# Dr Robert Nürnberg

Faculty of Natural SciencesDepartment of Mathematics

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### Contact

+44 (0)20 7594 8572robert.nurnberg

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### Location

6M14Huxley BuildingSouth Kensington Campus

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## Publications

Publication Type
Year
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78 results found

Garcke H, Nürnberg R, 2021, Structure-preserving discretizations of gradient flows for axisymmetric two-phase biomembranes, Publisher: Oxford University Press (OUP)

<jats:title>Abstract</jats:title> <jats:p>The form and evolution of multi-phase biomembranes are of fundamental importance in order to understand living systems. In order to describe these membranes, we consider a mathematical model based on a Canham–Helfrich–Evans two-phase elastic energy, which will lead to fourth-order geometric evolution problems involving highly nonlinear boundary conditions. We develop a parametric finite element method in an axisymmetric setting. Using a variational approach it is possible to derive weak formulations for the highly nonlinear boundary value problems such that energy decay laws, as well as conservation properties, hold for spatially discretized problems. We will prove these properties and show that the fully discretized schemes are well posed. Finally, several numerical computations demonstrate that the numerical method can be used to compute complex, experimentally observed two-phase biomembranes.</jats:p>

Working paper

Garcke H, Nurnberg R, 2021, Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes, IMA Journal of Numerical Analysis, Vol: 41, Pages: 1899-1940, ISSN: 0272-4979

The form and evolution of multi-phase biomembranes is of fundamental impor-tance in order to understand living systems. In order to describe these membranes,we consider a mathematical model based on a Canham–Helfrich–Evans two-phaseelastic energy, which will lead to fourth order geometric evolution problems involv-ing highly nonlinear boundary conditions. We develop a parametric finite elementmethod in an axisymmetric setting. Using a variational approach, it is possible toderive weak formulations for the highly nonlinear boundaryvalue problems suchthat energy decay laws, as well as conservation properties,hold for spatially discre-tised problems. We will prove these properties and show thatthe fully discretisedschemes are well-posed. Finally, several numerical computations demonstrate thatthe numerical method can be used to compute complex, experimentally observedtwo-phase biomembranes.

Journal article

Ebenbeck M, Garcke H, Nürnberg R, 2020, Cahn-Hilliard-Brinkman systems for tumour growth, Publisher: arXiv

A phase field model for tumour growth is introduced that is based on aBrinkman law for convective velocity fields. The model couples a convectiveCahn-Hilliard equation for the evolution of the tumour to areaction-diffusion-advection equation for a nutrient and to a Brinkman-Stokestype law for the fluid velocity. The model is derived from basicthermodynamical principles, sharp interface limits are derived by matchedasymptotics and an existence theory is presented for the case of a mobilitywhich degenerates in one phase leading to a degenerate parabolic equation offourth order. Finally numerical results describe qualitative features of thesolutions and illustrate instabilities in certain situations.

Working paper

Garcke H, Nürnberg R, 2020, Structure preserving discretisations of gradient flows for axisymmetric two-phase biomembranes, Publisher: arXiv

The form and evolution of multi-phase biomembranes is of fundamentalimportance in order to understand living systems. In order to describe thesemembranes, we consider a mathematical model based on a Canham--Helfrich--Evanstwo-phase elastic energy, which will lead to fourth order geometric evolutionproblems involving highly nonlinear boundary conditions. We develop aparametric finite element method in an axisymmetric setting. Using avariational approach, it is possible to derive weak formulations for the highlynonlinear boundary value problems such that energy decay laws, as well asconservation properties, hold for spatially discretised problems. We will provethese properties and show that the fully discretised schemes are well-posed.Finally, several numerical computations demonstrate that the numerical methodcan be used to compute complex, experimentally observed two-phase biomembranes.

Working paper

Barrett JW, Deckelnick K, Nurnberg R, 2020, A finite element error analysis for axisymmetric mean curvature flow, IMA Journal of Numerical Analysis, ISSN: 0272-4979

We consider the numerical approximation of axisymmetric mean curvature flow with the help of linear finite elements. In the case of a closed genus-1 surface, we derive optimal error bounds with respect to the L 2– and H1–norms for a fully discrete approximation. We perform convergence experiments to confirm the theoretical results, and also present numerical simulations for some genus-0 and genus-1 surfaces, including for the Angenent torus.

