Domain Perturbation Technique Research Articles

Shear flow over a surface containing a groove covered by an incompressible surfactant phase

We study shear-driven liquid flow over a planar surface with an embedded gas-filled groove, with the gas–liquid interface protruding slightly above or below the planar surface. The flow direction is along the groove, which is taken to be much longer than it is wide, and the gas–liquid interface is assumed to be covered by an incompressible surface fluid, representing a surfactant phase. Using the incompressibility condition for the surface fluid, the equations of motion and corresponding boundary conditions for the liquid phase are obtained by minimizing the dissipation rate. Assuming a moderate deformation of the interface, a domain perturbation technique with the maximal deformation as the small parameter is employed. The Stokes equation in the liquid phase under corresponding boundary conditions is solved to second order in the deformation using the Keldysh–Sedov formalism. The obtained analytical results are compared with numerical calculations of the same problem, allowing an assessment of the limits of validity of the expansion. While on a planar gas–liquid interface no flow is induced, a recirculating flow is observed on an interface protruding slightly above or below the planar surface. The study sheds light onto the mobility of curved gas–liquid interfaces in the presence of surfactants acting as an incompressible surface fluid.

Viscoelastic flow through a wavy curved channel

Time-dependent flow of a viscoelastic material through a wavy curved channel is studied. The flow traversing the wavy curved domain is modelled in the curvilinear coordinates. The viscoelastic fluid is described by the fluid relaxation and retardation times, and their variations are considered in relation to the flow characteristic time scale. The wavy walls of the curved channels are in sinusoidal form, with arbitrary phase difference. Using domain perturbation technique, the analytical solution of the model is obtained, including the velocity and the volumetric flow rate. The analytical solution is explained as a function of the flow domain structure, the viscoelastic fluid properties, and the flow generating force. Flow enhancement due to the flow domain is discussed; we have shown that the flow augmentation depends on the properties of the viscoelastic fluid. Furthermore, comparisons have been made with Newtonian fluid flow, and the prominent variations in the flow behaviours have been reported.

Dilute dispersion of compound particles: deformation dynamics and rheology

Compound particles are a class of composite systems in which solid particles encapsulated in a fluid droplet are suspended in another fluid. They are encountered in various natural and biological processes, for e.g. nucleated cells, hydrogels, microcapsules etc. Generation and transportation of such multiphase structures in microfluidic devices is associated with several challenges because of the poor understanding of their structural stability in a background flow and the rheological characteristics of their dispersions. Hence, in this work, we analyse the flow in and around a concentric compound particle and investigate the deformation dynamics of the confining drop and its stability against breakup in imposed linear flows. In the inertia-less limit (Reynolds number, $Re \ll 1$ ) and assuming that the surface tension force dominates the viscous forces (low capillary number, $Ca$ , limit), we obtain analytical expressions for the velocity and pressure fields up to ${O}(Ca)$ for a compound particle subjected to a linear flow using a domain perturbation technique. Simultaneously, we determine the deformed shape of the confining drop correct up to ${O}(Ca^2)$ , facilitating the following. (i) Since ${O}(Ca^2)$ calculations account for the rotation of the anisotropically deformed interface, the reorientation dynamics of the deformed compound particles is determined. (ii) Calculations involving the ${O}(Ca^2)$ shape of the confining interface are found to be important for compound particles as ${O}(Ca)$ calculations make qualitatively different predictions in generalised extensional flows. (iii) An ${O}(Ca)$ constitutive equation for the volume-averaged stress for a dilute dispersion of compound particles was developed to study both shear and extensional rheology in a unified framework. Our analysis shows that the presence of an encapsulated particle always enhances all the measured rheological quantities such as the effective shear viscosity, extensional viscosity and normal stress differences. (iv) Moreover, linear viscoelastic behaviour of a dilute dispersion of compound particles is characterised in terms of complex modulus by subjecting the dilute dispersion to a small-amplitude oscillatory shear (SAOS) flow. (v) Various expressions pertaining to a suspension of particles, drops, and particles coated with a fluid film are also derived as limiting cases of compound particles.

