Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

Building on our recent work on induced-charge electro-osmosis (ICEO) and electrophoresis (ICEP), as well as the Russian literature on spherical metal colloids, we examine the rich consequences of broken geometric and field symmetries upon the ICEO flow around conducting bodies. Through a variety of paradigmatic examples involving ideally polarizable (e.g. metal) bodies with thin double layers in weak fields, we demonstrate that spatial asymmetry generally leads to a net pumping of fluid past the body by ICEO, or, in the case of a freely suspended colloidal particle, translation and/or rotation by ICEP. We have chosen model systems that are simple enough to admit analysis, yet which contain the most important broken symmetries. Specifically, we consider (i) symmetrically shaped bodies with inhomogeneous surface properties, (ii) ‘nearly symmetric’ shapes (using a boundary perturbation scheme), (iii) highly asymmetric bodies composed of two symmetric bodies tethered together, (iv) symmetric conductors in electric-field gradients, and (v) arbitrarily shaped conductors in general non-uniform fields in two dimensions (using complex analysis). In non-uniform fields, ICEO flow and ICEP motion exist in addition to the more familiar dielectrophoretic forces and torques on the bodies (which also vary with the square of the electric field). We treat all of these problems in two and three dimensions, so our study has relevence for both colloids and microfluidics. In the colloidal context, we describe principles to ‘design’ polarizable particles which rotate to orient themselves and translate steadily in a desired direction in a DC or AC electric field. We also describe ‘ICEO spinners’ that rotate continuously in AC fields of arbitrary direction, although we show that ‘near spheres’ with small helical perturbations do not rotate, to leading order in the shape perturbation. In the microfluidic context, strong and steady flows can be driven by small AC potentials applied to systems containing asymmetric structures, which holds promise for portable or implantable self-powered devices. These results build upon and generalize recent studies in AC electro-osmosis (ACEO). Unlike ACEO, however, the inducing surfaces in ICEO can be physically distinct from the driving electrodes, increasing the frequency range and geometries available.

http://absimage.aps.org/image/DFD05/MWS_DFD05-2005-001488.pdf

Breaking symmetries in induced-charge electro-osmosis and electrophoresis

Elements of Gasdynamics.

Small objects can swim by generating around them fields or gradients which in turn induce fluid motion past their surface by phoretic surface effects. We quantify for arbitrary swimmer shapes and surface patterns, how efficient swimming requires both surface ``activity'' to generate the fields, and surface ``phoretic mobility.'' We show in particular that (i) swimming requires symmetry breaking in either or both of the patterns of "activity" and ``mobility,'' and (ii) for a given geometrical shape and surface pattern, the swimming velocity is size-independent. In addition, for given available surface properties, our calculation framework provides a guide for optimizing the design of swimmers.

Designing phoretic micro- and nano-swimmers

Calculations of the lift and lift-induced drag in steady incompressible high-Reynolds-number flows are proposed in terms of volume integration of the Lamb vector (cross product of the vorticity and the velocity vector). Although conventional far-field methods can compute the profile drag by the volume integral, induced drag is usually obtained by Maskell’s method, which requires a two-dimensional plane cut from three-dimensional data. Using the Lamb vector integration, plane data reconstruction is not necessary, and a definition of the lift-induced drag in viscous flows is obtained. Furthermore, an insight of the drag production is provided by the analysis of the Lamb vector field. Numerical solutions of the flow around arbitrary-shaped bodies can be analyzed by the present method. In particular, the analysis of an elliptic wing is here proposed. The acquired aerodynamic forces are evaluated in comparison with the results of classical methods.

