arXiv subjects

Dieter Bothe

Publications and source records attributed to Dieter Bothe.

At least 19 recordsLinked to original sources

arXiv2026

A Sharpened Entropy Principle for Two-Fluid Polymer Thermodynamics

A compressible, non-isothermal dilute polymer solution is formulated as a Class-II binary mixture with separate solvent and polymer mass and momentum balances and with total energy and entropy balances for the complete mixture. The entropy exploitation is sharpened by a balance-anchoring axiom introduced here: in a fixed balance representative, each selected binary dissipative product contains a constitutive factor from an unclosed balance flux or source. Polymer configuration is resolved first by a connector distribution and then by its conformation tensor. Population kinematics fix polymer transport and, by moments, yield a two-velocity upper-convected rate; objectivity verifies covariance rather than selecting it. Within the total entropy balance, configurational transport and deformation powers cancel their chemical-potential and elastic partial-stress counterparts when deformation and stress-decomposition weights match. A Gordon--Schowalter test independently requires the affine upper-convected choice for the stated dumbbell free energy and Kramers stress unless an additional reversible channel is supplied. Coordinated stress--interaction changes shift the local entropy flux/production pair by a divergence, exposing representation dependence of local mechanism-wise production. An entropy-invariant Class-II-to-Class-I reduction selects a descendant entropy flux preserving the parent production and yields thermo-chemical, configurational-stress and partial-viscous-stress diffusion terms. The omitted quadratic relative-inertia flux is paired with relative kinetic-energy storage and transport and is a reversible truncation, not missing entropy production. The resulting compressible non-isothermal Hookean stress and temperature equations reduce to Oldroyd-B/UCM only after one-velocity, incompressible and isothermal limits.

cond-mat.soft↗

arXiv2026

Global Strong Solutions for Maxwell-Stefan Diffusion with Additive Friction Coefficients

We study Maxwell-Stefan diffusion with additive friction coefficients $f_{ij}=g_i+g_j$. In mass fractions, the system isolates the constrained pair-friction block; in mole fractions, it is the classical ideal isothermal/isobaric Maxwell-Stefan system at constant total molar concentration. Additivity makes the constrained pair-friction dissipation species-diagonal; conversely, species-diagonality on one interior barycentric constraint space forces a pair-sum representation. At operator level, the positive constrained relaxation operator is a scalar shift of a compression of $G=diag[g_1,\ldots,g_N]$. Its scalar resolvent yields both an explicit constrained inverse and interlacing spectral roots, which form global real-analytic coordinates on the open simplex and whose differentials are left eigen-covectors. In root coordinates the principal part is diagonal, no self-square gradient term occurs, and scalar comparison yields invariant rectangles and separation from the simplex boundary. For regularity we introduce entropy-stabilized one-sided multi-EPD truncations: Euler-Poisson-Darboux entropies cancel mixed quadratic production, while for $N\ge4$ a truncation-weighted mixing-entropy correction supplies transverse coercivity. Caccioppoli and logarithmic estimates, shrinking, and critical mass yield H\"older continuity up to the Neumann boundary. The mixing entropy also symmetrizes the moment system; frozen conormal estimates give spatial Lipschitz bounds. Together with time H\"older control and short-interval maximal regularity, this yields global strong solvability on bounded $C^{2+\alpha}$ domains with $0<\alpha<1$, for each $N\ge2$, $d\ge2$, and $p>d+2$, for all uniformly positive, compatible initial concentrations in the natural trace class. Solutions become classical for positive times and converge exponentially to equilibrium in relative entropy, $L^2$, and $C^1(\bar\Omega)$.

math.AP↗

arXiv2026

Existence and qualitative theory for nonlinear accretive evolutions with Carath\'eodory forcing

