Yannis Dimakopoulos

Transient displacement of a viscoplastic material by air in straight and suddenly constricted tubes

Abstract We examine the transient displacement of a viscoplastic material from straight or suddenly constricted cylindrical tubes of finite length. Our general goal is to develop accurate and efficient numerical methods for the fundamental study of processes in which a gas is displacing a liquid from prototype geometries under various operating conditions. Such processes can be part of the Gas Assisted Injection Molding (GAIM) or enhanced oil recovery. To this end, we use the mixed finite element method coupled with a quasi-elliptic mesh generation scheme in order to follow the very large deformations of the fluid volume. The displacing fluid is gas at high pressure, which forms a bubble of increasing length and a shape that depends on the fluid properties, the flow conditions, and the tube geometry. The cross-section of the bubble is always smaller than that of the tube due to adherence of fluid on the tube walls. The thickness of the remaining film depends on the same parameters and for most of its length it behaves as unyielded material. Unyielded material also arises in front of the bubble, around the axis of symmetry of the tube(s) and in the case of a constricted tube near the recirculation corner, but not around the entrance of the secondary tube. The rate of growth of the ‘tip splitting’ instability, that arises at relatively large values of the Reynolds number for Newtonian fluids in straight tubes, decreases as the Bingham number increases and, eventually, the instability disappears. The resistance provided by the constricted tube downstream makes the bubble move at a nearly constant velocity only when the Bingham number is not large. When the bubble approaches the constriction it becomes more pointed, but after entering it, the bubble reassumes its well-developed profile. Depending on parameter values, the bubble in the secondary tube may periodically split, thus forming a train of smaller bubbles directed towards the exit of the tube, a phenomenon for which experimental evidence exists.

On the elliptic mesh generation in domains containing multiple inclusions and undergoing large deformations

We present an improved method to generate a sequence of structured meshes even when the physical domain contains deforming inclusions. This method belongs to the class of Arbitrary Lagrangian-Eulerian (ALE) methods for solving moving boundary problems. Its tools are either (a) separate mappings of the domain boundaries and enforcing the node distribution on lines emanating from singular points or (b) domain decomposition and separate mappings of each subdomain using suitable coordinate systems. The latter is shown to be more versatile and general. In both cases a set of elliptic equations is used to generate the grid extending in this way the method advanced by Dimakopoulos and Tsamopoulos [Y. Dimakopoulos, J.A. Tsamopoulos, A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations, J. Comput. Phys. 192 (2003) 494-522]. We shall present examples where this earlier method and all other mesh generating methods which are based on a conformal mapping or solving a quasi-elliptic set of PDEs fail to produce an acceptable mesh and accurate solutions in such geometries. Furthermore, in contrast to other methods, appropriate boundary conditions and constraints such as, orthogonality of specific mesh lines and prespecified node distributions on them, can be easily implemented along a specific part of the domain or its boundary. Hence, no attractive terms at specific corners or singular points are needed. To increase the mesh resolution around the moving interfaces while keeping low the memory requirements and the computational time, a local mesh refinement technique has been incorporated as well. The method is demonstrated in two challenging examples where no remeshing is required in spite of the large domain deformations. In the first one, the transient growth of two bubbles embedded in a viscoelastic filament undergoing stretching in the axial direction is examined, while in the second one the linear and non-linear dynamics of two bubbles in a viscous medium are determined in an acoustic field. The large elasticity of the filament in the first case or the large inertia in the second case coupled with the externally induced large deformations of the liquid domain requires the accurate calculation which is achieved by the method we propose herein. The governing equations are solved using the finite element/Galerkin method with appropriate modifications to solve the hyperbolic constitutive equation of a viscoelastic fluid. These are coupled with an implicit Euler method for time integration or with Arnoldis algorithm for normal mode analysis.

Transient displacement of a Newtonian fluid by air in straight or suddenly constricted tubes

We study the displacement of a viscous fluid by highly pressurized air in a straight or a suddenly constricted cylindrical tube of finite length. In contrast to previous efforts, the transient situation is examined. A long, narrower than the tube and round-ended bubble is created during the process. This is sometimes called “fingering instability” and is often encountered in several applications, but we will focus on process parameters that are relevant to the gas-assisted injection molding. For our numerical simulations we have combined the mixed finite element method with an appropriate system of elliptic partial differential equations and boundary conditions, capable of generating a boundary-fitted finite element mesh. The bubble front and the thickness of the deposited film on the tube wall are affected by the properties of the fluid being displaced and the flow conditions. Specifically, in straight tubes, the bubble keeps accelerating due to the decreasing fluid mass ahead of it. Increasing the Reyno...

