Using lubrication theory, a model to investigate spreading of solidifying melts is presented. We ... more Using lubrication theory, a model to investigate spreading of solidifying melts is presented. We concentrate on crust-formation at the free interface between melt and ambient gas. Crust-formation leads to a noslip condition between liquid melt and solid crust, whereas the problem without crusts has a free-slip condition at the free interface. This leads to a change in the kinematic boundary conditions and can have a significant influence on the spreading flow. Results show that crust-formation can increase or decrease the front velocity from what is expected in the case without crusts.
Proceedings in applied mathematics & mechanics, Oct 1, 2015
This article deals with a liquid displacing a gas in a thin heterogeneous porous material, which ... more This article deals with a liquid displacing a gas in a thin heterogeneous porous material, which occurs e.g. during the filling process of a lithium-ion battery with an electrolyte. The investigation is based upon the local volume-averaged Navier-Stokes equations, using a Volume-of-Fluid method to treat the interface. For the flow the wall effect and capillary forces have to be considered. Capillary rise experiments are used to determine the permeability. Since the layers are thin and the characteristic size of the particles is comparatively large, friction with the electrode is taken into account with respect to the mobility of the contact line. The implemented models are validated against analytical results, showing a good agreement.
Für die Ausbreitung eines dünnen rotierenden Tropfens wird, unter Ausnutzung der Schmierfilmappro... more Für die Ausbreitung eines dünnen rotierenden Tropfens wird, unter Ausnutzung der Schmierfilmapproximation, eine Entwicklungsgleichung zur Beschreibung der Position der freien Grenzfläche abgeleitet. Die dynamische Benetzung wird mit dem empirischen Tanner'schen Gesetz beschrieben und die Modellierung mit Experimenten aus der Literatur verglichen. Eine lineare Störungsanalyse liefert die instabilen Moden und das Einsetzen von Fingerbildung während der Ausbreitung. Betrachtet wird auch Konvektion aufgrund der temperaturabhängigen Oberflächenspannung und Dichte sowie deren Stabilität.
Proceedings in applied mathematics & mechanics, Dec 1, 2008
Disperse gas bubbles play an important role in many industrial applications. Knowing the rising v... more Disperse gas bubbles play an important role in many industrial applications. Knowing the rising velocity, the interfacial area, or the critical size for break-up or coalescence in different systems can be crucial for the process design. Hence, knowing the fundamental behaviour of a single bubble appears mandatory for the examination of bubble swarms and for the Euler-Lagrange or Euler-Euler modelling of disperse systems. In the present work a level-set-based volume-tracking method is implemented into the CFD-code OpenFOAM to follow the free interface of a single bubble. The volume-tracking method is coupled with a transport model for surfactants on the interface, including adsorption and desorption processes. The dependency of surface tension on the local surfactant concentration on the interface is modelled by a non-linear (Langmuir) equation of state. Marangoni forces, resulting from surface tension gradients, are included. The rise of a single air bubble (i) in pure water and (ii) in the presence of surfactants of different strengths is simulated. The results show good agreement with available correlations from literature.
We investigate the influence of internal electrodes onto the flow field, governed by electroosmos... more We investigate the influence of internal electrodes onto the flow field, governed by electroosmosis and electrophoresis in a modular rectangular microchannel. As internal electrodes can be positioned at lower distances, they can be operated at lower voltages and still ensure strong electrical field strength. Even at lower voltages, electrode reactions influence the species concentration fields, and the crucial question arises, whether at the electrodes all species can be kept in dissolution or whether some species are released in gaseous form. The position and charge of multiple internal electrodes is a further focus of our investigations: wall-tangential electrical field components are responsible for pumping, wall-normal electrical field components are responsible for mixing. Hence, an optimized position and charge of all electrodes will lead to an optimized electrical field, designed to fulfill the desired tasks of the modular microchannel. The mathematical model for the numerical treatment relies on a first-principle description of the EDL and the electrical forces caused by the electrical field between the internal electrodes. Hence, the so-called Debye-Hu¨ckel approximation is avoided. The governing system of equations consists of a Poisson equation for the electrical potential, the continuity and Navier-Stokes equations for the flow field, species transport equations, based on the Nernst-Planck equation, and a charge transport equation. Further, a model for the electrode reactions, based on the Butler-Volmer equation, is in place. The simulations are time-dependent and two-dimensional in nature and employ a FVM.