Let $A$ be $m$-accretive in a real Banach space $X$. Given closed sets $K(t)\subset X$, we develop an existence theory for mild solutions of the constrained problem \[ u'(t)+Au(t)\ni f(t,u(t)), \qquad u(t)\in K_A(t):=K(t)\cap\overline{D(A)}, \] where the forcing is Carath\'eodory on the effective tube $K_A$. We also assume joint measurability of $f$ on the moving graph or a Carath\'eodory extension to a fixed cylinder. Under a linear-growth hypothesis and the corresponding regular- and exceptional-time subtangential conditions, we reduce the problem to a bounded effective tube with a common integrable bound. % A first central result constructs a closed separable resolvent-invariant subspace preserving distances to the moving sets and the subtangential conditions, permitting use of the Scorza--Dragoni property for the reduced forcing. A second central ingredient is a two-level approximation. Mild solutions $u_\varepsilon$ driven by $w_\varepsilon$ are accompanied by auxiliary paths $v_\varepsilon$ and current-time selectors $x_\varepsilon$ satisfying $x_\varepsilon(t)\in K_A(t)$ and $w_\varepsilon(t)=f(t,x_\varepsilon(t))$ for almost every $t$ in a closed regularity set of almost full measure. For every $t$ there is also $\sigma_\varepsilon(t)\in[(t-\varepsilon)^+,t]$ with $v_\varepsilon(\sigma_\varepsilon(t))\in K_A(\sigma_\varepsilon(t))$. The current-time selectors identify the limiting forcing, while the lagged nodes guarantee viability. % This yields well-posedness under time-dependent constraints for such forcings $f$ when they are locally Lipschitz in the state variable. Further viability results follow under various compactness conditions. Application of the abstract viability theory yields comparison principles, nonautonomous Lyapunov pairs, periodic solutions, and time-dependent bounds for abstract reaction--diffusion systems.

math.FA↗

arXiv2026

Non-intrusive MEMS microphone sensing of acoustic field state in resonant acoustic levitators

Reliable operation of resonant acoustic levitators requires knowledge of the acoustic field state because the optimum transducer-reflector distance and resonant operating condition shift with wavelength, temperature, object insertion, and mechanical alignment. Existing adjustment methods are limited, especially for compact closed-loop operation and architectures without a passive reflector. Here, we investigate transducer-mounted microelectromechanical system (MEMS) microphones as off-axis external sensors that acquire relative acoustic signals without placing sensors inside the levitation cavity. Using a linear microphone configuration, we performed transducer-reflector distance sweeps over resonance modes n = 5-8 and compared microphone amplitude with acoustic radiation force measured by a precision balance and with peak-to-peak transducer current. The channel-mean microphone-voltage maxima occurred within two sampled distance increments, or at most 30 micrometers, of the force maxima. At the microphone-derived peak positions, at least 98.3% of the corresponding maximum force was retained. Microphone amplitude localized the force maximum more sharply than peak-to-peak transducer current. In one frequency-shift experiment, microphone phase provided a proof of principle for correction-direction estimation, while envelope modulation captured channel-resolved field changes during object oscillation. Ring measurements showed channel-dependent responses as transducer-reflector tilt was varied, but did not provide a calibrated or unique tilt estimate. These results show the potential of external MEMS microphones as relative acoustic observables for resonance-related field-state assessment and provide a basis for compact transducer-side feedback. The principle may also be transferable to transducer-transducer and array-based architectures.

physics.app-ph↗

arXiv2026

Co-moving volumes and the Reynolds transport theorem for two-phase flows

We consider the local kinematics at fluid interfaces in sharp-interface two-phase flows with phase change and interfacial slip. In this setting the governing velocity field is discontinuous at the phase boundary, with possible jumps of both normal and tangential components, and the associated kinematic initial value problems may fail to be uniquely solvable. A physically consistent example exhibits this non-uniqueness and, in addition, rapid loss of boundary regularity: smooth initial control volumes can instantaneously develop edges, while their phasewise parts may form cusps. Motivated by these phenomena, we use concepts from differential inclusions to define co-moving volumes as attainable sets. For such attainable-set co-moving volumes in three-dimensional two-phase flows, we prove the Reynolds transport theorem first in boundary-integral form and then in divergence form. A key ingredient is a boundary-integral form of the single-phase Reynolds transport theorem for families of compact regular closed sets whose space-time tubes are Lipschitz domains. We also provide a short proof of this single-phase result by applying the divergence theorem in space-time; this proof does not require the motion to be generated by an ambient velocity field.

math.AP↗

arXiv2025

A two-sided subgrid-scale model for mass transfer across fluid interfaces

The occurrence of extremely thin concentration boundary layers at fluid interfaces for high local P\'eclet numbers is a severe obstacle for efficient and accurate numerical simulation of mass transfer processes in two-phase fluid systems. Especially challenging are liquid-liquid systems, in which thin concentration boundary layers can appear on both sides of the fluid interface under convection-dominated conditions. In those cases, the one-sided species concentrations at the interface are a-priori not even known approximately, but are determined by a conjugate mass transfer problem governed by interfacial jump conditions. To the best of the authors' knowledge we for the first time introduce a two-sided Subgrid-Scale (SGS) boundary layer model for conjugate mass transfer at fluid interfaces. It accurately computes the local mass transfer rates on moderate or coarse mesh resolutions even when very high concentration gradients in interface vicinity occur. For this purpose, SGS modeling is applied on both sides of an interface transmissive to passive scalars, such as the interface in a two-phase fluid system, enabling the accurate capture of conjugate mass transfer across thin boundary layer on one or on both sides of the interface. We implement our approach in the unstructured Finite-Volume Arbitrary Lagrangian / Eulerian Interface-Tracking (ALE-IT) OpenFOAM module twoPhaseInterTrackFoam. We have made twoPhaseInterTrackFoam publicly available in our previous publication (Schwarzmeier et al., 2025).

