Papers by Vassil M. Vassilev
Mathematics, Sep 21, 2021
Assessing the Non-Linear Dynamics of a Hopf-Langford Type System.
AIP Conference Proceedings
Van der Waals interactions between planar substrate and tubular lipid membranes undergoing pearli... more Van der Waals interactions between planar substrate and tubular lipid membranes undergoing pearling instability
APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 12th International On-line Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’20
arXiv (Cornell University), Mar 23, 2018
We present an analytical solution of the Ginzburg's Ψ-theory for the behavior of the Casimir forc... more We present an analytical solution of the Ginzburg's Ψ-theory for the behavior of the Casimir force in a film of 4 He in equilibrium with its vapor near the superfluid transition point, and we revisit the corresponding experiments [1] and [2] in terms of our findings. We find reasonably good agreement between the Ψ-theory predictions and the experimental data. Our calculated force is attractive, and the largest absolute value of the scaling function is 1.848, while experiment yields 1.30. The position of the extremum is predicted to be at x = (L/ξ0)(T /T λ − 1) 1/ν = π, while experiment is consistent with x = 3.8. Here L is the thickness of the film, T λ is the bulk critical temperature and ξ0 is the correlation length amplitude of the system for T > T λ .
Willmore Energy and Willmore Conjecture, 2017
INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING ICCMSE 2020, 2021
We study a model of a critical fluid described by the standard φ4 Ginzburg-Landau Hamiltonian and... more We study a model of a critical fluid described by the standard φ4 Ginzburg-Landau Hamiltonian and determine the Casimir force in the case of Dirichlet-Dirichlet boundary conditions imposed along the finite dimension of the film. Physically, the fluid can be a simple fluid near its liquid-vapor critical point, or a binary liquid mixture close to the corresponding demixing point. Mathematically, the stable states of the system are determined by the minimizers of the functional which leads to finding the solution of the corresponding Euler-Lagrange nonlinear differential equations under the considered boundary conditions. We solve these equations in terms of Weierstrass and Jacobi elliptic functions and give analytic representation of the Casimir force as a function of both the temperature and ordering field. The derived expressions are then evaluated numerically in the plane of the temperature and the ordering external field.
MATEC Web of Conferences, 2018
The behaviour of the local and total susceptibilities of a fluid system bounded by different surf... more The behaviour of the local and total susceptibilities of a fluid system bounded by different surfaces is studied in the framework of the Ginsburg-Landau Ising type model. The case of a plain geometry, Neumann-infinity boundary conditions under variations of the temperature and an external ordering field is considered. Exact analytic expressions for the order parameter, local and total susceptibilities in such a system are presented. They are used to analyse the phase behaviour of fluids confined in regions close to the bulk critical point of the respective infinite system.
AIP Conference Proceedings, 2017
In the current article we study the behavior of the van der Waals force between a planar substrat... more In the current article we study the behavior of the van der Waals force between a planar substrate and an axisymmetric bilayer lipid membrane undergoing pearling instability, caused by uniform hydrostatic pressure difference. To do so, the recently suggested "surface integration approach" is used, which can be considered a generalization of the well known and widely used Derjaguin approximation. The static equilibrium shape after the occurrence of the instability is described in the framework of Helfrich's spontaneous curvature model. Some specific classes of exact analytical solutions to the corresponding shape equation are considered, and the components of the respective position vectors given in terms of elliptic integrals and Jacobi elliptic functions. The mutual orientation between the interacting objects is chosen such that the axis of revolution of the distorted cylinder be parallel to the plane bounding the substrate. Based on the discussed models and approaches we made some estimations for the studied force in real experimentally realizable systems, thus showing the possibility of pearling as an useful technique for reduction of the adhesion in variety of industrial processes using lipid membranes as carriers.
