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Efficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen‐Loeve expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss‐Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method.

The linear stability analysis of Rivlin–Ericksen fluids of second order is investigated for boundary layer flows, where a semi-infinite wedge is placed symmetrically with respect to the flow direction. Second order fluids belong to a larger family of fluids called order fluids, which is one of the first classes proposed to model departures from Newtonian behavior. Second order fluids can model non-zero normal stress differences, which is an essential feature of viscoelastic fluids. The linear stability properties are studied for both signs of the elasticity number K, which characterizes the non-Newtonian response of the fluid. Stabilization is observed for the temporal and spatial evolution of two-dimensional disturbances when K > 0 in terms of increase of critical Reynolds numbers and reduction of growth rates, whereas the flow is less stable when K < 0. By extending the analysis to three-dimensional disturbances, we show that a positive elasticity number K destabilizes streamwise independent waves, while the opposite happens for K < 0. We show that, as for Newtonian fluids, the non-modal amplification of streamwise independent disturbances is the most dangerous mechanism for transient energy growth, which is enhanced when K > 0 and diminished when K < 0.

This paper is concerned with the development of efficient iterative methods for solving the linear system of equations arising from stochastic FEMs for single-phase fluid flow in porous media. It is assumed that the conductivity coefficient varies randomly in space according to some given correlation function and is approximated using a truncated Karhunen–Loeve expansion. Distinct discretizations of the deterministic and stochastic spaces are required for implementations of the stochastic FEM. In this paper, the deterministic space is discretized using classical finite elements and the stochastic space using a polynomial chaos expansion. The highly structured linear systems which result from this discretization mean that Krylov subspace iterative solvers are extremely effective. The performance of a range of preconditioned iterative methods is investigated and evaluated in terms of robustness with respect to mesh size and variability of the conductivity coefficient. An efficient symmetric block Gauss–Seidel preconditioner is proposed for problems in which the conductivity coefficient has a large standard deviation.The companion paper, herein, referred to as Part 2, considers the situation in which Gaussian random fields are transformed into lognormal ones by projecting the truncated Karhunen–Loeve expansion onto a polynomial chaos basis. This results in a stochastic nonlinear problem because the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables.

The benchmark problem of flow of a viscoelastic fluid around a confined cylinder is considered. The governing equations are discretised using spectral/hp element methods. These allow the spatial and temporal variations in the solution that are characteristic of viscoelastic flows, to be resolved accurately and efficiently. A decoupled approach is employed in which the conservation equations are solved for velocity and pressure and the constitutive equation (Oldroyd-B and Giesekus) are solved for the polymeric component of the extra-stress tensor. The computations are stabilized using the DEVSS-G/DG formulation of the problem. Excellent agreement with the literature is achieved for the drag coefficient in the case of an Oldroyd-B fluid. Causes for the breakdown in numerical convergence with mesh refinement beyond some critical value of the Weissenberg number are explored. The high resolution property of spectral/hp approximations has enabled an instability that develops in the shear layer on the cylinder and is convected downstream to be identified. The onset of this instability is shown to occur at the critical value of the Weissenberg number predicted by the theory of Dou and Phan-Thien [9]. The influence of the Reynolds number and, for the Giesekus model, the mobility parameter on the drag coefficient is also investigated and discussed.

When a cavity forms near a solid boundary a liquid jet can form directed towards the boundary, causing the generation of high pressures at the wall (potentially causing damage) and the formation of a toroidal bubble. In this paper several recent developments in the boundary element modelling of the dynamics of cavitation bubbles in viscoelastic fluids are presented. The standard formulation of the boundary element method (BEM) is in terms of a boundary integral equation with a singular kernel. A reformulation of the BEM in terms of a non-singular kernel is shown to provide enhanced stability. In situations when a liquid jet forms and impacts the far side of the bubble there is a transition to a toroidal form. This topological singularity in bubble geometry is modelled by placing a vortex ring inside the bubble to account for the circulation in the fluid and the discontinuity in potential following jet impact. The bubble dynamics are dependent on the initial stand-off distance from the boundary as well as the viscous and elastic properties of the fluid. It is shown that, while the viscosity of the fluid inhibits jet formation, the dynamics are particularly dependent on the relative strength of viscous, elastic and inertial forces. In particular, if the Deborah number is large enough elastic effects effectively negate fluid viscosity and behaviour similar to the inviscid case is recovered in terms of liquid jet formation.