physics.comp-ph↗

arXiv2025

A continuum thermodynamic model of the influence of non-ionic surfactant on mass transfer from gas bubbles

Mass transfer of gaseous components from rising bubbles to the ambient liquid depends not only on the chemical potential difference of the transfer component but also on the interfacial free energy and composition. The latter is strongly affected by surface active agents that are present in many applications. Surfactants lead to local changes in the interfacial tension, which influence the mass transfer rates in two different ways. On the one hand, inhomogeneous interfacial tension leads to Marangoni stress, which can strongly change the local hydrodynamics. One the other hand, the coverage by surfactant molecules results in a mass transfer resistance. This hindrance effect is not included in current continuum physical models. The present work provides the experimental validation of a recently introduced extended sharp-interface model for two-phase flows with mass transfer that also accounts for the mass transfer hindrance due to adsorbed surfactant. The crucial feature is to account for area-specific concentrations not only of adsorbed constituents but also of transfer species, and to model mass transfer as a series of two bi-directional sorption-type bulk-interface exchange processes. The resulting model is shown to quantitatively describe experimental measurements on mass transfer reduction for the dissolution of CO$_2$ bubbles in different surfactant solutions.

physics.flu-dyn↗

arXiv2025

Multi-Velocity Sharp-Interface Continuum Thermodynamics of Fluid Systems with Adsorption

We revisit the sharp-interface continuum thermodynamics of two-phase multicomponent fluid systems, accounting for partial mass and partial momentum balances both in the bulk phases and on the interface. This allows to describe the transfer of species between the individual bulk phases and the interface, i.e. ad- and desorption processes. In fact, the transfer of any constituent between the two bulk-phases is considered as a series of ad- and desorption processes. In this framework, all species transfer processes are coupled via the interfacial thermodynamics. As a consequence, the influence of surface active species on the transfer of other constituents can be captured in detail. The derivation of this model class relies on an axiomatic form of the entropy principle which, at the same time, allows for an efficient closure process. This form of the entropy principle has been introduced for one-phase fluid systems in (Bothe, Dreyer, Acta Mechanica 226, 2015) as the result of intense joint work of the late Wolfgang Dreyer and the present author.

physics.flu-dyn↗

arXiv2024

Mathematical analysis of the velocity extension level set method

A passively advected sharp interface can be represented as the zero level set of a level set function $f$. The linear transport equation $\partial_tf+v\cdot \nabla f =0$ is the simplest governing equation for such a level set function. While the signed distance of the interface is a geometrically convenient function, e.g., the norm of the gradient is everywhere one, its time evolution is not governed by the linear transport equation. In computational fluid dynamics, several modifications of the simplest case have been proposed in order to compute the signed distance function or to stabilize the norm of the gradient of a level set function on the interface. The velocity extension method is a prominent method used for efficient numerical approximation of the local signed distance function of the interface. Our current paper presents a rigorous mathematical formulation of the velocity extension method and proves that the method provides the local signed distance function of the moving interface. A key is to derive a first-order fully nonlinear PDE that is equivalent to the linear transport equation with extended velocity. Then, wellposedness of the PDE is established in the class of $C^2$-smooth solutions, global in time and local in space, with the local signed distance property, as well as in the class of $C^0$-viscosity solutions, global in time and space. Furthermore, partial regularity of the viscosity solution is proven, thus confirming that, if initial data is smooth near the initial interface, the viscosity solution is smooth in a time-global tubular neighborhood of the interface, coinciding with the local-in-space $C^2$-smooth solution.

math.AP↗

arXiv2024

A consistent treatment of dynamic contact angles in the sharp-interface framework with the generalized Navier boundary condition

