Ivan B. Bazhlekov

Interaction of a deformable bubble with a rigid wall at moderate Reynolds numbers

The unsteady viscous flow induced by a deformable gas bubble approaching or receding away from a rigid boundary is investigated for moderate Reynolds numbers. The full Navier–Stokes equations were solved by means of a finite-element method. The bubble is driven by the buoyancy force. The performance of the numerical scheme is displayed for two different configurations of the flow: the bubble moves (i) in the half-space bounded by a rigid plate; (ii) in a spherical container filled with viscous fluid. Results are obtained for the evolution of the flow pattern and bubble shape for a number of values of Reynolds and Eotvos numbers: 2.2 × 10 −3

Viscoelastic flows with fractional derivative models: Computational approach by convolutional calculus of Dimovski

An initial-boundary value problem for the velocity distribution of a viscoelastic flow with generalized fractional Oldroyd-B constitutive model is studied. The model contains two Riemann-Liouville fractional derivatives in time. The eigenfunction expansion of the solution is constructed. The behavior of the time-dependent components of the solution is studied and the results are used to establish convergence of the series under some conditions. Further, applying the convolutional calculus approach proposed by Dimovski (I.H. Dimovski, Convolutional Calculus, Kluwer, Dordrecht (1990)), a Duhamel-type representation of the solution is found, containing two convolution products of particular solutions and the given initial and source functions. A non-classical convolution with respect to spatial variable is used. The obtained representation is applied for numerical computation of the solution in the case of a generalized second grade fluid. Numerical results for several one-dimensional examples are given and the present technique is compared to a finite difference method in terms of efficiency, accuracy, and CPU time.

Numerical investigation of the dynamic influence of the contact line region on the macroscopic meniscus shape

The influence of different boundary conditions applied in the contact line region on the outer meniscus shape is analysed by means of a finite-element numerical simulation of the steady movement of a liquid-gas meniscus in a capillary tube. The free-surface steady shape is obtained by solving the unsteady creeping-flow approximation of the Navier–Stokes equations starting from some initial shape. Comparisons of the outer solutions obtained using two different inner models, together with that published by Lowndes (1980), indicate the relative insensitivity of the outer solution to the type of model utilized in the contact line region.

Unidirectional flows of fractional Jeffreys fluids

A class of initialboundary value problems governing the velocity distribution of unidirectional flows of viscoelastic fluids is studied. The generalized fractional Jeffreys constitutive model is used to describe the viscoelastic properties. Thermodynamic constraints on the parameters of the model are derived from the monotonicity of the corresponding relaxation function. Based on these constraints, a subordination principle for the considered class of problems is established. It gives an integral representation of the solution in terms of a probability density function and the solution of a related wave equation. Explicit representation of the probability density function is derived from the solution of the Stokes first problem. Numerical verification of the obtained analytical results is provided.

Numerical method for viscous hydrodynamic problems with dynamic contact lines

Abstract A numerical method is presented for computational modelling of viscous monophase and multiphase flows in presence of dynamic contact lines. Both general cases of liquid-fluid-solid and liquid-liquid-fluid contact lines are included. The method could also be considered as a mathematical tool for free-surface hydrodynamics and interface science. The modelling is based on unsteady Navier-Stokes equations and full force balance at the deformable liquid-liquid or liquid-gas interfaces in presence of surface tension. The mathematical model is transient, nonlinear and involves a free and internal surface with singularity at the contact lines. The numerical method belongs to the class of finite element ones of divergence-free type and is developed in Lagrangian approach. Comparisons with other theoretical and experimental studies are presented that confirm the consistency of the numerical method. A number of numerical examples of unsteady fluid motion in a capillary, spreading or hanging of a drop on a horizontal plate, rise of 35 compound drop and engulfing of a drop by the liquid from another drop in a third quiescent ambient liquid are included.