Improved asymptotic predictions for the effective slip over a corrugated topography

• Significantly improved asymptotic predictions on effective slip in comparison to literature. • Systematic study on convergence radii of asymptotic series. • Novel use of series improvement and acceleration techniques. • Novel use of spectral analysis and symbolic computation. Accurate analytical prediction of the effective slip boundary condition in shear-driven Stokes flows directed longitudinally and transversely to a one-dimensional sinusoidal no-slip topography is explored. First, the domain perturbation technique is extended through spectral analysis and symbolic computations to obtain polynomial approximations (Taylor polynomials) of arbitrary specifiable order for the effective slip length. However, when assessed for numerical accuracy against fully-resolved numerical simulations using the finite-element-method, higher order Taylor polynomials give progressively inferior predictions in comparison to lower-order ones, unless the product of amplitude and wave-number is restricted below unity. From Domb–Sykes plots, the reason for the poor accuracy of higher order Taylor polynomials is assessed to be the finite convergence radii, approximately equaling unity, of the asymptotic power series for both longitudinal and transverse flows. For either of the flows, application of Euler transformation to the expansion parameter provide polynomial-form approximations that are accurate for amplitude values exceeding the convergence radius. The slow convergence of the Euler-transformed series can be remedied through Shanks transformation, at the cost of losing the convenience of closed forms. Finally, Padé approximants are shown to provide even more accurate but still closed-form alternatives to polynomials that work accurately at amplitudes much exceeding the above-identified convergence radii.

Perturbation Sensitivity Analysis and Dynamic Topology Optimization for Heat Conduction Structure

Topology optimization is a fundamental problem for steady heat conduction. Due to the unidirectional treatment for the intermediate density elements in the traditional topology optimization, we propose a Bi-Directional Interpolation Model (BDIM) that intermediate density elements are judged by a threshold, and establish optimization model. Besides, considering the generation of new hole boundary in the process of heat conduction enhancement topology optimization, we study the perturbation sensitivity analysis with respect to new hole, and present an adaptive dynamic boundary method for the new boundary in the optimization process. Furthermore, based on the domain perturbation technique and Lebesgue differential theory, the topological derivative formulas with different objective functions subjected to three kinds of boundary conditions are derived for the control system of Poisson equation. Finally, a number of numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method for designing the heat conduction structure.

Vertically stratified two-phase flow in a curved channel: Insights from a domain perturbation analysis

In this work, we investigate the fully developed flow field of two vertically stratified fluids (one phase flowing above the other) in a curved channel of rectangular cross section. The domain perturbation technique is applied to obtain an analytical solution in the asymptotic limit of low Reynolds numbers and small curvature ratios (the ratio of the width of the channel to its radius of curvature). The accuracy of this solution is verified by comparison with numerical simulations of the nonlinear equations. The flow is characterized by helical vortices within each fluid, which are driven by centrifugal forces. The number of vortices and their direction of circulation varies with the parameters of the system (the volume fraction, viscosity ratio, and Reynolds numbers). We identify nine distinct flow patterns and organize the parameter space into corresponding flow regimes. We show that the fully developed interface between the fluids is not horizontal, in general, but is deformed by normal stresses associated with the circulatory flow. The results are especially significant for flows in microchannels, where the Reynolds numbers are small. The mathematical results in this paper include an analytical solution to two coupled biharmonic partial differential equations; these equations arise in two-phase, two-dimensional Stokes flows.

Effects of interface deformation on flow through microtubes containing superhydrophobic surfaces with longitudinal ribs and grooves

This paper presents analytical and numerical results for Poiseuille flow through microtubes patterned with superhydrophobic surfaces which consist of alternative ribs and grooves aligned longitudinally with the flow direction. The superhydrophobic surface prevents the flowing liquid from penetrating the grooves and the liquid–gas interface experiences deformation as a consequence of a pressure difference across the interface. Employing a domain perturbation technique, the effects of a small interface deformation on the effective slip behavior are analytically quantified. For large interface deformations, numerical studies are performed to predict the effects of interface protrusion on the effective slip behavior of the superhydrophobic microtube. Comparisons are made between the effective slip behavior for tube and channel flows patterned with superhydrophobic surfaces containing alternating longitudinal ribs and grooves.

