An Analytical Solution of Two-Dimensional Flow and Deformation Coupling Due to a Point Source Within a Finite Poroelastic Media

2011 ◽  
Vol 78 (6) ◽  
Author(s):  
Peichao Li ◽  
Detang Lu
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Point Source ◽  
Transient Behavior ◽  
Finite Domain ◽  
Two Dimensional ◽  
Pressure Derivative ◽  
Poroelastic Media ◽  
Two Dimensional Flow ◽  
Dependent Flow

An analytical solution is derived for the time-dependent flow and deformation coupling of a saturated isotropic homogeneous incompressible poroelastic media within a two-dimensional (2D) finite domain due to a point source at some arbitrary position. In this study, the pore pressure field is assumed to conform to the second type of boundary conditions. Boundary conditions of the displacement field are chosen with care to match the appropriate finite sine and cosine transforms and simplify the resulting solution. It is found that the analytical solution is always independent of the Poisson’s ratio. The detailed solutions are given for the case of a periodic point source with zero pressure derivatives on the boundaries and for an imposed pressure derivative on the lower edge in the absence of a source. The presented analytical solutions are highly applicable for calibrating numerical codes, and meanwhile they can be used to further investigate the transient behavior of flow and deformation coupling induced by fluid withdrawal within a 2D finite poroelastic media.

Multiple solutions and flow limitation in collapsible channel flows

2000 ◽  
Vol 420 ◽  
pp. 301-324 ◽  
Author(s):  
X. Y. LUO ◽  
T. J. PEDLEY
Keyword(s):  
Boundary Conditions ◽  
Channel Model ◽  
Control Method ◽  
Flow Limitation ◽  
Two Dimensional ◽  
Solution Branch ◽  
Two Dimensional Flow ◽  
Unstable Solutions ◽  
Steady Solutions ◽  
The Stability

Steady and unsteady numerical simulations of two-dimensional flow in a collapsible channel were carried out to study the flow limitation which typically occurs when the upstream transmural pressure is held constant while flow rate and pressure gradient along the collapsible channel can vary independently. Multiple steady solutions are found for a range of upstream transmural pressures and Reynolds number using an arclength control method. The stability of these steady solutions is tested in order to check the correlation between flow limitation and self-excited oscillations (the latter being a consequence of unstable steady solutions). Both stable and unstable solutions are found when flow is limited. Self-excited oscillations and divergence instabilities are observed in certain solution branches. The instability of the steady solutions seems to depend on the unsteady boundary conditions used, i.e. on which parameters are allowed to vary. However, steady solutions associated with the solution branch before flow limitation where the membrane wall bulges are found to be stable for each of the three different boundary conditions employed. We conclude that there is no one to one correlation between the two phenomena in this two dimensional channel model.

Boundary conditions at a liquid/air interface in lubrication flows

1982 ◽  
Vol 119 ◽  
pp. 107-120 ◽  
Author(s):  
K. J. Ruschak
Keyword(s):  
Finite Element Method ◽  
Boundary Conditions ◽  
Problem Solution ◽  
Lubrication Approximation ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
Air Interface ◽  
Outer Expansion ◽  
Two Dimensional Flow ◽  
Representative Problem

A difficulty in applying the lubrication approximation to flows where a liquid/air interface forms lies in supplying boundary conditions at the point of formation of the interface that are consistent with the lubrication approximation. The method of matched asymptotic expansions is applied to the flow between partially submerged, counter-rotating rollers, a representative problem from this class, and the lubrication approximation is found to generate the first term of an outer expansion of the problem solution. The first term of an inner expansion describes the two-dimensional flow in the vicinity of the interface, and approximate results are found by the finite-element method. Matching between the inner and outer solutions determines boundary conditions on the pressure and the pressure gradient at the point of formation of the interface which allow the solution to the outer, lubrication flow to be completed.

Two and Three-Dimensional Flow using Finite Elements

1969 ◽  
Vol 73 (707) ◽  
pp. 961-964 ◽  
Author(s):  
J. H. Argyris ◽  
G. Mareczek ◽  
D. W. Scharpf
Keyword(s):  
Finite Elements ◽  
Boundary Conditions ◽  
Stream Function ◽  
Three Dimensional ◽  
Finite Domain ◽  
Flow Problems ◽  
Dimensional Flow ◽  
Two Dimensional Flow ◽  
Triangular Elements ◽  
Compressibility Effects

The method of finite elements is in certain cases advantageous when dealing with flow problems in a finite domain. This is particularly so when attempting to include subcritical compressibility effects. In the present note we first consider the two-dimensional flow using TRIC and TRIM-like triangular elements in conjunction with the concept of the stream function, which is assigned to the nodal points of the elements. The application of the stream function allows a direct and exact satisfaction of the boundary conditions. Strictly the elements in question could also be used in conjunction with the potential function but the observance of the boundary conditions is then cumbersome.