This paper is concerned with the development of a high-order numerical scheme for two-phase viscoelastic flows. In the companion paper, herein referred to as Part 1, the scheme is applied to the modelling of two-phase Newtonian flows. The particular problem of the collapse of a 2D bubble in the vicinity of a rigid boundary is considered. Attention is given to the construction of the most general form of the compressible Oldroyd B model that is consistent with the compressible Newtonian and upper-convected Maxwell models in the appropriate limits. The governing equations are discretized using the spectral element method, and the two phases are modelled using a marker particle method. A comprehensive set of results is presented for the problem of bubble collapse near a rigid wall, and qualitative agreement is obtained with other numerical studies and experimental observations. Viscoelastic effects that are predicted include increased bubble oscillation with increasing Weissenberg number and considerable bubble deformation and cusping near the wall. Most importantly, it has been shown that viscoelasticity has the ability to prevent jet formation and therefore is likely to have a mitigating effect on cavitation damage.

Thermodynamical considerations have largely been avoided in the modelling of complex fluids by invoking the assumption of incompressibility. This approximation allows pressure to be defined as a Lagrange multiplier, and therefore its natural connection with other thermodynamic variables such as density and temperature is irretrievably lost. Relaxing this condition to allow more realistic modelling involves much more than prescribing an equation of state. Even for a simple isothermal viscoelastic model, as explored in this paper, the transition to a compressible model is non-trivial. This paper shows that pressure enters the governing equations in a non-intuitive way. Furthermore, a fluid volume element, which is no longer constant, radically changes the way the basic element of the constitutive equations is viewed—stress is no longer the fundamental constitutive link between the momentum equations and velocity. The importance of geometry in fluid modelling is emphasised through the use of the Lie derivative, which is of a more fundamental character than the “upper” and “lower” convected derivatives prevalent in the literature and which are found to be almost redundant for a compressible fluid. There is now a strong body of non-equilibrium thermodynamics theory for flowing systems, which proves indispensible for this development. These fundamental principles are described herein using methodology and examples, that are sometimes conflicting, from the literature. The main conflict arises from the relationship between thermodynamic pressure and the trace of Cauchy stress, where the current preferred choice is (up to a constant) to set them equal—this is shown to be incorrect. Other issues such as the dependence of viscosity on density, bulk viscosity, integral modelling, the principle of objectivity and convected derivatives, are also clarified and resolved.

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius

The numerical solution of the one-dimensional Fokker-Planck equation for describing the evolution of the configuration probability density function associated with kinetic theory models in polymer dynamics is presented. The finitely extensible non-linear elastic (FENE) model is considered and the spectral element discretisation is applied using an adaptive reduced basis technique. This technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection. The reduced basis method is extended to the high-dimensional Fokker-Planck equation, using the d-dumbbell FENE model, by consideration of a high-dimensional singular value decomposition. Some numerical results are presented to demonstrate the efficiency of the numerical scheme.

This paper presents a modification to the numerical treatment of integral constitutive equations initiated by Peters et al. [E.A.J.F. Peters, M.A. Hulsen, B.H.A.A. van den Brule, Instationary Eulerian viscoelastic flow simulations using time separable Rivlin–Sawyers constitutive equations. J. Non-Newtonian Fluid Mech., 89 (2000) 209–228] that enables computational stability to be achieved for higher Weissenberg number for steady flows. Increased efficiency is also achieved so that computational times are of the same order (2 or 3 times) as equivalent differential models and of similar accuracy. The memory cost for storage of the deformation fields is approximately 300 times that of the stress tensor for each grid point to yield approximations that lie within 0.01% of predictions of equivalent differential models. The good agreement between equivalent integral and differential models found here suggests that more general integral models with no differential equivalent may be used reliably for steady flow simulations. A modification of the deformation fields method that avoids the introduction of a cut-off time and which is applicable to transient flows is also presented.