Journal article

Barrett JW, Deckelnick K, Nürnberg R, 2019, A finite element error analysis for axisymmetric mean curvature flow, Publisher: arXiv

We consider the numerical approximation of axisymmetric mean curvature flowwith the help of linear finite elements. In the case of a closed genus-1surface, we derive optimal error bounds with respect to the $L^2$-- and$H^1$--norms for a fully discrete approximation. We perform convergenceexperiments to confirm the theoretical results, and also present numericalsimulations for some genus-0 and genus-1 surfaces.

Working paper

Nurnberg R, Agnese M, 2019, Fitted front tracking methods for two-phase incompressible Navier--Stokes flow: Eulerian and ALE finite element discretizations, International Journal of Numerical Analysis and Modeling, ISSN: 1705-5105

We investigate novel fitted finite element approximations for two-phase Navier–Stokes flow. In particular, we consider both Eulerian and Arbitrary Lagrangian–Eulerian (ALE) finite element formulations. The moving interface is approximatedwith the help of parametric piecewise linear finite element functions. The bulk meshis fitted to the interface approximation, so that standard bulk finite element spacescan be used throughout. The meshes describing the discrete interface in general donot deteriorate in time, which means that in numerical simulations a smoothing ora remeshing of the interface mesh is not necessary. We present several numericalexperiments, including convergence experiments and benchmark computations, forthe introduced numerical methods, which demonstrate the accuracy and robustnessof the proposed algorithms. We also compare the accuracy and efficiency of theEulerian and ALE formulations.

Journal article

Barrett JW, Garcke H, Nürnberg R, 2019, Stable approximations for axisymmetric Willmore flow for closed and opensurfaces, Publisher: arXiv

For a hypersurface in ${\mathbb R}^3$, Willmore flow is defined as the$L^2$--gradient flow of the classical Willmore energy: the integral of thesquared mean curvature. This geometric evolution law is of interest indifferential geometry, image reconstruction and mathematical biology. In thispaper, we propose novel numerical approximations for the Willmore flow ofaxisymmetric hypersurfaces. For the semidiscrete continuous-in-time variants weprove a stability result. We consider both closed surfaces, and surfaces with aboundary. In the latter case, we carefully derive suitable boundary conditions.Furthermore, we consider many generalizations of the classical Willmore energy,particularly those that play a role in the study of biomembranes. In thegeneralized models we include spontaneous curvature and area differenceelasticity (ADE) effects, Gaussian curvature and line energy contributions.Several numerical experiments demonstrate the efficiency and robustness of ourdeveloped numerical methods.

Working paper

Barrett JW, Garcke H, Nürnberg R, 2019, Stable discretizations of elastic flow in Riemannian manifolds, SIAM Journal on Numerical Analysis, ISSN: 0036-1429

The elastic flow, which is the $L^2$-gradient flow of the elastic energy, hasseveral applications in geometry and elasticity theory. We present stablediscretizations for the elastic flow in two-dimensional Riemannian manifoldsthat are conformally flat, i.e.\ conformally equivalent to the Euclidean space.Examples include the hyperbolic plane, the hyperbolic disk, the elliptic planeas well as any conformal parameterization of a two-dimensional manifold in${\mathbb R}^d$, $d\geq 3$. Numerical results show the robustness of themethod, as well as quadratic convergence with respect to the spacediscretization.

Journal article

Dorfler W, Nurnberg R, 2019, Discrete gradient flows for general curvature energies, SIAM Journal on Scientific Computing, Vol: 41, Pages: A2012-A2036, ISSN: 1064-8275

We consider the numerical approximation of theL2–gradient flow of general curvatureenergies∫G(|~κ|) for a curve inRd,d≥2. Here the curve can be either closed, or it can be open andclamped at the end points. These general curvature energies, and the considered boundary conditions,appear in the modelling of the power loss within an optical fibre. We present two alternative finiteelement approximations, both of which admit a discrete gradient flow structure. Apart from beingstable, in addition, one of the methods satisfies an equidistribution property. Numerical resultsdemonstrate the robustness and the accuracy of the proposedmethods.