Steady two-dimensional viscous flow in a jet

1972 ◽  
Vol 55 (1) ◽  
pp. 49-63 ◽  
Author(s):  
K. Capell
Keyword(s):  
Point Source ◽  
Viscous Flow ◽  
Closed Form ◽  
Stream Function ◽  
Asymptotic Expansions ◽  
Far Field ◽  
Two Dimensional ◽  
Complete Structure ◽  
Dimensional Flow ◽  
Two Dimensional Flow

An idealized two-dimensional flow due to a point source ofxmomentum is discussed. In the far field the flow is modelled by a jet region of large vorticity outside which the flow is potential. After use of the transformation\[ \zeta^3 = (\xi + i\eta)^3 = x + iy, \]the equations suggest naively obvious asymptotic expansions for the stream function in these two regions, namely\[ \sum_{n=0}^{\infty}\xi^{1-n}f_n(\eta)\quad {\rm and}\quad\sum_{n=0}^{\infty}\xi^{1-n}F_n(\eta/\xi) \]respectively. Consistency in matching these expansions is achieved by including logarithmic terms associated with the occurrence of eigensolutions.Fnis easy to find andJncan be found in closed form so the inner and outer eigensolutions may be fully determined along with the complete structure of the expansions.

Lubrication Theory for Micropolar Fluids

1971 ◽  
Vol 38 (3) ◽  
pp. 646-650 ◽  
Author(s):  
S. J. Allen ◽  
K. A. Kline
Keyword(s):  
Flow Field ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Two Dimensional ◽  
Slider Bearing ◽  
Governing Equations ◽  
Micropolar Fluids ◽  
Linear Ordinary Differential Equations ◽  
Two Dimensional Flow ◽  
Order Of Magnitude

The equations governing the flow of a fluid with rigid, spherical substructure are summarized. A two-dimensional flow field is considered and applied to the geometry of a slider bearing. Order-of-magnitude arguments are used which reduce the governing equations to a system of coupled, linear, ordinary differential equations. The equations are solved subject to appropriate boundary conditions and the effects of substructure discussed with the help of a specific numerical example.

A Localized Finite-Element Method for Two-Dimensional Steady Potential Flows with a Free Surface

1978 ◽  
Vol 22 (04) ◽  
pp. 216-230
Author(s):  
Kwang June Bai
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Boundary Conditions ◽  
Complex Geometry ◽  
The Body ◽  
Finite Domain ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Test Functions ◽  
Infinite Domain

A numerical method is presented for solving two-dimensional uniform flow problems with a linearized free-surface boundary condition. The boundary-value problem governed by Laplace's equation is replaced by a weak formulation (also known as Galerkin's method) with certain essential boundary conditions. The infinite domain of the fluid is reduced to a finite domain by utilizing known solution spaces in certain subdomains. The bases for the trial and test functions are chosen from the same subspace of the polynomial function space in the reduced subdomain. The essential boundary conditions are properly taken into account by an unconventional choice of the basis for the trial functions, which is different from that for the test functions in other subdomains. This method is applied to two-dimensional steady flow past a submerged elliptic section, a hydrofoil at an arbitrary angle of attack, and a bump on the bottom. In each example the body boundary condition is satisfied exactly. Both subcritical and supercritical flows are treated. We present the numerical results of wave resistance, lift force, moment, circulation strength, and flow blockage parameter. The computed pressure distributions on the hydrofoil and wave profiles are shown. The test results obtained by the present method agree very well with existing results. The main advantage of this method is that any complex geometry of the boundary can be easily accommodated.

scholarly journals Notes on superposed incremental deformations in the mechanics of lipid membranes

2017 ◽  
Vol 24 (1) ◽  
pp. 181-194 ◽  
Author(s):  
Mahdi Zeidi ◽  
Chun IL Kim
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Lipid Membranes ◽  
Membrane System ◽  
Parametric Representation ◽  
Radius Of Convergence ◽  
Complete Analysis ◽  
Finite Domain ◽  
Morphological Transitions ◽  
Excellent Stability

We present an analysis of the superposed incremental deformations of lipid membranes in contact with a circular substrate. A complete analytical solution describing the morphological transitions of lipid membranes is obtained via Monge parametric representation and admissible linearization. The corresponding solution demonstrates smooth and bounded behavior within the finite domain of interest (annular) and, more importantly, shows excellent stability as it approaches the boundary of the circular substrate with the radius of convergence compatible with a few nanometers. Under the prescription of the superposed incremental deformations, a complete analysis of the necessary boundary conditions, the anchoring condition of the lipid molecules on an edge, and other geometrical quantities of the membrane is illustrated for the case of the circular substrate–membrane system.

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