Injection of a viscoplastic material inside a tube or between two parallel disks: Conditions for wall detachment of the advancing front

The injection of a viscoplastic material, driven by a constant pressure drop, inside a pipe or between two parallel coaxial disks under creeping flow conditions is examined. The transient nature of both flow arrangements requires solving a time-dependent problem and fully accounting for the advancing liquid/air interface. Material viscoplasticity is described by the Papanastasiou constitutive equation. A quasi-elliptic grid generation scheme is employed for the construction of the mesh, combined with local mesh refinement near the material front and, periodically, full mesh reconstruction. All equations are solved using the mixed finite element/Galerkin formulation coupled with the implicit Euler method. For a viscoplastic fluid, the flow field changes qualitatively from that of a Newtonian fluid because the material gets detached from the walls. For small Bingham numbers, the contact line moves in the flow direction, so that initially the flow resembles that of a Newtonian fluid, but even in that case de...

Numerical simulation of multiple bubbles growing in a Newtonian liquid filament undergoing stretching

Our recent finite element-based study of the deformation of a single bubble in a Newtonian or viscoelastic filament undergoing stretching is extended here to the case of multiple bubbles simultaneously growing in the stretched medium. The filament, having initially the shape of a cylinder with uniform radius, is confined between two disks and is continuously stretched by pulling the upper disk along the filament axis with a constant velocity; the lower disk is assumed stationary. All bubbles are taken to lie along the axis of symmetry of the filament and undergo deformation and/or growth with the medium being stretched. The governing equations are solved by a finite element/Galerkin method coupled with an implicit Euler method for the time integration, using an adaptive time step. The problem of the multiple bubble-liquid interfaces is addressed by a robust mesh-generation scheme that solves a set of elliptic differential equations for the locations of the nodal points. The resulting numerical scheme is a...

Stress-gradient induced migration of polymers in corrugated channels

We study the flow of a dilute polymer solution in a wavy channel under steady-state flow conditions by employing the nonequilibrium thermodynamics two-fluid model [Mavrantzas and Beris, Phys. Rev. Lett. 69, 273–276 (1992)], allowing for the coupling between polymer concentration and polymer stresses. The resulting highly complex system of partial differential equations describing inhomogeneous transport phenomena in the fluid are solved with an efficient implementation of the mixed finite-element method. We present numerical results for polymer concentration, stress, velocity, and fluxes of polymer as a function of the nondimensional parameters of the problem (the Deborah number De, the Peclet number Pe, the Reynolds number Re, the ratio of the solvent viscosity to the total fluid viscosity β, and the constriction ratio of the channel width cr). We find that the constricted part of the wall is depleted of polymer, when the polymer diffusion length scale, expressed by the ratio of De/Pe, increases. The mig...

Hemodynamics in stenotic vessels of small diameter under steady state conditions: Effect of viscoelasticity and migration of red blood cells

BACKGROUND In microcirculation, the non-Newtonian behavior of blood and the complexity of the microvessel network are responsible for the high flow resistance and the large reduction of the blood pressure. Red blood cell aggregation along with inward radial migration are two significant mechanisms determining the former. Yet, their impact on hemodynamics in non-straight vessels is not well understood. OBJECTIVE In this study, the steady state blood flow in stenotic rigid vessels is examined, employing a sophisticated non-homogeneous constitutive law. The effect of red blood cells migration on the hydrodynamics is quantified and the constitutive models accuracy is evaluated. METHODS A numerical algorithm based on the two-dimensional mixed finite element method and the EVSS/SUPG technique for a stable discretization of the mass and momentum conservation equations in addition to the constitutive model is employed. RESULTS The numerical simulations show that a cell-depleted layer develops along the vessel wall with an almost constant thickness for slow flow conditions. This causes the reduction of the drag force and the increase of the pressure gradient as the constriction ratio decreases. CONCLUSIONS Viscoelastic effects in blood flow were found to be responsible for steeper decreases of tube and discharge hematocrits as decreasing function of constriction ratio.