In this work, we revisit the Generalized Navier Boundary condition (GNBC) introduced by Qian et al.\ in the sharp interface Volume-of-Fluid context. We replace the singular uncompensated Young stress by a smooth function with a characteristic width $\varepsilon > 0$ that is understood as a physical parameter of the model. Therefore, we call the model the ``Contact Region GNBC'' (CR-GNBC). We show that the model is consistent with the fundamental kinematics of the contact angle transport described by Fricke, K\"ohne and Bothe. We implement the model in the geometrical Volume-of-Fluid solver Basilisk using a ``free angle'' approach. This means that the dynamic contact angle is not prescribed but reconstructed from the interface geometry and subsequently applied as an input parameter to compute the uncompensated Young stress. We couple this approach to the two-phase Navier Stokes solver and study the withdrawing tape problem with a receding contact line. It is shown that the model allows for grid-independent solutions and leads to a full regularization of the singularity at the moving contact line, which is in accordance with the thin-film equation subject to this boundary condition. In particular, it is shown that the curvature at the moving contact line is finite and mesh converging. As predicted by the fundamental kinematics, the parallel shear stress component vanishes at the moving contact line for quasi-stationary states (i.e. for $\dot{\theta}_d=0$) and the dynamic contact angle is determined by a balance between the uncompensated Young stress and an effective contact line friction. Furthermore, a non-linear generalization of the model is proposed, which aims at reproducing the Molecular Kinetic Theory of Blake and Haynes for quasi-stationary states.

physics.flu-dyn↗

arXiv2024

The initial acceleration of a buoyant spherical bubble revisited

An analytical derivation of the buoyancy-induced initial acceleration of a spherical gas bubble in a host liquid is presented. The theory makes no assumptions further than applying the two-phase incompressible Navier-Stokes equations, showing that neither the classical approach using potential theory nor other simplifying assumptions are needed. The result for the initial bubble acceleration as a function of the gas and liquid densities, classically built on potential theory, is retained. The result is reproduced by detailed numerical simulations. The accelerated, although stagnant state of the bubble induces a pressure distribution on the bubble surface which is different from the result related to the Archimedean principle, emphasizing the importance of the non-equilibrium state for the force acting on the bubble.

physics.flu-dyn↗

arXiv2024

Persistence exponents via perturbation theory: MA(1)-processes

For the moving average process $X_n=\rho \xi_{n-1}+\xi_n$, $n\in\mathbb{N}$, where $\rho\in\mathbb{R}$ and $(\xi_i)_{i\ge -1}$ is an i.i.d. sequence of normally distributed random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots, X_N\ge 0)$, for $N\to\infty$. We exploit that the exponential decay rate $\lambda_\rho$ of that quantity, called the persistence exponent, is given by the leading eigenvalue of a concrete integral operator. This makes it possible to study the problem with purely functional analytic methods. In particular, using methods from perturbation theory, we show that the persistence exponent $\lambda_\rho$ can be expressed as a power series in $\rho$. Finally, we consider the persistence problem for the Slepian process, transform it into the moving average setup, and show that our perturbation results are applicable.

math.PR↗

arXiv2024

On the problem of optimal fluid transport in capillaries

In this note, we revisit the problem of the pressure-driven transport of a meniscus through a narrow cylindrical capillary or pore. This generic process finds many applications in science and technology. As it is known that Direct Numerical Simulations of moving contact line problems are highly demanding in terms of computational costs, simplified models in the form of ordinary differential equations offer an interesting alternative to perform a mathematical optimization of the flow. Blake and De Coninck studied the pressure-driven transport of a meniscus and identified two major competing mechanisms. While a hydrophilic surface is favorable to enhance the spontaneous imbibition into the pore, the friction is known to be significantly reduced on a hydrophobic surface. Blake and De Coninck showed that, depending on the applied pressure difference, there exists an optimal wettability that minimizes the time required to move the meniscus over a certain distance. We revisit this problem and derive analytical solutions in the limiting cases of negligible inertia and negligible contact line friction.

physics.flu-dyn↗

arXiv2024

Scale-bridging within a complex model hierarchy for investigation of a metal-fueled circular energy economy by use of Bayesian model calibration with model error quantification

Metal energy carriers recently gained growing interest in research as a promising storage and transport material for renewable electricity. Within the development of a metal-fueled circular energy economy, research involves a model hierarchy spanning from micro to macro scales, making the transfer of information among different levels of complexity a crucial task for the implementation of the new technology. Chemical reactor networks (CRNs) are models of reduced complexity and a promising approach to accomplish the scale-bridging task. This holds if valid information from CRNs can be obtained on a much denser set of operating conditions than available from experiments and elaborated simulation methods like Computational Fluid Dynamics (CFD). An approach for CRN calibration from recent literature, including model error quantification, is further developed to construct a CRN model of a laboratory reactor for flash ironmaking, using data from the literature. By introducing a meta model of a CRN parameter, a simple CRN model on an extended set of operating conditions has successfully been calibrated. This way, the employed coupled calibration and uncertainty quantification framework has proven promising for the task of scale-bridging in the model hierarchy under investigation.

