Georgios C. Georgiou

Compressible viscous flow in slits with slip at the wall

We study the time‐dependent compressible flow of a Newtonian fluid in slits using an arbitrary nonlinear slip law relating the shear stress to the velocity at the wall. This slip law exhibits a maximum and a minimum and so does the flow curve. According to one‐dimensional stability analyses, the steady‐state solutions are unstable if the slope of the flow curve is negative. The two‐dimensional flow problem is solved using finite elements for the space discretization and a standard fully implicit scheme for the time discretization. When compressibility is taken into account and the volumetric flow rate at the inlet is in the unstable regime, we obtain self‐sustained oscillations of the pressure drop and of the mass flow rate at the exit, similar to those observed with the stick‐slip instability. The effects of compressibility and of the length of the slit on the amplitude and the frequency of the oscillations are also examined.

Hindcast of oil-spill pollution during the Lebanon crisis in the Eastern Mediterranean, July–August 2006

MOON (Mediterranean Operational Oceanography Network http://www.moon-oceanforecasting.eu) provides near-real-time information on oil-spill detection (ocean color and SAR) and predictions [ocean forecasts (MFS and CYCOFOS) and oil-spill predictions (MEDSLIK)]. We employ this system to study the Lebanese oil-pollution crisis in summer 2006 and thus to assist regional and local decision makers in Europe, regionally and locally. The MEDSLIK oil-spill predictions obtained using CYCOFOS high-resolution ocean fields are compared with those obtained using lower-resolution MFS hydrodynamics, and both are validated against satellite observations. The predicted beached oil distributions along the Lebanese and Syrian coasts are compared with in situ observations. The oil-spill predictions are able to simulate the northward movement of the oil spill, with the CYCOFOS predictions being in better agreement with satellite observations. Among the free MEDSLIK parameters tested in the sensitivity experiments, the drift factor appears to be the most relevant to improve the quality of the results.

Laminar jets of Bingham‐plastic liquids

The steady and transient behavior of jets generated by circular and slit nozzles are analyzed by the Galerkin finite‐element method with free‐surface parametrization and Newton iteration. A novel constitutive equation is used to approximate Bingham liquids that is valid uniformly in yielded and unyielded domains and which approximates the ideal Bingham model and the Newtonian liquid in its two limiting behaviors. At steady state the influence of yield stress on the die swell is equivalent to that of surface tension; that is, suppression of jet diameter at low Reynolds numbers and necking at high Reynolds number. The predictions at high Reynolds numbers agree with the asymptotic behavior at infinite Reynolds number of the jet far downstream. In the transient analysis, surface tension destabilizes round jets and increases the size of satellite drops. Yield stress was found to retard jet breakup times in addition to producing smaller satellites. Shear thinning was found to result in shorter collapse times th...

The time-dependent, compressible Poiseuille and extrudate-swell flows of a Carreau fluid with slip at the wall

Abstract We solve the time-dependent, compressible Poiseuille and extrudate-swell flows of a shear-thinning fluid that obeys the Carreau constitutive model, using finite elements in space and a fully-implicit scheme in time. Slip is assumed to occur along the die wall following a non-monotonic slip equation that relates the wall shear stress to the slip velocity and is based on experimental measurements with polyethylene melts. Thus, the resulting flow curve is also non-monotonic, and consists of two stable positive-slope branches and a linearly unstable negative-slope branch. The steady-state numerical results compare well with certain analytical solutions for Poiseuille flow. The time-dependent calculations at fixed volumetric flow rates demonstrate the existence of periodic solutions in the unstable regime, due to the combination of compressibility and slip. Self-sustained oscillations of the pressure-drop and of the mass-flow rate are obtained. In the extrudate region, high-frequency, small amplitude waves are generated on the free-surface, which also oscillates radially. The wavelength and the amplitude of the free-surface waves and the amplitude of the oscillations in the radial direction are reduced, as the Reynolds number is decreased and approaches the conditions of the experiments.

On the stability of the simple shear flow of a Johnson–Segalman fluid

We solve the time-dependent simple shear flow of a Johnson‐Segalman fluid with added Newtonian viscosity. We focus on the case where the steady-state shear stress:shear rate curve is not monotonic. We show that, in addition to the standard smooth linear solution for the velocity, there exists, in a certain range of the velocity of the moving plate, an uncountable infinity of steady-state solutions in which the velocity is piecewise linear, the shear stress is constant and the other stress components are characterized by jump discontinuities. The stability of the steady-state solutions is investigated numerically. In agreement with linear stability analysis, it is shown that steady-state solutions are unstable only if the slope of a linear velocity segment is in the negative-slope regime of the shear stress:shear rate curve. The time-dependent solutions are always bounded and converge to a stable steady state. The number of the discontinuity points and the final value of the shear stress depend on the initial perturbation. No regimes of self-sustained oscillations have been found.

