The boundary integral equation for curved solid/liquid interfaces propagating into a binary liquid with convection

2021 ◽  
Author(s):  
Ekaterina Titova ◽  
Dmitri Alexandrov
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Curvilinear Coordinates ◽  
Forced Flow ◽  
Function Technique ◽  
Liquid Interfaces ◽  
Convective Boundary ◽  
Solid Liquid

Abstract The boundary integral method is developed for unsteady solid/liquid interfaces propagating into undercooled binary liquids with convection. A single integrodifferential equation for the interface function is derived using the Green function technique. In the limiting cases, the obtained unsteady convective boundary integral equation (CBIE) transforms into a previously developed theory. This integral is simplified for the steady-state growth in arbitrary curvilinear coordinates when the solid/liquid interface is isothermal (isoconcentration). Finally, we evaluate the boundary integral for a binary melt with a forced flow and analyze how the melt undercooling depends on P\'eclet and Reynolds numbers.

Analysis of Crack Propagation in Roller Bearings Using the Boundary Integral Equation Method—A Mixed-Mode Loading Problem

1988 ◽  
Vol 110 (3) ◽  
pp. 408-413 ◽  
Author(s):  
L. J. Ghosn
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Crack Propagation ◽  
Boundary Integral Equation ◽  
Stress Intensity Factors ◽  
High Speed ◽  
Roller Bearing ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Intensity Factors

Crack propagation in a rotating inner raceway of a high-speed roller bearing is analyzed using the boundary integral method. The model consists of an edge plate under plane strain condition upon which varying Hertzian stress fields are superimposed. A multidomain boundary integral equation using quadratic elements was written to determine the stress intensity factors KI and KII at the crack tip for various roller positions. The multidomain formulation allows the two faces of the crack to be modeled in two different subregions making it possible to analyze crack closure when the roller is positioned on or close to the crack line. KI and KII stress intensity factors along any direction were computed. These calculations permit determination of crack growth direction along which the average KI times the alternating KI is maximum.

scholarly journals The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

2018 ◽  
Vol 376 (2113) ◽  
pp. 20170218 ◽  
Author(s):  
Peter K. Galenko ◽  
Dmitri V. Alexandrov ◽  
Ekaterina A. Titova
Keyword(s):  
Binary Systems ◽  
Computational Modelling ◽  
Selection Criterion ◽  
Type Equation ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Isotropic Case ◽  
Liquid Interfaces ◽  
Stationary Growth ◽  
Solid Liquid

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay–Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

scholarly journals A boundary integral method with volume-changing objects for ultrasound-triggered margination of microbubbles

2017 ◽  
Vol 836 ◽  
pp. 952-997 ◽  
Author(s):  
Achim Guckenberger ◽  
Stephan Gekle
Keyword(s):  
Drug Delivery ◽  
Integral Equation ◽  
Boundary Integral Equation ◽  
Integral Method ◽  
Three Dimensional ◽  
Formal Proof ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Drug Uptake ◽  
Periodic Domains

A variety of numerical methods exist for the study of deformable particles in dense suspensions. None of the standard tools, however, currently include volume-changing objects such as oscillating microbubbles in three-dimensional periodic domains. In the first part of this work, we develop a novel method to include such entities based on the boundary integral method. We show that the well-known boundary integral equation must be amended with two additional terms containing the volume flux through the bubble surface. We rigorously prove the existence and uniqueness of the solution. Our proof contains as a subset the simpler boundary integral equation without volume-changing objects (such as red blood cell or capsule suspensions) which is widely used but for which a formal proof in periodic domains has not been published to date. In the second part, we apply our method to study microbubbles for targeted drug delivery. The ideal drug delivery agent should stay away from the biochemically active vessel walls during circulation. However, upon reaching its target it should attain a near-wall position for efficient drug uptake. Though seemingly contradictory, we show that lipid-coated microbubbles in conjunction with a localized ultrasound pulse possess precisely these two properties. This ultrasound-triggered margination is due to hydrodynamic interactions between the red blood cells and the oscillating lipid-coated microbubbles which alternate between a soft and a stiff state. We find that the effect is very robust, existing even if the duration in the stiff state is more than three times lower than the opposing time in the soft state.

scholarly journals The boundary integral equation method for the solution of a class of problems in anisotropic elasticity

1981 ◽  
Vol 22 (4) ◽  
pp. 394-407 ◽  
Author(s):  
David L. Clements ◽  
Oscar A. C. Jones
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Integral Method ◽  
Integral Equation Method ◽  
Anisotropic Elasticity ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Important Class ◽  
Numerical Examples ◽  
Integral Procedure

AbstractA boundary integral procedure for the solution of an important class of problems in anisotropic elasticity is outlined. Specific numerical examples are considered in order to provide a comparison with the standard boundary integral method.

scholarly journals Shielding properties of a conducting bar calculated with a boundary integral method

2005 ◽  
Vol 3 ◽  
pp. 119-123
Author(s):  
L. O. Fichte ◽  
S. Lange ◽  
Th. Steinmetz ◽  
M. Clemens
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Vector Potential ◽  
Low Frequency ◽  
Integral Equation Method ◽  
Fourier Integral ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Frequency Magnetic Field ◽  
Shielding Properties

Abstract. A plane rectangular bar of conducting and permeable material is placed in an external low-frequency magnetic field. The shielding properties of this object are investigated by solving the given plane eddy current problem for the vector potential with the boundary integral equation method. The vector potential inside the rectangle is governed by Helmholtz' equation, which in our case is solved by separation. The solution is inserted into the remaining boundary integral equation for the exterior vector potential in the domain surrounding the bar. By expressing its logarithmic kernel as a Fourier integral the overall solution inside and outside the bar is calculated using analytical means only.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

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