Journal article

Antonopoulou D, Banas L, Nürnberg R, Prohl Aet al., 2019, Numerical approximation of the stochastic Cahn-Hilliard equation nearthe sharp interface limit

We consider the stochastic Cahn-Hilliard equation with additive noise term$\varepsilon^\gamma g\, \dot{W}$ ($\gamma >0$) that scales with the interfacialwidth parameter $\varepsilon$. We verify strong error estimates for a gradientflow structure-inheriting time-implicit discretization, where$\varepsilon^{-1}$ only enters polynomially; the proof is based onhigher-moment estimates for iterates, and a (discrete) spectral estimate forits deterministic counterpart. For $\gamma$ sufficiently large, convergence inprobability of iterates towards the deterministic Hele-Shaw/Mullins-Sekerkaproblem in the sharp-interface limit $\varepsilon \rightarrow 0$ is shown.These convergence results are partly generalized to a fully discrete finiteelement based discretization. We complement the theoretical results by computational studies to providepractical evidence concerning the effect of noise (depending on its 'strength'$\gamma$) on the geometric evolution in the sharp-interface limit. For thispurpose we compare the simulations with those from a fully discrete finiteelement numerical scheme for the (stochastic) Mullins-Sekerka problem. Thecomputational results indicate that the limit for $\gamma\geq 1$ is thedeterministic problem, and for $\gamma=0$ we obtain agreement with a (new)stochastic version of the Mullins-Sekerka problem.

Working paper

Barrett JW, Garcke H, Nürnberg R, 2019, Parametric finite element approximations of curvature driven interface evolutions, Publisher: arXiv

Parametric finite elements lead to very efficient numerical methods forsurface evolution equations. We introduce several computational techniques forcurvature driven evolution equations based on a weak formulation for the meancurvature. The approaches discussed, in contrast to many other methods, havegood mesh properties that avoid mesh coalescence and very non-uniform meshes.Mean curvature flow, surface diffusion, anisotropic geometric flows,solidification, two-phase flow, Willmore and Helfrich flow as well asbiomembranes are treated. We show stability results as well as resultsexplaining the good mesh properties.

Working paper

Barrett JW, Garcke H, Nürnberg R, 2019, Numerical approximation of curve evolutions in Riemannian manifolds, IMA Journal of Numerical Analysis, ISSN: 0272-4979

We introduce variational approximations for curve evolutions intwo-dimensional Riemannian manifolds that are conformally flat, i.e.\conformally equivalent to the Euclidean space. Examples include the hyperbolicplane, the hyperbolic disk, the elliptic plane as well as any conformalparameterization of a two-dimensional surface in ${\mathbb R}^d$, $d\geq 3$. Inthese spaces we introduce stable numerical schemes for curvature flow and curvediffusion, and we also formulate a scheme for elastic flow. Variants of theschemes can also be applied to geometric evolution equations for axisymmetrichypersurfaces in ${\mathbb R}^d$. Some of the schemes have very good propertieswith respect to the distribution of mesh points, which is demonstrated with thehelp of several numerical computations.

Journal article

Barret JW, Garcke H, Nürnberg R, 2019, Variational discretization of axisymmetric curvature flows, Numerische Mathematik, ISSN: 0029-599X

We present natural axisymmetric variants of schemes for curvature flowsintroduced earlier by the present authors and analyze them in detail. Althoughnumerical methods for geometric flows have been used frequently in axisymmetricsettings, numerical analysis results so far are rare. In this paper, we presentstability, equidistribution, existence and uniqueness results for theintroduced approximations. Numerical computations show that these schemes arevery efficient in computing numerical solutions of geometric flows as only aspatially one-dimensional problem has to be solved. The good mesh properties ofthe schemes also allow them to compute in very complex axisymmetric geometries.

Journal article

Barrett JW, Garcke H, Nürnberg R, 2019, Finite element methods for fourth order axisymmetric geometric evolutionequations, Journal of Computational Physics, Vol: 376, Pages: 733-766, ISSN: 0021-9991

Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce and analyze several new finite element schemes for fourth order geometric evolution equations in an axisymmetric setting, and for selected schemes we will show existence, uniqueness and stability results. The presented schemes have very good mesh and stability properties, as will be demonstrated by several numerical examples.