The particle deficiency problem in the presence of a rigid wall for smoothed particle hydrodynamics (SPH) is considered. The problem arises from insufficient information being available to perform accurate interpolation of data at particles located nearer to the boundary than the support of the interpolation kernel. The standard method for overcoming this problem is based on the introduction of image particles to populate the deficient regions and the use of linear extrapolation to determine the velocity of these image particles from that of fluid particles. A consistent treatment of boundary conditions, utilising the momentum equation to obtain approximations to the velocity of image particles, is described. The method ensures second order approximation of the boundary conditions. It is validated for Poiseuille and Couette flow, for which analytical series solutions exist and shows second order convergence under certain conditions.

We present stable and accurate spectral element methods for predicting the steady-state flow of branched polymer melts past a confined cylinder. The fluid is modelled using a modification of the pom-pom model known as the single eXtended Pom-Pom (XPP) model, where we have included a multi-mode model of a commercial low-density polyethylene. We have analyzed the XPP model and found interesting multiple solutions for certain choices of the parameters which indicate possible problems with the model. The operator-integration-factor-splitting technique is used to discretize the governing equations in time, while the spectral element method is used in space. An iterative solution algorithm that decouples the computation of velocity and pressure from that of stress is used to solve the discrete equations. Appropriate preconditioners are developed for the efficient solution of these problems. Local upwinding factors are used to stabilize the computations. Numerical results are presented demonstrating the performance of the algorithm and the predictions of the model. The influence of the model parameters on the solution is described and, in particular, the dependence of the drag on the cylinder as function of the Weissenberg number.

Compressibility plays a significant role in the load-bearing capacity of a journal bearing. This paper offers more realistic modelling of the lubricant than presented in an earlier paper (Int. J. Numer. Meth. Fluids 2007; 55(11):1091–1120) by including variable sound speed, piezoviscosity and both temperature and shear thinning. The load-bearing capacity of the journal bearing is sensitive to all of these attributes of the model, but piezoviscosity is found to be the most significant. The equations of motion are adapted to a moving frame to explore the stability of the journal in a more dynamic setting and it is found that a free journal using this model will spiral outward exhibiting half speed whirl. The model is discretized in 2D, semi-implicitly in time and using the spectral element method in space. Numerical results are presented that highlight the contributions of the different elements in the model to journal stability.

The influence of viscoelasticity on the performance of statically and dynamically loaded journal bearings is considered. The lubricant in the system is modelled using either the Oldroyd-B or linear PTT models. Significant viscoelastic effects are presented for both moderate and narrow gap journal bearing configurations. The dynamical behaviour of the journal bearing system is shown to be dependent on the fluid model, the relaxation time and also the gap size.

The problem of droplet deformation and break-up is considered. A hybrid Eulerian-Lagrangian method is used in which the velocity and pressure are discretized on a fixed mesh and Lagrangian particles are used to implicitly track the interface between the two phases. The Navier-Stokes equations are solved using an approximate Godunov projection method, collectively called the Godunov marker-particle projection scheme. The results show good qualitative agreement with previous research as well as demonstrating the efficacy of the method.

The extended pom-pom (XPP) model was first introduced to eradicate perceived deficiencies of the original pom-pom model such as the lack of a second normal stress difference and a discontinuity in the derivative of the extensional viscosity. However, in this paper it is shown that the XPP model itself possesses some disconcerting attributes. In simple steady shear and uniaxial extensional flow, multiple solutions are found. Furthermore, the extended pom-pom model can predict positive as well as negative values for the second normal stress difference. Experimentally, the second normal stress difference is negative for polymer melts.