We investigate the electrokinetic flow and mass transport in a microchannel junction, serving as ... more We investigate the electrokinetic flow and mass transport in a microchannel junction, serving as an injector of a microelectrophoresis device. In order to consider all essential features of this complex system, the electrical situation, the fluid dynamics, and the (physical) chemistry is ...
Proceedings in applied mathematics & mechanics, Dec 1, 2011
Thin-film flows are involved in many coating processes, where it is desirable to achieve thin and... more Thin-film flows are involved in many coating processes, where it is desirable to achieve thin and homogeneous fluid layers. In the present investigations, we treat droplets, spreading on rotating solid substrates. Micro-scale effects appear, firstly, at the wetting front, where the film height tends to zero. Secondly, micro-scale effects may appear at other locations, where the free liquid/gas-interface approaches the solid substrate, as e.g. at film rupture. For such situations, molecular effects need to be considered, e.g. in form of the disjoining pressure (DJP), to get physically-correct solutions. Otherwise, the spreading can be modeled within the frame of continuum mechanics, augmented by the (empirical) law of Tanner to capture the contact-line dynamics. We present, on the one hand, an overview of several interesting issues, as (i) spreading with and without considering the DJP, (ii) spreading after central rupture, including hysteresis effects, and (iii) non-isothermal spreading, including temperaturedependent surface tension (Marangoni effect) and temperature-dependent density (Rayleigh-Bénard effect). On the other hand, we present results for the instability of the contact line, for which the contact line gets corrugated (under isothermal conditions). This instability goes along with a transition from (rotationally-symmetric) two-dimensional to three-dimensional behavior.
Bulletin of the American Physical Society, Nov 22, 2005
Submitted for the DFD05 Meeting of The American Physical Society Countering capillary pressure wi... more Submitted for the DFD05 Meeting of The American Physical Society Countering capillary pressure with electroosmotic pressure at small scales PAUL STEEN, MICHAEL VOGEL, Cornell University, PETER EHRHARD, IKET, FzK Germany-Electroosmosis, originating in the doublelayer of a small liquid-filled pore (size R) and driven by a voltage V , is shown to be effective in pumping liquid against the capillary pressure of a larger liquid droplet (size B) provided the dimensionless parameter R 2 σ/ε|ζ|V B is small enough. Here σ is surface tension of the droplet liquid/gas interface, ε the liquid dielectric constant, and ζ the zeta potential of the solid/liquid pair. As droplet size diminishes, the voltage required to pump eletroosmotically scales as V ∼ R 2 /B. Accordingly, the voltage needed to pump against smaller higher-pressure droplets can actually decrease provided the pump pore-size scales down with droplet size appropriately. In this talk, we shall focus on the electroosmotic droplet-switch, two droplets coupled by an electroosmotic pump. For millimeter-size droplets and micron-size pores, 5 volts yields switching times under 5 seconds in experiment. The down-scaling of this voltage and switching-time are of interest.
Proceedings in applied mathematics & mechanics, Dec 1, 2014
Particles that are placed in a laminar pipe flow rotate and migrate transversally to a radial equ... more Particles that are placed in a laminar pipe flow rotate and migrate transversally to a radial equilibrium position. This so called Segré-Silberberg effect is used in a new method for size separation of particles. The particles to be separated are placed in a pipe flow and subsequently enter an expansion chamber, where the flow is split and the particles are divided into two fractions. This paper reports the results of two-dimensional Euler-Lagrange simulations of the motion of neutrally-buoyant particles inside the expansion chamber. The simulation results agree well with experimental data on the separation and show that the Saffman force has a significant impact onto the particle trajectories.
Proceedings in applied mathematics & mechanics, 2021
We investigate how the walls of cylindrical capillaries affect the velocity of rising gas bubbles... more We investigate how the walls of cylindrical capillaries affect the velocity of rising gas bubbles of various diameters. Of course, as the capillary diameter increases, the velocity of the rising bubble will approach the case of free rising. Such systematic experiments on bubble rise in capillaries, in which the ratio of bubble diameter and capillary diameter is varied from one towards smaller values, can hardly be found in literature. First orienting experiments within the system water/air have been conducted and will be discussed in this paper.