Journal article

Garcke H, Lam KF, Nurnberg R, Sitka Eet al., 2018, A multiphase Cahn--Hilliard--Darcy model for tumour growth with necrosis, Mathematical Models and Methods in Applied Sciences (M3AS), Vol: 28, Pages: 525-577, ISSN: 0218-2025

We derive a Cahn–Hilliard–Darcy model to describe multiphase tumour growth taking interactions with multiple chemical species into account as well as the simultaneous occurrence of proliferating, quiescent and necrotic regions. A multitude of phenomena such as nutrient diffusion and consumption, angiogenesis, hypoxia, blood vessel growth, and inhibition by toxic agents, which are released for example by the necrotic cells, are included. A new feature of the modelling approach is that a volume-averaged velocity is used, which dramatically simplifies the resulting equations. With the help of formally matched asymptotic analysis we develop new sharp interface models. Finite element numerical computations are performed and in particular the effects of necrosis on tumour growth are investigated numerically. In particular, for certain modelling choices, we obtain some form of focal and patchy necrotic growth that have been observed in experiments.

Journal article

Barrett JW, Garcke H, Nürnberg R, 2018, Gradient flow dynamics of two-phase biomembranes: Sharp interface variational formulation and finite element approximation, SMAI Journal of Computational Mathematics, Vol: 4, Pages: 151-195, ISSN: 2426-8399

A finite element method for the evolution of a two-phase membrane in a sharp interfaceformulation is introduced. The evolution equations are given as anL2–gradient flow of an energy involvingan elastic bending energy and a line energy. In the two phases Helfrich-type evolution equations are prescribed,and on the interface, an evolving curve on an evolving surface, highly nonlinear boundary conditions have tohold. Here we consider bothC0– andC1–matching conditions for the surface at the interface. A new weakformulation is introduced, allowing for a stable semidiscrete parametric finite element approximation of thegoverning equations. In addition, we show existence and uniqueness for a fully discrete version of the scheme.Numerical simulations demonstrate that the approach can deal with a multitude of geometries. In particular,the paper shows the first computations based on a sharp interface description, which are not restricted tothe axisymmetric case.

Journal article

Barrett JW, Garcke H, Nürnberg R, 2017, Finite Element Approximation for the Dynamics of Fluidic Two-Phase Biomembranes, ESAIM: Mathematical Modelling and Numerical Analysis (ESAIM: M2AN), Vol: 51, Pages: 2319-2366, ISSN: 0764-583X

Biomembranes and vesicles consisting of multiple phases can attain amultitude of shapes, undergoing complex shape transitions. We study aCahn--Hilliard model on an evolving hypersurface coupled to Navier--Stokesequations on the surface and in the surrounding medium to model thesephenomena. The evolution is driven by a curvature energy, modelling theelasticity of the membrane, and by a Cahn--Hilliard type energy, modelling lineenergy effects. A stable semidiscrete finite element approximation isintroduced and, with the help of a fully discrete method, several phenomenaoccurring for two-phase membranes are computed.

Journal article

Barrett JW, Garcke H, Nurnberg R, 2017, Stable variational approximations of boundary value problems for Willmore flow with Gaussian curvature, IMA JOURNAL OF NUMERICAL ANALYSIS, Vol: 37, Pages: 1657-1709, ISSN: 0272-4979

Journal article

Aland S, Hahn A, Kahle C, Nurnberg Ret al., 2017, Comparative simulations of Taylor Flow with surfactants based on sharp- and diffuse-interface methods, Transport Processes at Fluidic Interfaces, Editors: Bothe, Reusken, Publisher: Birkhäuser, Pages: 639-679, ISBN: 978-3-319-56602-3

We present a quantitative comparison of simulations based on diffuse- and sharp-interface models for two-phase flows with soluble surfactants. The test scenario involves a single Taylor bubble in a counter-current flow. The bubble assumes a stationary position as liquid inflow and gravity effects cancel each other out, which makes the scenario amenable to high resolution experimental imaging. We compare the accuracy and efficiency of the different numerical models and four different implementations in total.