A compressible viscous isothermal model is presented for studying journal-bearing lubrication. The viscosity in the model thickens with increasing density. The governing equations are written in terms of velocity, the natural logarithm of the density and the kinematic extra-stress tensor. A semi-Lagrangian treatment of the material derivatives is combined with a spectral element discretization in space. The roles of the speed of sound and the eccentricity ratio on the load-bearing capacity of the journal bearing are investigated. Compressibility is shown to enhance the load-bearing capacity and this effect is amplified as the eccentricity ratio approaches unity. It is shown that for speeds of sound in the region of those of multigrade oils, the dominant component of the force on the journal acts along the line joining the centres of the bearing and journal and in the direction away from the narrow gap.

Spectral element methods are developed for simulating viscoelastic flows based on a micro–macro modelling approach in which the polymer dynamics is described by the evolution of an ensemble of Brownian configuration fields. The application of the technique to the start-up of Couette flow and flow between eccentrically rotating cylinders are described. Comparisons are made between simulations based on the Oldroyd B constitutive model and those based on Brownian configuration fields using Hookean dumbbells. Comparisons are also made between simulations based on different force laws. The performance of the method is discussed.

This paper is concerned primarily with the numerical prediction of the pressure field associated with the flow of an Oldroyd-B fluid through 4:1 contractions, 1:4 expansions and combined 4:1:4 contraction/expansions. Particular interest lies in the effect of the ratio of solvent to total viscosity parameter on the profile of the pressure gradient and on the entry and exit development lengths. Although most of the results refer to planar (2D) flows, some results are presented to illustrate how the 3D case convergences towards two-dimensional flow as the relevant aspect ratio of the geometries becomes large. Some results are presented for certain aspects related to the flow kinematics, in order to illustrate the link between the pressure field and the resulting flow field.

In this article we propose and analyze an a posteriori error estimator for a three-field model of a generalized Stokes problem. The components of the a posteriori error estimator are defined via a non-linear projection of the residues of the variational equations. Both upper and lower bounds for the approximation error are derived in terms of the components of the a posteriori error estimator. The non-linear projections do not need to be explicitly computed to construct the a posteriori error estimates.

This paper considers the spectral element approximation of the Stokes problem and the conditioning of the resulting discrete problem. The well-posedness of the variational formulation of the Stokes problem and, in particular, the uniqueness of the pressure has been demonstrated when the subspace of square-integrable functions having vanishing mean is chosen for the pressure space. In the discrete setting a conforming subspace is chosen for the discrete pressure space and the analysis of the discrete problem is based on this choice. However, this space is not used in practical implementations of finite or spectral element methods for the Stokes problem. In this paper it is shown how the zero volume condition on the pressure may be accommodated within the trial space by modifying the continuity equation in a consistent manner. The dependence of the condition number of the Uzawa operator and the accuracy of the spectral element approximation on the weighting factor is investigated.

Stable and accurate spectral element methods for predicting the flow of branched polymer melts past a confined cylinder are presented. The fluid is modelled using a modification of the pom–pom model known as the extended pom–pom (XPP) model. Steady and transient flows are considered in this paper. The operator integration factor splitting technique is used to discretize the governing equations in time, while the spectral element method is used in space. An iterative solution algorithm that decouples the computation of velocity and pressure from that of stress is used to solve the discrete equations. Appropriate preconditioners are developed for the efficient solution of these problems. Local upwinding factors are used to stabilize the computations. Numerical results are presented demonstrating the performance of the algorithm and the predictions of the model. The influence of the model parameters on the solution is described and, in particular, the dependence of the drag on the cylinder as function of the Weissenberg number.