The method of fundamental solutions for three-dimensional elastostatics problems

Abstract We consider the application of the method of fundamental solutions to isotropic elastostatics problems in three space dimensions. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity, which are expressed in terms of sources placed outside the domain of the problem under consideration. The final positions of the sources and the coefficients of the fundamental solutions are determined by enforcing the satisfaction of the boundary conditions in a least squares sense. The applicability of the method is demonstrated on two test problems. The numerical experiments indicate that accurate results can be obtained with relatively few degrees of freedom.

Time-dependent Compressible Extrudate-swell Problem With Slip At the Wall

We solve the time‐dependent compressible Newtonian extrudate‐swell problem with slip at the wall, in an attempt to simulate the stick‐slip extrusion instability. An arbitrary nonlinear slip model relating the shear stress to the velocity at the wall is employed, such that the flow curve consists of two stable branches separated by an unstable negative‐slope branch. Finite elements are used for the space discretization and a standard fully implicit scheme for the time discretization. When the volumetric flow rate at the inlet is in the unstable regime and compressibility is taken into account, self‐sustained periodic oscillations of the pressure drop and of the mass flow rate at the exit are observed and the extrudate surface becomes wavy, as is the case in stick‐slip instability. Results are presented for different values of the compressibility number. As compressibility is reduced, the frequency of the oscillations becomes higher, the amplitude of the pressure drop oscillations decreases, and the amplitu...

A SINGULAR FUNCTION BOUNDARY INTEGRAL METHOD FOR THE LAPLACE EQUATION

The authors present a new singular function boundary integral method for the numerical solution of problems with singularities which is based on approximation of the solution by the leading terms of the local asymptotic expansion. The essential boundary conditions are weakly enforced by means of appropriate Lagrange multipliers. The method is applied to a benchmark Laplace-equation problem, the Motz problem, giving extremely accurate estimates for the leading singular coefficients. The method converges exponentially with the number of singular functions and requires a low computational cost. Comparisons are made to the analytical solution and other numerical methods.

The time-dependent extrudate-swell problem of an Oldroyd-B fluid with slip along the wall

We demonstrate that viscoelasticity combined with nonlinear slip acts as a storage of elastic energy generating oscillations of the pressure drop similar to those observed experimentally in extrusion instabilities. We consider the time-dependent axisymmetric incompressible Poiseuille and extrudate-swell flows of an Oldroyd-B fluid. We assume that slip occurs along the wall of the die following a slip equation which relates the shear stress to the velocity at the wall and exhibits a maximum and a minimum. We first study the stability of the one-dimensional axisymmetric Poiseuille flow by means of a one-dimensional linear stability analysis and time-dependent calculations. The numerically predicted instability regimes agree well with the linear stability ones. The calculations reveal that periodic solutions are obtained when an unstable steady-state is perturbed and that the amplitude and the period of the oscillations are increasing functions of the Weissenberg number. We then continue to numerically solve the time-dependent two-dimensional axisymmetric Poiseuille and extrudate-swell flows using the elastic-viscous split stress method for the integration of the constitutive equation. Again, oscillations are observed in the unstable regime; consequently, the surface of the extrudate is wavy. However, the amplitude and the period of the pressure drop oscillations are considerably smaller than in the one-dimensional flow. The most important phenomenon revealed by our two-dimensional calculations is that the flow in the die is periodic in the axial direction

An efficient finite element method for treating singularities in Laplace's equation

We present a new finite element method for solving partial differential equations with singularities caused by abrupt changes in boundary conditions or sudden changes in boundary shape. Terms from the local solution supplement the ordinary basis functions in the finite element solution. All singular contributions reduce to boundary integrals after a double application of the divergence theorem to the Galerkin integrals, and the essential boundary conditions are weakly enforced using Lagrange multipliers. The proposed method eliminates the need for high-order integration, improves the overall accuracy, and yields very accurate estimates for the singular coefftcients. It also accelerates the convergence with regular mesh refinement and converges rapidly with the number of singular functions. Although here we solve the Laplace equation in two dimensions, the method is applicable to a more general class of problems.