Book chapter

Banas L, Nurnberg R, 2017, Numerical approximation of a non-smooth phase-field model for multicomponent incompressible flow, ESAIM: Mathematical Modelling and Numerical Analysis, Vol: 51, Pages: 1089-1117, ISSN: 0399-0516

We present a phase-field model for multiphase flow for an arbitrary number of immiscible incompressible fluids with variable densities and viscosities. The model consists of a system of the Navier−Stokes equations coupled to multicomponent Cahn−Hilliard variational inequalities. The proposed formulation admits a natural energy law, preserves physically meaningful constraints and allows for a straightforward modelling of surface tension effects. We propose a practical fully discrete finite element approximation of the model which preserves the energy law and the associated physical constraints. In the case of matched densities we prove convergence of the numerical scheme towards a weak solution of the continuous model. The convergence of the numerical approximations also implies the existence of weak solutions. Furthermore, we propose a convergent iterative fixed-point algorithm for the solution of the discrete nonlinear system of equations and present several computational studies of the proposed model.

Journal article

Barrett JW, Garcke H, Nurnberg R, 2016, Erratum: Numerical computations of the dynamics of fluidic membranes and vesicles [Phy. Rev. E 92, 052704, (2015)], Physical Review E, Vol: 94, ISSN: 1539-3755

Journal article

Nurnberg R, Barrett JW, Garcke H, 2016, A stable numerical method for the dynamics of fluidic membranes, Numerische Mathematik, Vol: 134, Pages: 783-822, ISSN: 0029-599X

We develop a finite element scheme to approximate the dynamics of two andthree dimensional fluidic membranes in Navier–Stokes flow. Local inextensibilityof the membrane is ensured by solving a tangential Navier–Stokes equation, takingsurface viscosity effects of Boussinesq–Scriven type into account. In our approachthe bulk and surface degrees of freedom are discretized independently, which leads toan unfitted finite element approximation of the underlying free boundary problem.Bending elastic forces resulting from an elastic membrane energy are discretized usingan approximation introduced by Dziuk (2008). The obtained numerical schemecan be shown to be stable and to have good mesh properties. Finally, the evolutionof membrane shapes is studied numerically in different flow situations in two andthree space dimensions. The numerical results demonstrate the robustness of themethod, and it is observed that the conservation properties are fulfilled to a highprecision.

Journal article

Nurnberg R, Tucker E, 2016, Stable finite element approximation of a Cahn--Hilliard--Stokes system coupled to an electric field, European Journal of Applied Mathematics, Vol: 28, Pages: 470-498, ISSN: 1469-4425

We consider a fully practical finite element approximation of the Cahn–Hilliard–Stokes systemγ∂u∂t + βv · ∇ u − ∇ · (∇ w) = 0 , w = −γ∆u + γ−1Ψ′(u) −12αc′(·, u)|∇ φ|2,∇ · (c(·, u)∇ φ) = 0 ,(−∆v + ∇ p = ςw∇ u,∇ · v = 0,subject to an initial condition u0(.) ∈ [−1, 1] on the conserved order parameter u ∈[−1, 1], and mixed boundary conditions. Here γ ∈ R>0 is the interfacial parameter,α ∈ R≥0 is the field strength parameter, Ψ is the obstacle potential, c(·, u) is thediffusion coefficient, and c′(·, u) denotes differentiation with respect to the secondargument. Furthermore, w is the chemical potential, φ is the electro-static potentialand (v, p) are the velocity and pressure. The system has been proposed to modelthe manipulation of morphologies in organic solar cells with the help of an appliedelectric field and kinetics.

Journal article

Barrett JW, Garcke H, Nurnberg R, 2016, Finite element approximation for the dynamics of asymmetric fluidic biomembranes, Mathematics of Computation, Vol: 86, Pages: 1037-1069, ISSN: 1088-6842

We present a parametric finite element approximation of a fluidic membrane,whose evolution is governed by a surface Navier–Stokes equation coupled to bulkNavier–Stokes equations. The elastic properties of the membrane are modelled withthe help of curvature energies of Willmore and Helfrich type. Forces stemming fromthese energies act on the surface fluid, together with a forcing from the bulk fluid.Using ideas from PDE constrained optimization, a weak formulation is derived,which allows for a stable semi-discretization. An important new feature of thepresent work is that we are able to also deal with spontaneous curvature and anarea difference elasticity contribution in the curvature energy. This allows for themodelling of asymmetric membranes, which compared to the symmetric case leadto quite different shapes. This is demonstrated in the numerical computationspresented.