We are concerned with the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We take our earlier work for transient start-up flow in a channel and extend this beyond Oldroyd-B modelling to consider a different fluid model of the pom-pom class. This includes Single Extended form of the pom-pom model (SXPP), comparing the results of two different finite volume schemes. The numerical techniques employed are time-stepping algorithms, one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum/continuity equations by a fractional-staged Taylor–Galerkin/pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. This draws upon fluctuation distribution schemes (upwinding), different combinations of ‘flux’ and ‘median-dual-cell’ spatial discretisations and time-term treatments. Here, unstructured and structured meshes may be used, based largely on triangular grids. A comparison of the two finite volume approaches will be presented, concentrating upon the new features posed by the pom-pom class of models.

This paper concerns the modelling of dynamically loaded journal bearing systems using a moving spectral element method. The moving grid method employed in this paper is the arbitrary Lagrangian-Eulerian (ALE) method. The ALE methodology is compared with a quasi-Eulerian approach in the context of dynamically loaded journal bearings and the advantages of adopting the ALE formulation are highlighted. A comparison with the predictions of lubrication theory is also presented and the limitations of the lubrication approximation are demonstrated when inertial effects are significant. A comprehensive set of results is presented illustrating the salient features of the spectral element mesh generation schemes described in the paper and the way in which these impinge on the efficiency of the iterative solution of the discrete equations at each time step. Copyright © 2005 John Wiley & Sons, Ltd.

The paper is concerned with the numerical prediction of the flow of polymer melts using a pom-pom model [J. Rheol. 42 (1998) 81]. The pom-pom model is a coarse-grained molecular model that was developed for describing branched polymers. A description of the configuration distribution is given in terms of the orientation and stretch of the pom-pom molecule. The variant of the pom-pom model used in this paper is the extended pom-pom model [J. Rheol. 45 (4) (2001) 823]. The model has been rewritten to make it suitable for implementation in an existing spectral element algorithm, which solves the constitutive equation together with the momentum and continuity equations The temporal discretization of the convective terms in the momentum equation and the constitutive equation is performed using a second-order time-splitting technique. Spatial discretization is performed using spectral element techniques. The system of discretized equations is solved using a preconditioned conjugate gradient method, and details of the preconditioners are given. Results are presented for the start-up of Poiseuille flow in a planar channel, and flow past a confined cylinder. The influence of the model parameters on the solution is described.

In this paper, the differences in the development of vortex structure with increasing elasticity between creeping and inertial flows in both planar and axisymmetric contraction configurations are studied for an Oldroyd B fluid. The numerical calculations are performed using a semi-Lagrangian finite volume method described in a paper by Phillips and Williams [J. Non-Newtonian Fluid Mech. 87 (1999) 214]. This method combines the advantages of fixed grids inherent in Eulerian methods with modifications to treat the convective terms using particle tracking methods. The method alleviates some of the difficulties encountered when discretizing differential constitutive equations of hyperbolic type for viscoelastic models. The semi-Lagrangian technique is applied to the convective terms in the momentum and constitutive equations. It is a natural way of treating these terms without resorting to upwinding techniques.

In dynamically loaded journal bearings it is important to predict and assess the performance of lubricants within the bearing with respect to friction, wear and load bearing capacity over a wide range of operating conditions. In this paper, a mathematical model is described which allows for a systematic study of the effects of shear-thinning, pressure-thickening and temperature-thinning viscosity on the dynamics and stability of a journal bearing. It is also shown that the thermal conductivity of the bearing can have a major effect on the minimum oil film thickness attained in the bearing.

A semi-Lagrangian finite volume scheme for solving viscoelastic flow problems is presented. A staggered grid arrangement is used in which the dependent variables are located at different mesh points in the computational domain. The convection terms in the momentum and constitutive equations are treated using a semi-Lagrangian approach in which particles on a regular grid are traced backwards over a single time-step. The method is applied to the 4 : 1 planar contraction problem for an Oldroyd B fluid for both creeping and inertial flow conditions. The development of vortex behaviour with increasing values of We is analyzed.