Viscoplastic flow in an extrusion damper

Abstract Numerical simulations of the flow in an extrusion damper are performed using a finite volume method. The damper is assumed to consist of a shaft, with or without a spherical bulge, oscillating axially in a containing cylinder filled with a viscoplastic material of Bingham type. The response of the damper to a forced sinusoidal displacement is studied. In the bulgeless case the configuration is the annular analogue of the well-known lid-driven cavity problem, but with a sinusoidal rather than constant lid velocity. Navier slip is applied to the shaft surface in order to bound the reaction force to finite values. Starting from a base case, several problem parameters are varied in turn in order to study the effects of viscoplasticity, slip, damper geometry and oscillation frequency to the damper response. The results show that, compared to Newtonian flow, viscoplasticity causes the damper force to be less sensitive to the shaft velocity; this is often a desirable damper property. The bulge increases the required force on the damper mainly by generating a pressure difference across itself; the latter is larger the smaller the gap between the bulge and the casing is. At high yield stresses or slip coefficients the amount of energy dissipation that occurs due to sliding friction at the shaft-fluid interface is seen to increase significantly. At low frequencies the flow is in quasi steady state, dominated by viscoplastic forces, while at higher frequencies the fluid kinetic energy storage and release also come into the energy balance, introducing hysteresis effects.

Theoretical study of the flow in a fluid damper containing high viscosity silicone oil: Effects of shear-thinning and viscoelasticity

The flow inside a fluid damper where a piston reciprocates sinusoidally inside an outer casing containing high-viscosity silicone oil is simulated using a Finite Volume method, at various excitation frequencies. The oil is modelled by the Carreau-Yasuda (CY) and Phan-Thien \& Tanner (PTT) constitutive equations. Both models account for shear-thinning but only the PTT model accounts for elasticity. The CY and other generalised Newtonian models have been previously used in theoretical studies of fluid dampers, but the present study is the first to perform full two-dimensional (axisymmetric) simulations employing a viscoelastic constitutive equation. It is found that the CY and PTT predictions are similar when the excitation frequency is low, but at medium and higher frequencies the CY model fails to describe important phenomena that are predicted by the PTT model and observed in experimental studies found in the literature, such as the hysteresis of the force-displacement and force-velocity loops. Elastic effects are quantified by applying a decomposition of the damper force into elastic and viscous components, inspired from LAOS (Large Amplitude Oscillatory Shear) theory. The CY model also overestimates the damper force relative to the PTT, because it underpredicts the flow development length inside the piston-cylinder gap. It is thus concluded that (a) fluid elasticity must be accounted for and (b) theoretical approaches that rely on the assumption of one-dimensional flow in the piston-cylinder gap are of limited accuracy, even if they account for fluid viscoelasticity. The consequences of using lower-viscosity silicone oil are also briefly examined.

Discretization of three-dimensional free surface flows and moving boundary problems via elliptic grid methods based on variational principles

Abstract A new boundary-fitted technique to describe free surface and moving boundary problems is presented. We have extended the 2D elliptic grid generator developed by Dimakopoulos and Tsamopoulos (2003) [19] and further advanced by Chatzidai et al. (2009) [18] to 3D geometries. The set of equations arises from the fulfillment of the variational principles established by Brackbill and Saltzman (1982) [21] , and refined by Christodoulou and Scriven (1992) [22] . These account for both smoothness and orthogonality of the grid lines of tessellated physical domains. The elliptic-grid equations are accompanied by new boundary constraints and conditions which are based either on the equidistribution of the nodes on boundary surfaces or on the existing 2D quasi-elliptic grid methodologies. The capabilities of the proposed algorithm are first demonstrated in tests with analytically described complex surfaces. The sequence in which these tests are presented is chosen to help the reader build up experience on the best choice of the elliptic grid parameters. Subsequently, the mesh equations are coupled with the Navier–Stokes equations, in order to reveal the full potential of the proposed methodology in free surface flows. More specifically, the problem of gas assisted injection in ducts of circular and square cross-sections is examined, where the fluid domain experiences extreme deformations. Finally, the flow-mesh solver is used to calculate the equilibrium shapes of static menisci in capillary tubes.