Journal article

Nurnberg R, Barrett JW, Garcke H, 2016, Computational parametric Willmore flow with spontaneous curvature and area difference elasticity effects, SIAM Journal on Numerical Analysis, Vol: 54, Pages: 1732-1762, ISSN: 0036-1429

A new stable continuous-in-time semi-discrete parametric finite element method forWillmore flow is introduced. The approach allows for spontaneous curvature and area differenceelasticity (ADE) effects, which are important for many applications, in particular, in the context ofmembranes. The method extends ideas from Dziuk and the present authors to obtain an approximationthat allows for a tangential redistribution of mesh points, which typically leads to better meshproperties. Moreover, we consider volume and surface area preserving variants of these schemes and,in particular, we obtain stable approximations of Helfrich flow. We also discuss fully discrete variantsand present several numerical computations.

Journal article

Nurnberg R, Agnese M, 2016, Fitted finite element discretization of two--phase stokes flow, International Journal for Numerical Methods in Fluids, Vol: 82, Pages: 709-729, ISSN: 1097-0363

We propose a novel fitted finite element method for two-phase Stokesflow problems that uses piecewise linear finite elements to approximate themoving interface. The method can be shown to be unconditionally stable.Moreover, spherical stationary solutions are captured exactly by the numericalapproximation. In addition, the meshes describing the discrete interfacein general do not deteriorate in time, which means that in numerical simulationsa smoothing or a remeshing of the interface mesh is not necessary.We present several numerical experiments for our numerical method, whichdemonstrate the accuracy and robustness of the proposed algorithm.

Journal article

Barrett JW, Garcke H, Nurnberg R, 2015, Numerical computations of the dynamics of fluidic membranes and vesicles, Physical Review E, Vol: 92, ISSN: 1539-3755

Vesicles and many biological membranes are made of two monolayers of lipid molecules and formclosed lipid bilayers. The dynamical behaviour of vesicles is very complex and a variety of formsand shapes appear. Lipid bilayers can be considered as a surface fluid and hence the governingequations for the evolution include the surface (Navier–)Stokes equations, which in particular takethe membrane viscosity into account. The evolution is driven by forces stemming from the curvatureelasticity of the membrane. In addition, the surface fluid equations are coupled to bulk(Navier–)Stokes equations.We introduce a parametric finite element method to solve this complex free boundary problem, andpresent the first three dimensional numerical computations based on the full (Navier–)Stokes systemfor several different scenarios. For example, the effects of the membrane viscosity, spontaneouscurvature and area difference elasticity (ADE) are studied. In particular, it turns out, that even inthe case of no viscosity contrast between the bulk fluids, the tank treading to tumbling transitioncan be obtained by increasing the membrane viscosity. Besides the classical tank treading andtumbling motions, another mode (called the transition mode in this paper, but originally called thevacillating-breathing mode and subsequently also called trembling, transition and swinging mode)separating these classical modes appears and will be studied by us numerically. We also studyhow features of equilibrium shapes in the ADE and spontaneous curvature models, like buddingbehaviour or starfish forms, behave in a shear flow.

Journal article

Nurnberg R, Sacconi A, 2015, A fitted finite element method for the numerical approximation of void electro-stress migration, Applied Numerical Mathematics, Vol: 104, Pages: 204-217, ISSN: 1873-5460

Microelectronic circuits usually contain small voids or cracks, and if those defects are large enough to sever the line, they cause an open circuit. A fully practical finite element method for the temporal analysis of the migration of voids in the presence of surface diffusion, electric loading and elastic stress is presented. We simulate a bulk–interface coupled system, with a moving interface governed by a fourth-order geometric evolution equation and a bulk where the electric potential and the displacement field are computed. The method presented here follows a fitted approach, since the interface grid is part of the boundary of the bulk grid. A detailed analysis, in terms of experimental order of convergence (when the exact solution to the free boundary problem is known) and coupling operations (e.g., smoothing/remeshing of the grids, intersection between elements of the two grids), is carried out. A comparison with a previously introduced unfitted approach (where the two grids are totally independent) is also performed, along with several numerical simulations in order to test the accuracy of the methods.

Journal article

Nurnberg R, Barrett JW, Garcke H, 2015, Stable Finite Element Approximations of Two-Phase Flow with SolubleSurfactant, Journal of Computational Physics, ISSN: 1090-2716

Journal article

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