The effects of non-Newtonian lubricants on the dynamics of a 3D journal bearing are investigated using a moving spectral element method. Comparisons are made with the findings reported for the 2D case. The variation of L/D, the ratio of the length of the bearing to its diameter, is shown to have a significant effect on the stability properties of the journal. Copyright © 1999 John Wiley & Sons, Ltd.

The flow of an Oldroyd‐B fluid through an undulating tube is considered. The effect of elasticity and inertia on flow resistance is investigated numerically using a time‐splitting technique. This technique can be used to solve both steady and transient viscoelastic flows. A pseudospectral method based on mixed Fourier–Chebyshev expansions is used to represent the flow variables in space. An approximation space for pressure is constructed which is compatible with that for the velocity. This is achieved by removing the spurious modes using a singular value decomposition. A projection method ensures that mass is conserved identically at the collocation points. Numerical results are presented in such a way as to highlight the inertial and elastic effects. To this end two sets of results are given: the first, inertialess viscoelastic flow; the second, flow at nonzero Reynolds number holding the Weissenberg number constant. The two cases are shown to have quite opposite effects upon the flow rate and resistance.

The flow past a sphere in an infinite expanse of viscoelastic fluid is simulated numerically using a pseudospectral technique. The problem is formulated using primitive variables. A time-splitting scheme is used that ensures that the velocity field is divergence-free at the end of each time step. The flow domain is treated without truncation using rational Chebyshev polynomials as basis functions in the radial direction. The drag on the sphere is computed for a range of values of the non-dimensional elasticity number We . A quantitative comparison of the drag coefficient with that obtained by a perturbation analysis is given. As We is initially increased, a reduction in the drag from its Newtonian value is observed. For larger values of We , drag enhancement is detected. In addition, for small values of We the streamlines are shifted slightly downstream in agreement with experimental observations. The numerical algorithm is described in detail and the numerical results compared qualitatively with experiments.

A spectral domain decomposition method for solving the planar stick-slip problem for an Oldroyd-B fluid is presented. The flow domain is decomposed into two semi-infinite rectangular subdomains. A comparison of the treatment of these subdomains by domain truncation and algebraic mapping is given. A fixed point method is used to linearize the constitutive equation. The linearized systems are solved in a manner sympathetic towards the structure of the coefficient matrices at each iterative step. The algorithm employed in solving the stick-slip problem is shown to be efficient in its demand upon array storage and CPU time. The numerical results demonstrate the necessity of adequate entry and exit lengths for the non-Newtonian problem and the adverse effect that the singularity has upon the range of Weissenberg numbers over which converged solutions are possible. Results generated from a regularized stick-slip geometry where the boundary singularity is smoothed out provide evidence that it is possible to extend this range by solving a slightly modified problem in which the effect of the singularity is weakened.

A three-dimensional spectral algorithm for the solution of Stokes flow between eccentrically rotating cylinders is described. Included in the model are pressure boundary conditions at the two ends of the finite length cylinders and the effect of a fluid line source on the inner cylinder. A comparison of results for the load and couple on the inner cylinder is made with those available from lubrication theory in the absence of a line source. Good agreement is shown for long, short and finite journal bearings when the various geometrical assumptions inherent in the lubrication analysis are satisfied.

The dynamics of bubbles immersed in a viscoelastic fluid directly beneath an initially plane free surface is modelled using the boundary integral method. The model predicts a range of dynamics that is dependent on the Deborah number, the Reynolds number and the proximity of the bubble to the free surface. The motion of the free surface jet caused by the collapse of a bubble in a viscoelastic fluid can be significantly retarded compared with the Newtonian case. The axial jet predicted in many instances in the Newtonian case is not observed when the inertial forces are sufficiently small. In this case an annular jet forms that can penetrate the bubble. At high Deborah numbers, there is a return to Newtonian-like dynamics since the effects of viscosity are abated by elasticity to such an extent that inertia is the prevailing influence on bubble dynamics.