Abstract. In this note we prove the existence of an inertial manifold, i.e., a global invariant, exponentially attracting, finite-dimensional smooth manifold, for two different sub-grid scale α-models of turbulence: the simplified Bardina model and the modified Leray-α model, in two-dimensional space. That is, we show the existence of an exact rule that parameterizes the dynamics of small spatial scales in terms of the dynamics of the large ones. In particular, this implies that the long-time dynamics of these turbulence models is equivalent to that of a finite-dimensional system of ordinary differential equations.MSC Classification: 35Q30, 37L30, 76BO3, 76D03, 76F20, 76F55, 76F65

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Using the standard weak-field approximation, we analyse the steady-state electrokinetic flow about an uncharged ideally polarizable spherical particle for the case of a Debye thickness which is large compared with the particle size. The dimensionless problem is governed by two parameters: β, the applied field magnitude (normalized with the thermal scale), and λ, the Debye thickness (normalized with particle size). The double limit β 1 and λ 1 is singular, and the resolution of the flow field requires the use of inner-outer asymptotic expansions in the spirit of Proudman & Pearson (J. Fluid Mech., vol. 2, 1957, p. 237). Two asymptotic limits are identified: the 'moderately thick' limit βλ 1, in which the outer domain is characterized by the Debye thickness, and the 'super-thick' limit βλ 1, in which the outer domain represents the emergence of electro-migration in the leading-order ionic-transport process. The singularity is stronger in the comparable two-dimensional flow about a circular cylinder, where a switchback mechanism in the moderately thick limit modifies the familiar O(β 2 ) leading-order flow to O(β 2 ln λ).

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