Electrohydrodynamics of a liquid drop in confined domains

The steady-state electrohydrodynamics of a leaky dielectric drop in confined domains is investigated analytically. The governing electrohydrodynamic equations are solved for Newtonian and immiscible fluids in the framework of leaky dielectric theory and for the creeping flow regime. The domain confinement strengthens or weakens the electric field, depending on R>1 or R<1, respectively, where R=σi/σo is the ratio of electric conductivity of the drop to that of the surrounding fluid. Similarly, the flow intensity decreases for R<1, but it remains unchanged or increases for R>1, depending on the interplay of electric and hydrodynamic effects. An expression for the drop deformation for small distortion from the spherical shape is found using the domain perturbation technique. It is shown that below a threshold domain size the confinement effect will lead to the reversal of the tendency of the net normal hydrodynamic stress in deforming the drop to an oblate or a prolate shape, and that below a critical domain size the necessary condition for having an oblate drop will be opposite to the classical one for an unbounded domain.

Electrohydrodynamics of a liquid column under a transverse electric field in confined domains

The steady state electrohydrodynamics of a leaky dielectric fluid column in confined domains is investigated. The governing electrohydrodynamic equations are solved, in a plane normal to the column axis, for Newtonian and immiscible fluids in the framework of leaky-dielectric theory and for creeping flow regime. The domain confinement strengthens or weakens the electric field, depending on R>1 or R<1, respectively, R=σi/σo being the ratio of electric conductivity of the column to that of the surrounding fluid. Similarly, the flow intensity decreases for R<1, but it remains unchanged or increases for R>1, depending on the interplay of electric and hydrodynamic effects. An expression for the interface deformation for small distortion from the circular shape is found using the domain perturbation technique. It is shown that below a threshold domain size the confinement effect will lead to the reversal of the tendency of the net normal hydrodynamic stress in deforming the interface to an oblate or a prolate shape, and that below a critical domain size the necessary condition for having an oblate column will transfer from R<S to R>S, where S=ϵi/ϵo is the permittivity ratio.

Creeping flow of a second grade fluid in a corner

A variant of Taylor’s (1962) [23] scraper problem, in which, the lower plate rotates is considered. The non-linear partial differential equations governing the flow of a second grade fluid are modeled and solved by using the domain perturbation technique considering the angular velocity of the rotating plate as a small parameter. Also the rheology of the second grade fluid is examined by depicting the profiles of the velocity, stream function, pressure and stress fields.

Time dependent finite element analysis of the linear stability of viscoelastic flows with interfaces

In this paper we present a new time marching scheme for the time dependent simulation of viscoelastic flows governed by constitutive equations of differential form. Based on the ideas of Carvalho and Scriven [J. Comput. Phys. 151 (1999) 534], a domain perturbation technique is introduced that can be applied to viscoelastic flows with fluid/fluid interfaces or free surfaces. This work mainly focuses on the development and, consequently, benchmarking of finite element algorithms (FEM) that can efficiently handle the stability problems of complex viscoelastic flows. Since spurious or non-physical solutions are easily generated for this type of analysis using finite element techniques, both the new time stepping scheme and the domain perturbation technique are benchmarked in simple shear flows of upper convected Maxwell (UCM) fluids. Both single and two layer flows are considered for which the dominating mode and associated growth rate of a perturbation are solutions of the one-dimensional generalized eigenvalue problem (GEVP). We show that both the growth rate and the most dangerous eigenmode of the simple shear flows can be accurately captured by our transient algorithm.