AIP Conference Proceedings, 2017
Van der Waals interactions between planar substrate and tubular lipid membranes undergoing pearli... more Van der Waals interactions between planar substrate and tubular lipid membranes undergoing pearling instability
AIP Conference Proceedings, 2014
Journal of Physics: Condensed Matter, 2020
We study the behaviour of the non-retarded van der Waals force between a planar substrate and a s... more We study the behaviour of the non-retarded van der Waals force between a planar substrate and a singlewalled carbon nanotube, assuming that the system is immersed in a liquid medium which exerts hydrostatic pressure on the tube's surface, thereby altering its cross section profile. The shape of the latter is described as a continual structure characterized by its symmetry index n. Two principle mutual positions of the tube with respect to the substrate are studied: when one keeps constant the minimal separation between the surfaces of the interacting objects; when the distance from the tube's axis to the substrates bounding surface is fixed. Within these conditions, using the technique of the surface integration approach, we derive in integral form the expressions which give the dependance of the commented force on the applied pressure.
The present study is concerned with thin isotropic shallow shells interacting with inviscid fluid... more The present study is concerned with thin isotropic shallow shells interacting with inviscid fluid flow of constant velocity. It is assumed that the dynamic behaviour of the shells is governed by the Marguerre-von Karman equations. The influence of the fluid flow is taken into account by introduc ing additional differential and external load terms in the shell equations. It is shown that the system of equations governing such a fluid-structure inter action is equivalent to the von Karman equations. Thus, the symmetries and conservation laws of the considered fluid-structure system are established.
Mathematics, Sep 15, 2022
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Geometry Integrability and Quantization, 2020
Geometry Integrability and Quantization, 2020
The work deals with a family of nonlinear differential equations of Lane-Emden-Fowler type. The o... more The work deals with a family of nonlinear differential equations of Lane-Emden-Fowler type. The original Lane-Emden equation was used to model the thermal behavior of spherical clouds of gas within the framework of classical thermodynamics. Slightly modified, it describes phase transitions in critical thermodynamic systems, cylindrical equilibrium shapes of lipid membranes and many other physical processes and phenomena. The aim of the present work is to obtain approximate analytical solutions of the regarded equations. The problem is formulated in terms of nonlinear Volterra integral equations of the second kind. The solutions are sought by He's homotopy perturbation technique, as series expansion in the independent variable, and by Picard's method of successive approximations.
AIP Conference Proceedings, 2019
This work is concerned with the determination of stresses and strains in cell membranes subjected... more This work is concerned with the determination of stresses and strains in cell membranes subjected to micro-injections. For that purpose, a suitable variational statement of the problem is developed within a continuum mechanics approach to the analysis of cell membrane geometry and physics. In this setting, the cell membrane is regarded as an axially symmetric surface in the three-dimensional Euclidean space providing a stationary value of the bending energy functional under the constraint of fixed total area. The Euler-Lagrange equations and the natural boundary conditions associated with the foregoing variational problem are derived, analyzed and used to express the stresses and moments in the membrane. Several examples of such surfaces representing possible shapes of cell membranes subjected to micro injection are determined numerically.
Journal of Statistical Mechanics: Theory and Experiment, 2016
When massless excitations are limited or modified by the presence of material bodies one observes... more When massless excitations are limited or modified by the presence of material bodies one observes a force acting between them generally called Casimir force. Such excitations are present in any fluid system close to its true bulk critical point. We derive exact analytical results for both the temperature and external ordering field behavior of the thermodynamic Casimir force within the mean-field Ginzburg-Landau Ising type model of a simple fluid or binary liquid mixture. We investigate the case when under a film geometry the boundaries of the system exhibit strong adsorption onto one of the phases (components) of the system. We present analytical and numerical results for the (temperature-field) relief map of the force in both the critical region of the film close to its finite-size or bulk critical points as well as in the capillary condensation regime below but close to the finite-size critical point.
MATEC Web of Conferences, 2018
In this article we consider a critical thermodynamic system with the shape of a thin film confine... more In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.
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Papers by Vassil M. Vassilev