-Drag model are then needed to obtain correct results. In a recent paper, Thornber et al. (2017, “Late-Time Growth Rate, Mixing, and Anisotropy in the Multimode Narrowband Richtmyer–Meshkov Instability: The θ-Group Collaboration,” Phys. Fluids, 29, p. 105107), a range of three-dimensional simulation techniques was used to calculate the evolution of the mixing zone integral width due to single-shock Richtmyer–Meshkov mixing from narrowband initial random perturbations. Further analysis of the results of these simulations gives greater insight into the transition from the initial linear behavior to late-time self-similar mixing and provides a way of modifying the Buoyancy-Drag model to treat the initial conditions accurately. Higher-resolution simulations are used to calculate the early time behavior more accurately and compare with a multimode model based on the impulsive linear theory. The analysis of the iLES data also gives a new method for estimating the growth exponent, θ (mixing zone width ∼ tθ), which is suitable for simulations which do not fully reach the self-similar state. The estimates of θ are consistent with the theoretical model of Elbaz and Shvarts (2018, “Modal Model Mean Field Self-Similar Solutions to the Asymptotic Evolution of Rayleigh-Taylor and Richtmyer-Meshkov Instabilities and Its Dependence on the Initial Conditions,” Phys. Plasmas, 25, p. 062126).

Sign in / Sign up

Export Citation Format

Share Document