Numerical simulation of drop coalescence in the presence of drop soluble surfactant

Numerical method is presented for simulation of the deformation, drainage and rupture of axisymmetric film (gap) between colliding drops in the presence of film soluble surfactants under the influence of van der Waals forces at small capillary and Reynolds numbers and small surfactant concentrations. The mathematical model is based on the lubrication equations in the gap between drops and the creeping flow approximation of Navier-Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. A non-uniform surfactant concentration on the interfaces, related with that in the film, leads to a gradient of the interfacial tension which in turn leads to additional tangential stress on the interfaces (Marangoni effects). Both film and interface surfactant concentrations, related via adsorption isotherm, are governed by a convection-diffusion equation. The numerical method consists of: Boundary integral method for the flow in the drops; Finite difference method for the flow...

Numerical simulation of dynamic contact-line problems

The presence of a three-phase region, where three immiscible phases are in mutual contact, causes additional difficulties in the investigation of many fluid mechanical problems. To surmount these difficulties some assumptions or specific hydrodynamic models have been used in the contact region (inner region). In the present paper an approach to the numerical solution of dynamic contact-line problems in the outer region is described. The influence of the inner region upon the outer one is taken into account by means of a solution of the integral mass and momentum conservation equations there. Both liquid–fluid–liquid and liquid–fluid–solid dynamic contact lines are considered. To support the consistency of this approach tests and comparisons with a number of experimental results are performed by means of finite-element numerical simulations.

Mathematical Modelling of the Effect of Biosurfactants on the Surface Tension

In this study a continuum mathematical model is developed to describe the effect of biosurfactants on the surface tension. The model includes generalized reaction-advection-diffusion equations for the concentration of the substrate, biomass and the product (biosurfactant). The reaction (bioprocess) is modeled using classical Monod kinetics for the growth of the biomass and biosurfactant. In the reaction-advection-diffusion equation for the biomass the bacterial chemotaxis is taken into account using KellerSegel model [1]. The evolution of the concentration of biosurfactants on the surface is governed by advection-diffusion equation where the flux bulk-surface is given by Fick’s low. The dependence bulk-surface concentration of biosurfactants is given by adsorption isotherm. The dependence of the surface tension on the biosurfactant concentration is modeled by the corresponding equation of state. Different nonlinear adsorption isotherms, respectively equations of state, are considered. In the case of flat surface in the absence of fluid flow and initially uniformly distributed substrate and biomass the model can be considered as one-dimensional. In this case the evolution of the surface tension is given by a generalization of the Ward-Tordai integral equation. Numerical results for the 1D model are presented. The authors are partially supported by Grant DFNI-I02/9/12.12.2014 from the Bulgarian National Science Fund.

Numerical Simulation of Drop Coalescence in the Presence of Inter-Phase Mass Transfer

A numerical method for simulation of the deformation and drainage of an axisymmetric film between colliding drops in the presence of inter-phase solute transfer at small capillary and Reynolds numbers and small solute concentration variations is presented. The drops are considered to approach each other under a given interaction force. The hydrodynamic part of the mathematical model is based on the lubrication equations in the gap between the drops and the Stokes equations in the drops, coupled with velocity and stress boundary conditions at the interfaces. Both drop and film solute concentrations, related via mass flux balance across the interfaces, are governed by convection-diffusion equations. These equations for the solute concentration in the drops and the film are solved simultaneously by a semi-implicit finite difference method. Tests and comparisons are performed to show the accuracy and stability of the presented numerical method.

Contour-Integral Representation of Single and Double Layer Potentials for Axisymmetric Problems

Based on recently proposed non-singular contour-integral representations of single and double layer potentials for 3D surfaces, formulas in the axisymmetric case are derived. They express explicitly the singular layer potentials in terms of elliptic integrals. The presented expressions are non-singular, satisfy exactly very important conservation principles and directly take into account the multivaluedness of the double layer potential. The results are compared with another method for calculating the single and double layer potentials. The comparison demonstrates higher accuracy and better performance of the presented formulas.