Proceedings in applied mathematics & mechanics, Nov 1, 2019
The beads in a packed bed are usually distributed randomly. Therefore, on a macroscopic scale, a ... more The beads in a packed bed are usually distributed randomly. Therefore, on a macroscopic scale, a fluid flows uniformly through such a porous medium. For thin porous media, deviations are known near the regions of confinement. This so-called wall effect is extensively investigated in literature for circular confinements. In this work, the wall effect and the flow through porous media in a rectangular domain is studied. This has great importance during the initial filling process of lithium-ion batteries with the liquid electrolyte. Such batteries consist of collector foils, on which the porous electrodes are attached. Their thickness is only 5-10 times a particle diameter dP and may be modeled as a monodisperse bed of spheres. The filling flow is extremely slow and may be in the pre-Darcy flow region, wherefore the validity of Darcy's law is experimentally checked. Then, the local porosity function depending on the distance to the wall is experimentally determined. For very thin porous media, the wall effect of each wall may overlap each other, which is taken into account by using an overlapping model from the literature. The widely used model of Carman-Kozeny [1] relates the permeability with the porosity. Its validity in the near-wall region is checked by comparing experimentally-obtained mean pore diameters to those related to the model of Carman-Kozeny. The flow through a porous media is computed and compared to experimental results from literature for circular confinement and preliminary optically-matched PIV measurements.
Proceedings in applied mathematics & mechanics, Dec 1, 2009
We investigate the influence of flow field and electrode reactions onto an electrical double laye... more We investigate the influence of flow field and electrode reactions onto an electrical double layer (EDL), which is located in the immediate vicinity of the walls of a rectangular microchannel. The precise knowledge of the EDL appears to be important for many technical applications in microchannels of small width, since the electrokinetic effects, as electroosmosis or electrophoresis, in such cases depend on the detailed charge distribution. Furthermore, the influence of internal electrodes onto the flow field is of interest. The mathematical model for the numerical treatment relies on a first-principle description of the EDL and the electrical forces, caused by the electrical field between internal electrodes. Hence, the so-called Debye-Hueckel approximation is avoided. The simulations are time-dependent and two-dimensional (plane) in nature and employ a finite-volume method.
Proceedings in applied mathematics & mechanics, Dec 1, 2006
The rapid mixing of small quantities, with the goal of reducing analysis and reaction times, is o... more The rapid mixing of small quantities, with the goal of reducing analysis and reaction times, is one of the key steps within innovative miniaturized total analysis systems (µTAS). To achieve wellmixed liquids with homogenous properties within an acceptable process time, the ...
Using lubrication theory, a model to investigate spreading of solidifying melts is presented. We ... more Using lubrication theory, a model to investigate spreading of solidifying melts is presented. We concentrate on crust-formation at the free interface between melt and ambient gas. Crust-formation leads to a noslip condition between liquid melt and solid crust, whereas the problem without crusts has a free-slip condition at the free interface. This leads to a change in the kinematic boundary conditions and can have a significant influence on the spreading flow. Results show that crust-formation can increase or decrease the front velocity from what is expected in the case without crusts.
Proceedings in applied mathematics & mechanics, Oct 1, 2015
This article deals with a liquid displacing a gas in a thin heterogeneous porous material, which ... more This article deals with a liquid displacing a gas in a thin heterogeneous porous material, which occurs e.g. during the filling process of a lithium-ion battery with an electrolyte. The investigation is based upon the local volume-averaged Navier-Stokes equations, using a Volume-of-Fluid method to treat the interface. For the flow the wall effect and capillary forces have to be considered. Capillary rise experiments are used to determine the permeability. Since the layers are thin and the characteristic size of the particles is comparatively large, friction with the electrode is taken into account with respect to the mobility of the contact line. The implemented models are validated against analytical results, showing a good agreement.
Für die Ausbreitung eines dünnen rotierenden Tropfens wird, unter Ausnutzung der Schmierfilmappro... more Für die Ausbreitung eines dünnen rotierenden Tropfens wird, unter Ausnutzung der Schmierfilmapproximation, eine Entwicklungsgleichung zur Beschreibung der Position der freien Grenzfläche abgeleitet. Die dynamische Benetzung wird mit dem empirischen Tanner'schen Gesetz beschrieben und die Modellierung mit Experimenten aus der Literatur verglichen. Eine lineare Störungsanalyse liefert die instabilen Moden und das Einsetzen von Fingerbildung während der Ausbreitung. Betrachtet wird auch Konvektion aufgrund der temperaturabhängigen Oberflächenspannung und Dichte sowie deren Stabilität.
