scholarly journals From diffusive mass transfer in Stokes flow to low Reynolds number Marangoni boats

2021 ◽  
Vol 44 (1) ◽  
Author(s):  
Hendrik Ender ◽  
Jan Kierfeld
Keyword(s):  
Symmetry Breaking ◽  
Boundary Conditions ◽  
Spontaneous Symmetry Breaking ◽  
Stokes Flow ◽  
Peclet Number ◽  
Swimming Velocity ◽  
Marangoni Flow ◽  
Péclet Number ◽  
Constant Flux ◽  
Flux Boundary Conditions

AbstractWe present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air–water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion–advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air–water–swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry.

scholarly journals Surfactant-loaded capsules as Marangoni microswimmers at the air–water interface: Symmetry breaking and spontaneous propulsion by surfactant diffusion and advection

2021 ◽  
Vol 44 (2) ◽  
Author(s):  
Hendrik Ender ◽  
Ann-Kathrin Froin ◽  
Heinz Rehage ◽  
Jan Kierfeld
Keyword(s):  
Symmetry Breaking ◽  
Peclet Number ◽  
Reynolds Numbers ◽  
Water Soluble ◽  
Swimming Velocity ◽  
Water Interface ◽  
Péclet Number ◽  
Air Interface ◽  
Alginate Capsules ◽  
Air Water Interface

Abstract We present a realization of a fast interfacial Marangoni microswimmer by a half-spherical alginate capsule at the air–water interface, which diffusively releases water-soluble spreading molecules (weak surfactants such as polyethylene glycol (PEG)), which act as “fuel” by modulating the air–water interfacial tension. For a number of different fuels, we can observe symmetry breaking and spontaneous propulsion although the alginate particle and emission are isotropic. The propulsion mechanism is similar to soap or camphor boats, which are, however, typically asymmetric in shape or emission to select a swimming direction. We develop a theory of Marangoni boat propulsion starting from low Reynolds numbers by analyzing the coupled problems of surfactant diffusion and advection and fluid flow, which includes surfactant-induced fluid Marangoni flow, and surfactant adsorption at the air–water interface; we also include a possible evaporation of surfactant. The swimming velocity is determined by the balance of drag and Marangoni forces. We show that spontaneous symmetry breaking resulting in propulsion is possible above a critical dimensionless surfactant emission rate (Peclet number). We derive the relation between Peclet number and swimming speed and generalize to higher Reynolds numbers utilizing the concept of the Nusselt number. The theory explains the observed swimming speeds for PEG–alginate capsules, and we unravel the differences to other Marangoni boat systems based on camphor, which are mainly caused by surfactant evaporation from the liquid–air interface. The capsule Marangoni microswimmers also exhibit surfactant-mediated repulsive interactions with walls, which can be qualitatively explained by surfactant accumulation at the wall. Graphic Abstract

scholarly journals Heat or mass transfer from a sphere in Stokes flow at low Péclet number

2013 ◽  
Vol 26 (4) ◽  
pp. 392-396 ◽  
Author(s):  
Christopher G. Bell ◽  
Helen M. Byrne ◽  
Jonathan P. Whiteley ◽  
Sarah L. Waters
Keyword(s):  
Mass Transfer ◽  
Stokes Flow ◽  
Peclet Number ◽  
Péclet Number

ON THE EFFECTS OF THE NEUMANN BOUNDARY CONDITIONS IN THE COLEMAN–WEINBERG MECHANISM

2010 ◽  
Vol 25 (07) ◽  
pp. 1389-1403 ◽  
Author(s):  
F. N. FAGUNDES ◽  
T. L. ANTONACCI OAKES ◽  
B. B. DILEM ◽  
J. A. NOGUEIRA
Keyword(s):  
Vector Field ◽  
Scalar Field ◽  
Symmetry Breaking ◽  
Boundary Conditions ◽  
Spontaneous Symmetry Breaking ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Finite Region

We investigate the effects of the homogeneous Neumann boundary conditions in the scalar electrodynamics with self-interaction. We show that if the length of the finite region is small enough ([Formula: see text], where Mϕ is the mass of the scalar field generated by the Coleman–Weinberg mechanism) the spontaneous symmetry breaking will not be induced and the vector field will not develop mass, however the scalar field will.