Lift and Lift-Induced Drag Computation by Lamb Vector Integration

Fundamental theories of aerodynamic force in viscous and compressible complex flows

We report our systematic development of a general and exact theory for diagnosis of total force and moment exerted on a generic body moving and deforming in a calorically perfect gas. The total force and moment consist of a longitudinal part (L-force) due to compressibility and irreversible thermodynamics, and a transverse part (T-force) due to shearing. The latter exists in incompressible flow but is now modulated by the former. The theory represents a full extension of a unified incompressible diagnosis theory of the same type developed by J. Z. Wu and coworkers to compressible flow, with Mach number ranging from low-subsonic to moderate-supersonic flows. Combined with computational fluid dynamics (CFD) simulation, the theory permits quantitative identification of various complex flow structures and processes responsible for the forces, and thereby enables rational optimal configuration design and flow control. The theory is confirmed by a numerical simulation of circular-cylinder flow in the range of free-stream Mach number between 0.2 and 2.0. The L-drag and T-drag of the cylinder vary with in different ways, the underlying physical mechanisms of which are analysed. Moreover, each L-force and T-force integrand contains a universal factor of local Mach number . Our preliminary tests suggest that the possibility of finding new similarity rules for each force constituent could be quite promising.

Longitudinal-transverse aerodynamic force in viscous compressible complex flow

This paper studies the lift and drag experienced by a body in a two-dimensional, viscous, compressible and steady flow. By a rigorous linear far-field theory and the Helmholtz decomposition of the velocity field, we prove that the classic lift formula , originally derived by Joukowski in 1906 for inviscid potential flow, and the drag formula , derived for incompressible viscous flow by Filon in 1926, are universally true for the whole field of viscous compressible flow in a wide range of Mach number, from subsonic to supersonic flows. Here, and denote the circulation of the longitudinal velocity component and the inflow of the transverse velocity component, respectively. We call this result the Joukowski–Filon theorem (J–F theorem for short). Thus, the steady lift and drag are always exactly determined by the values of and , no matter how complicated the near-field viscous flow surrounding the body might be. However, velocity potentials are not directly observable either experimentally or computationally, and hence neither are the J–F formulae. Thus, a testable version of the J–F formulae is also derived, which holds only in the linear far field. Due to their linear dependence on the vorticity, these formulae are also valid for statistically stationary flow, including time-averaged turbulent flow. Thus, a careful RANS (Reynolds-averaged Navier–Stokes) simulation is performed to examine the testable version of the J–F formulae for a typical airfoil flow with Reynolds number and free Mach number . The results strongly support and enrich the J–F theorem. The computed Mach-number dependence of and and its underlying physics, as well as the physical implications of the theorem, are also addressed.

Lift and drag in two-dimensional steady viscous and compressible flow

http://iopscience.iop.org/article/10.1088/0169-5983/46/6/061417/pdf

A dynamic counterpart of Lamb vector in viscous compressible aerodynamics

We theoretically and numerically investigate the instabilities driven by diffusiophoretic flow, caused by a solutal concentration gradient along a reacting surface. The important control parameter is the Peclet number Pe, which quantifies the ratio of the solutal advection rate to the diffusion rate. First, we study the diffusiophoretic flow on a catalytic plane in two dimensions. From a linear stability analysis, we obtain that for Pe larger than 8pi, mass transport by convection overtakes that by diffusion, and a symmetry-breaking mode arises, which is consistent with numerical results. For even larger Pe, non-linear terms become important. For Pe larger than 16pi, multiple concentration plumes are emitted from the catalytic plane, which eventually merges into a single larger one. When Pe is even larger, there are continuous emissions and merging events of the concentration plumes. This newly-found flow state reflects the non-linear saturation of the system. The critical Peclet number for the transition to this state depends on Schmidt number Sc. In the second part of the paper, we conduct three-dimensional simulations for spherical catalytic particles, and beyond a critical Peclet number again find continuous plume emission and plume merging, now leading to chaotic motion of the phoretic particle. Our results thus help to understand the experimentally observed chaotic motion of catalytic particles in the high Pe regime.

/pdf/instabilities-driven-by-diffusio-phoretic-flow-on-catalytic-1auervyhik.pdf

Luoqin Liu

Papers

Fundamental theories of aerodynamic force in viscous and compressible complex flows

Longitudinal-transverse aerodynamic force in viscous compressible complex flow

Lift and drag in two-dimensional steady viscous and compressible flow

A dynamic counterpart of Lamb vector in viscous compressible aerodynamics

Instabilities driven by diffusio-phoretic flow on catalytic surfaces