Comparison of Theoretical and Experimental Characteristics of Oscillating Bubbles

Mass transfer in typical areation devices such as sieve plate reactors and bubble columns takes place predominantly within the oscillating bubble regime. Mass-transfer rates from oscillating bubbles can be several times larger than mass-transfer rates from rigid bubbles. The hydrodynamics of oscillating bubbles has been described by Tsamopoulos and Brown (J. Fluid Mech. 1983, 127, 519-537) and later by Feng (SIAM J. Appl. Math. 1992, 52 (1), 1-14), using the domain perturbation technique first developed by Joseph (Arch. Ration. Mech. Anal. 1973, 51, 295-303). A high-speed video imaging system was used to determine the shapes of oscillating bubbles for a wide range of fluids and operating conditions. Air bubbles introduced through an orifice were recorded while rising inside a liquid. When the liquid was water, a hydrogen bubble technique was used to determine pathlines around the rising bubble. The experimental data on shapes and pathlines were compared with bubble shapes and pathlines created using theoretical models.

Mass transfer from oscillating bubbles in bioreactors

In bioreactors, the purpose of aeration is to transfer oxygen from air bubbles into the liquid phase where the biological reactions occur. In some cases, aeration is also used as a mixing tool. In all aeration devices there is a continuous liquid phase and, generally, a dispersed gas phase in the form of bubbles. The most common aeration devices are stirred tank reactors, bubble columns and sieve plate reactor-towers. Bubble shape, bubble volume, and associated liquid flow patterns are key aspects of bubble dynamics in sieve plates. The most prevalent bubble shapes are spherical, oscillating (wobbling), ellipsoidal, spherical-cap, and skirted. Bubble volume and bubble shapes determine the surface to volume ratios, a very important parameter in determining overall mass transfer rates. Mass transfer in sieve plate reactors takes place, predominantly, within the oscillating bubble regime. Mass transfer rates from oscillating bubbles can be orders of magnitude larger than mass transfer rates from spherical bubbles. A fundamental correlation for mass transfer from single, oscillating bubbles was developed based on a solution of the mass transfer equations following the domain perturbation technique first outlined by Joseph (1973) and the hydrodynamics results of Tsamopoulos and Brown (1983). The correlation derived here entirely from hydrodynamics and mass transfer concepts, introduces the effect of interfacial tension in bubble mass transfer from fundamental concepts, has no adjustable parameters, and agrees very well with experimental data.

Electrically charged conducting drops revisited

Since its publication in 1882, Rayleigh’s work on electrically charged conducting drops has been widely quoted, but its rigorous derivation has not been given in the literature. By means of the domain perturbation technique, this work presents a rigorous derivation of Rayleigh’s results, following his approach with Lagrange’s equation. With the systematic procedure, it becomes explicit that the first-order surface deformations result in a deviation of the drop surface potential of only second-order significance. Besides providing mathematical details, this work also reveals an apparent error in Rayleigh’s original result for two-dimensional (cylindrical) drops.

Dynamics of a conducting drop in a time-periodic electric field

The nonlinear dynamical behaviour of a conducting drop in a time-periodic electric field is studied. Taylor's (1964) theory on the equilibrium shape is extended to derive a dynamical equation in the form of an ordinary differential equation for a conducting drop in an arbitrary time-dependent, uniform electric field based on a spheroidal approximation for the drop shape and the weak viscosity effect. The dynamics is then investigated via the classical two-timing analysis and the Poincaré map analysis of the resulting dynamical equation. The analysis reveals that in the neighbourhood of a stable steady solution, an time-dependent change of drop shape can be obtained from an O(ε) resonant forcing. It is also shown that the probability of drop breakup via chaotic oscillation can be maximized by choosing an optimal frequency for a fixed forcing amplitude. As a preliminary analysis, the effect of weak viscosity on the oscillation frequency modification in a steady electric field is also studied by using the domain perturbation technique. Differently from other methods based on the theory of viscous dissipation, the viscous pressure correction is directly obtained from a consideration of the perturbed velocity field due to weak viscosity.

Free Convection in a Trench with Radiative Wall Conditions

AbstractFree convection is considered in a rectangular trench with an assigned temperature distribution at the bottom and radiative vertical walls. Following a domain perturbation technique, the problem reduces to the Stokes' equation. After proving the existence and uniqueness of the solution, from a functional point of view, the stream function is expressed in closed form via a bi‐harmonic series and the shape of the free surface is also given. An application follows in order to show the convergence characteristics of the solution and the flow structure.