physics.comp-ph↗

arXiv2024

Gradient and Hessian regularity in elliptic transmission problems near a point cusp

We consider elliptic transmission problems in several space dimensions near an interface which is $C^{1,1}$ diffeomorphic to an axisymmetric reference-interface with a singular point of cusp type. We establish the regularity of the gradient and of the Hessian in $L^p$ spaces up to the cusp point for local weak solutions. We obtain regularity thresholds which are different according to whether the cusp is inward or outward to the subdomain, and which depend explicitly on the opening of the interface at the cusp. Our results allow for source terms in the bulk and on the interface.

math.AP↗

arXiv2024

twoPhaseInterTrackFoam: an OpenFOAM module for Arbitrary Lagrangian/Eulerian Interface Tracking with Surfactants and Subgrid-Scale Modeling

We provide an implementation of the unstructured Finite-Volume Arbitrary Lagrangian / Eulerian (ALE) Interface-Tracking method for simulating incompressible, immiscible two-phase flows as an OpenFOAM module. In addition to interface-tracking capabilities that include tracking of two fluid phases, an implementation of a Subgrid-Scale (SGS) modeling framework for increased accuracy when simulating sharp boundary layers is enclosed. The SGS modeling framework simplifies embedding subgrid-scale profiles into the unstructured Finite Volume discretization. Our design of the SGS model library significantly simplifies adding new SGS models and applying SGS modeling to Partial Differential Equations (PDEs) in OpenFOAM.

physics.comp-ph↗

arXiv2024

An unstructured finite-volume level set / front tracking method for two-phase flows with large density-ratios

We extend the unstructured LEvel set / froNT tracking (LENT) method for handling two-phase flows with strongly different densities (high-density ratios) by providing the theoretical basis for the numerical consistency between the mass and momentum conservation in the collocated Finite Volume discretization of the single-field two-phase Navier-Stokes equations. Our analysis provides the theoretical basis for the mass conservation equation introduced by Ghods and Herrmann [3] and used in [4, 5, 6, 7, 8]. We use a mass flux that is consistent with mass conservation in the implicit Finite Volume discretization of the two-phase momentum convection term, and solve the single-field Navier-Stokes equations with our SAAMPLE segregated solution algorithm [2]. The proposed $ρ$LENT method recovers exact numerical stability for the two-phase momentum advection of a spherical droplet with density ratios ranging in $[1, 10^4]$. Numerical stability is demonstrated for in terms of the relative $L_\infty$ velocity error norm, for density-ratios in the range of $[1, 10^4]$, dynamic viscosity-ratios in the range of $[1, 10^4]$ and very strong surface tension forces, for challenging mercury/air and water/air fluid pairings. In addition, the solver performs well in cases characterized by strong interaction between two phases, i.e., oscillating droplets and rising bubbles. The proposed $ρ$LENT method is applicable to any other two-phase flow simulation method that discretizes the single-field two-phase Navier-Stokes Equations using the collocated unstructured Finite Volume Method but does not solve an advection equation for the phase indicator using a flux-based approach, by adding the proposed geometrical approximation of the mass flux and the auxiliary mass conservation equation to the solution algorithm.

physics.flu-dyn↗

arXiv2023

An analytical study of capillary rise dynamics: Critical conditions and hidden oscillations

The rise of a liquid column inside a thin capillary against the action of gravity is a prototypical example of a dynamic wetting process and plays an important role for applications but also for fundamental research in the area of multiphase fluid dynamics. Since the pioneering work by Lucas and Washburn, many research articles have been published which aim at a simplified description of the capillary rise dynamics using complexity-reduced models formulated as ordinary differential equations. Despite the fact that these models are based on profound simplifications, they may still be able to describe the essential physical mechanisms and their interplay. In this study, we focus on the phenomenon of oscillations of the liquid column. The latter has been observed experimentally for liquids with sufficiently small viscosity leading to comparably small viscous dissipation. Back in 1999, Quere et al. formulated a condition for the appearance of rise height oscillations for an ODE model introduced by Bosanquet in 1923. This model has later been extended to include further dissipative mechanisms. In this work, we extend the mathematical analysis to a larger class of models including additional channels of dissipation. We show that Quere's critical condition is generalized to $Ω+ β< 2$, where $Ω$ was introduced earlier and $β$ is an additional non-dimensional parameter describing, e.g., contact line friction. A quantitative prediction of the oscillation dynamics is achieved from a linearization of the governing equations. We apply the theory to experimental data by Quere et al. and, in particular, reveal the oscillatory behavior of dynamics for the nearly critically damped case of ethanol.

physics.flu-dyn↗