Proceedings in applied mathematics & mechanics, Dec 1, 2008
Disperse gas bubbles play an important role in many industrial applications. Knowing the rising v... more Disperse gas bubbles play an important role in many industrial applications. Knowing the rising velocity, the interfacial area, or the critical size for break-up or coalescence in different systems can be crucial for the process design. Hence, knowing the fundamental behaviour of a single bubble appears mandatory for the examination of bubble swarms and for the Euler-Lagrange or Euler-Euler modelling of disperse systems. In the present work a level-set-based volume-tracking method is implemented into the CFD-code OpenFOAM to follow the free interface of a single bubble. The volume-tracking method is coupled with a transport model for surfactants on the interface, including adsorption and desorption processes. The dependency of surface tension on the local surfactant concentration on the interface is modelled by a non-linear (Langmuir) equation of state. Marangoni forces, resulting from surface tension gradients, are included. The rise of a single air bubble (i) in pure water and (ii) in the presence of surfactants of different strengths is simulated. The results show good agreement with available correlations from literature.
We investigate the influence of internal electrodes onto the flow field, governed by electroosmos... more We investigate the influence of internal electrodes onto the flow field, governed by electroosmosis and electrophoresis in a modular rectangular microchannel. As internal electrodes can be positioned at lower distances, they can be operated at lower voltages and still ensure strong electrical field strength. Even at lower voltages, electrode reactions influence the species concentration fields, and the crucial question arises, whether at the electrodes all species can be kept in dissolution or whether some species are released in gaseous form. The position and charge of multiple internal electrodes is a further focus of our investigations: wall-tangential electrical field components are responsible for pumping, wall-normal electrical field components are responsible for mixing. Hence, an optimized position and charge of all electrodes will lead to an optimized electrical field, designed to fulfill the desired tasks of the modular microchannel. The mathematical model for the numerical treatment relies on a first-principle description of the EDL and the electrical forces caused by the electrical field between the internal electrodes. Hence, the so-called Debye-Hu¨ckel approximation is avoided. The governing system of equations consists of a Poisson equation for the electrical potential, the continuity and Navier-Stokes equations for the flow field, species transport equations, based on the Nernst-Planck equation, and a charge transport equation. Further, a model for the electrode reactions, based on the Butler-Volmer equation, is in place. The simulations are time-dependent and two-dimensional in nature and employ a FVM.
We investigate the electrokinetic flow and mass transport in a microchannel junction, serving as ... more We investigate the electrokinetic flow and mass transport in a microchannel junction, serving as an injector of a microelectrophoresis device. In order to consider all essential features of this complex system, the electrical situation, the fluid dynamics, and the (physical) chemistry is ...
Proceedings in applied mathematics & mechanics, Dec 1, 2011
Thin-film flows are involved in many coating processes, where it is desirable to achieve thin and... more Thin-film flows are involved in many coating processes, where it is desirable to achieve thin and homogeneous fluid layers. In the present investigations, we treat droplets, spreading on rotating solid substrates. Micro-scale effects appear, firstly, at the wetting front, where the film height tends to zero. Secondly, micro-scale effects may appear at other locations, where the free liquid/gas-interface approaches the solid substrate, as e.g. at film rupture. For such situations, molecular effects need to be considered, e.g. in form of the disjoining pressure (DJP), to get physically-correct solutions. Otherwise, the spreading can be modeled within the frame of continuum mechanics, augmented by the (empirical) law of Tanner to capture the contact-line dynamics. We present, on the one hand, an overview of several interesting issues, as (i) spreading with and without considering the DJP, (ii) spreading after central rupture, including hysteresis effects, and (iii) non-isothermal spreading, including temperaturedependent surface tension (Marangoni effect) and temperature-dependent density (Rayleigh-Bénard effect). On the other hand, we present results for the instability of the contact line, for which the contact line gets corrugated (under isothermal conditions). This instability goes along with a transition from (rotationally-symmetric) two-dimensional to three-dimensional behavior.
Bulletin of the American Physical Society, Nov 22, 2005
Submitted for the DFD05 Meeting of The American Physical Society Countering capillary pressure wi... more Submitted for the DFD05 Meeting of The American Physical Society Countering capillary pressure with electroosmotic pressure at small scales PAUL STEEN, MICHAEL VOGEL, Cornell University, PETER EHRHARD, IKET, FzK Germany-Electroosmosis, originating in the doublelayer of a small liquid-filled pore (size R) and driven by a voltage V , is shown to be effective in pumping liquid against the capillary pressure of a larger liquid droplet (size B) provided the dimensionless parameter R 2 σ/ε|ζ|V B is small enough. Here σ is surface tension of the droplet liquid/gas interface, ε the liquid dielectric constant, and ζ the zeta potential of the solid/liquid pair. As droplet size diminishes, the voltage required to pump eletroosmotically scales as V ∼ R 2 /B. Accordingly, the voltage needed to pump against smaller higher-pressure droplets can actually decrease provided the pump pore-size scales down with droplet size appropriately. In this talk, we shall focus on the electroosmotic droplet-switch, two droplets coupled by an electroosmotic pump. For millimeter-size droplets and micron-size pores, 5 volts yields switching times under 5 seconds in experiment. The down-scaling of this voltage and switching-time are of interest.