The Different Nature of the Interdecadal Variability of the Thermohaline Circulation under Mixed and Flux Boundary Conditions

2006 ◽  
Vol 36 (9) ◽  
pp. 1703-1718 ◽  
Author(s):  
Olivier Arzel ◽  
Thierry Huck ◽  
Alain Colin de Verdière
Keyword(s):  
Heat Flux ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Interdecadal Variability ◽  
Flux Boundary Condition ◽  
Constant Flux ◽  
Eddy Heat Flux ◽  
Flux Boundary Conditions ◽  
Density Variance

Abstract The differences between the interdecadal variability under mixed and constant flux boundary conditions are investigated using a coarse-resolution ocean model in an idealized flat-bottom single-hemisphere basin. Objective features are determined that allow one type of oscillation to be distinguished versus the other. First, by performing a linear stability analysis of the steady state obtained under restoring boundary conditions, it is shown that the interdecadal variability under constant flux and mixed boundary conditions arises, respectively, from the instability of a linear mode around the mean stratification and circulation and from departure from the initial state. Based on the budgets of density variance, it is shown next that the two types of oscillations have different energy sources: Under the constant-flux boundary condition (the thermal mode), the downgradient meridional eddy heat flux in the western boundary current regions sustains interdecadal variability, whereas under mixed boundary conditions (the salinity mode), a positive feedback between convective adjustment and restoring surface heat flux is at the heart of the existence of the decadal oscillation. Furthermore, the positive correlations between temperature and salinity anomalies in the forcing layer are shown to dominate the forcing of density variance. In addition, the vertical structure of perturbations reveals vertical phase lags at different depths in all tracer fields under constant flux, while under mixed boundary conditions only the temperature anomalies show a strong dipolar structure. The authors propose that these differences will allow one to identify which type of oscillation, if any, is at play in the more exhaustive climate models.

scholarly journals Spontaneous symmetry breaking in free theories with boundary potentials

2021 ◽  
Vol 11 (2) ◽  
Author(s):  
Vladimir Prochazka ◽  
Alexander Söderberg
Keyword(s):  
Fixed Point ◽  
Symmetry Breaking ◽  
Boundary Conditions ◽  
Spontaneous Symmetry Breaking ◽  
Phase Boundary ◽  
Critical Point ◽  
Effective Potential ◽  
Flow Diagram ◽  
Conformal Boundary ◽  
Effective Potentials

Patterns of symmetry breaking induced by potentials at the boundary of free O(N)O(N)-models in d=3- \epsilond=3−ϵ dimensions are studied. We show that the spontaneous symmetry breaking in these theories leads to a boundary RG flow ending with N - 1N−1 Neumann modes in the IR. The possibility of fluctuation-induced symmetry breaking is examined and we derive a general formula for computing one-loop effective potentials at the boundary. Using the \epsilonϵ-expansion we test these ideas in an O(N)\oplus O(N)O(N)⊕O(N)-model with boundary interactions. We determine the RG flow diagram of this theory and find that it has an IR-stable critical point satisfying conformal boundary conditions. The leading correction to the effective potential is computed and we argue the existence of a phase boundary separating the region flowing to the symmetric fixed point from the region flowing to a symmetry-broken phase with a combination of Neumann and Dirchlet boundary conditions.

Forced Heat and Mass Transfer From a Slightly Deformed Sphere at Small but Finite Peclet Numbers in Stokes Flow

2013 ◽  
Vol 135 (8) ◽  
Author(s):  
Zhi-Gang Feng
Keyword(s):  
Mass Transfer ◽  
Nusselt Number ◽  
Singular Perturbation ◽  
Perturbation Method ◽  
Heat And Mass Transfer ◽  
Stokes Flow ◽  
Peclet Number ◽  
Fundamental Problem ◽  
Singular Perturbation Method ◽  
Péclet Number

The fundamental problem of heat and mass transfer from a slightly deformed sphere at low but finite Peclet numbers in Stokes flow is solved by a combined regular and singular perturbation method. The deformed sphere is assumed to be axisymmetric and its shape is described by a power series in a small parameter; the correction to the Nusselt number due to the deformation of the sphere is obtained through a regular perturbation with respect to this parameter. On the contrary, the correction to the Nusselt number due to the small Peclet number is derived by applying a singular perturbation method. The analytical solution is derived for the averaged Nusselt number in terms of the Peclet number and the deformation parameter.

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