Proceedings in applied mathematics & mechanics, Dec 1, 2014
Particles that are placed in a laminar pipe flow rotate and migrate transversally to a radial equ... more Particles that are placed in a laminar pipe flow rotate and migrate transversally to a radial equilibrium position. This so called Segré-Silberberg effect is used in a new method for size separation of particles. The particles to be separated are placed in a pipe flow and subsequently enter an expansion chamber, where the flow is split and the particles are divided into two fractions. This paper reports the results of two-dimensional Euler-Lagrange simulations of the motion of neutrally-buoyant particles inside the expansion chamber. The simulation results agree well with experimental data on the separation and show that the Saffman force has a significant impact onto the particle trajectories.
Proceedings in applied mathematics & mechanics, 2021
We investigate how the walls of cylindrical capillaries affect the velocity of rising gas bubbles... more We investigate how the walls of cylindrical capillaries affect the velocity of rising gas bubbles of various diameters. Of course, as the capillary diameter increases, the velocity of the rising bubble will approach the case of free rising. Such systematic experiments on bubble rise in capillaries, in which the ratio of bubble diameter and capillary diameter is varied from one towards smaller values, can hardly be found in literature. First orienting experiments within the system water/air have been conducted and will be discussed in this paper.
Proceedings in applied mathematics & mechanics, Nov 1, 2019
The beads in a packed bed are usually distributed randomly. Therefore, on a macroscopic scale, a ... more The beads in a packed bed are usually distributed randomly. Therefore, on a macroscopic scale, a fluid flows uniformly through such a porous medium. For thin porous media, deviations are known near the regions of confinement. This so-called wall effect is extensively investigated in literature for circular confinements. In this work, the wall effect and the flow through porous media in a rectangular domain is studied. This has great importance during the initial filling process of lithium-ion batteries with the liquid electrolyte. Such batteries consist of collector foils, on which the porous electrodes are attached. Their thickness is only 5-10 times a particle diameter dP and may be modeled as a monodisperse bed of spheres. The filling flow is extremely slow and may be in the pre-Darcy flow region, wherefore the validity of Darcy's law is experimentally checked. Then, the local porosity function depending on the distance to the wall is experimentally determined. For very thin porous media, the wall effect of each wall may overlap each other, which is taken into account by using an overlapping model from the literature. The widely used model of Carman-Kozeny [1] relates the permeability with the porosity. Its validity in the near-wall region is checked by comparing experimentally-obtained mean pore diameters to those related to the model of Carman-Kozeny. The flow through a porous media is computed and compared to experimental results from literature for circular confinement and preliminary optically-matched PIV measurements.
Proceedings in applied mathematics & mechanics, Dec 1, 2009
We investigate the influence of flow field and electrode reactions onto an electrical double laye... more We investigate the influence of flow field and electrode reactions onto an electrical double layer (EDL), which is located in the immediate vicinity of the walls of a rectangular microchannel. The precise knowledge of the EDL appears to be important for many technical applications in microchannels of small width, since the electrokinetic effects, as electroosmosis or electrophoresis, in such cases depend on the detailed charge distribution. Furthermore, the influence of internal electrodes onto the flow field is of interest. The mathematical model for the numerical treatment relies on a first-principle description of the EDL and the electrical forces, caused by the electrical field between internal electrodes. Hence, the so-called Debye-Hueckel approximation is avoided. The simulations are time-dependent and two-dimensional (plane) in nature and employ a finite-volume method.
Proceedings in applied mathematics & mechanics, Dec 1, 2006
The rapid mixing of small quantities, with the goal of reducing analysis and reaction times, is o... more The rapid mixing of small quantities, with the goal of reducing analysis and reaction times, is one of the key steps within innovative miniaturized total analysis systems (µTAS). To achieve wellmixed liquids with homogenous properties within an acceptable process time, the ...
Uploads
Papers by Peter Ehrhard