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1. Asymmetric thermocapillarity-based pump: Concept and exactly solved model

   https://doi.org/10.1103/PhysRevFluids.8.094201  

   Crowdy, Darren; Mayer, Michael; Hodes, Marc (September 2023, Physical Review Fluids)

   Lauga, Eric; McKeon, Beverly (Ed.)

   A mechanism for a microfluidic pump that leverages alternating adverse and favorable thermocapillary stresses along menisci in a (periodically) fully developed transverse flow in a microchannel is exemplified. The transverse ridges are the interdigitated teeth of cold and hot (isothermal) “combs” and free surface menisci span the interstitial regions between them. The teeth are asymmetrically positioned so that the widths of adjacent menisci differ. This architecture is essentially that of the theoretical pump proposed by Adjari [Phys. Rev. E 61, R45(R) (2000)] but exploits thermocapillarity rather than electro-osmotic slip to drive unidirectional pumping. A theoretical model of the multiphysics pumping mechanism is given that is solved in closed form. Two explicit formulas for the pumping speed are provided. One is derived from the exact solution to the full problem; the other follows from the reciprocal theorem for Stokes flow combined with an exact solution to a distinct problem resolving apparent slip over superhydrophobic surfaces [D. G. Crowdy, Phys. Fluids 23, 072001 (2011)]. A conceptual design of the pump is also outlined; this involves no moving parts, requires no external driving pressure, and pumps a continuous stream of liquid through a microchannel, as opposed to a series of discrete droplets. Since there is only a periodic component of the pressure field the microchannel could be made arbitrarily long and the menisci, which would be essentially flat, are more robust than for conventional pressure-driven flow. 

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2. Unsteady Incompressible Thermal Laminar Boundary Layers Subject to Streamwise Pressure Gradients

   Ramirez M. and Araya G. (September 2023, Proceedings of the 21st International Conference of Numerical Analysis and Applied Mathematics)

   This paper focuses on the laminar boundary layer startup process (momentum and thermal) in incompressible flows. The unsteady boundary layer equations can be solved via similarity analysis by normalizing the stream-wise (x), wall-normal (y) and time (t) coordinates by a variable η and τ, respectively. The resulting ODEs are solved by a finite difference explicit algorithm. This can be done for two cases: flat plate flow where the change in pressure are zero (Blasius solution) and wedge or Falkner-Skan flow where the changes in pressure can be favorable (FPG) or adverse (APG). In addition, transient passive scalar transport is examined by setting several Prandtl numbers in the governing equation at two different wall thermal conditions: isothermal and isoflux. Numerical solutions for the transient evolution of the momentum and thermal boundary layer profiles are compared with analytical approximations for both small times (unsteady flow) and large (steady-state flow) times. 

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3. Stability Analysis of Unsteady Laminar Boundary Layers Subject to Streamwise Pressure Gradient

   https://doi.org/10.3390/fluids10040100  

   Ramirez, Miguel; Araya, Guillermo (April 2025, Fluids)

   A transient stability flow analysis is performed using the unsteady laminar boundary layer equations. The flow dynamics are studied via the Navier–Stokes equations. In the case of external spatially developing flow, the differential equations are reduced via Prandtl or boundary-layer assumptions, consisting of continuity and momentum conservation equations. Prescription of streamwise pressure gradients (decelerating and accelerating flows) is carried out by an impulsively started Falkner–Skan (FS) or wedge-flow similarity flow solution in the case of flat plate or a Blasius solution for particular zero-pressure gradient case. The obtained mean streamwise velocity and its derivatives from FS flows are then inserted into the well-known Orr–Sommerfeld equation of small disturbances at different dimensionless times (τ). Finally, the corresponding eigenvalues are dynamically computed for temporal stability analysis. A finite difference algorithm is effectively applied to solve the Orr–Sommerfeld equations. It is observed that flow acceleration or favorable pressure gradients (FPGs) lead to a significantly shorter transient period before reaching steady-state conditions, as the developed shear layer is notably thinner compared to cases with adverse pressure gradients (APGs). During the transient phase (i.e., for τ<1), the majority of the flow modifications are confined to the innermost 20–25% of the boundary layer, in proximity to the wall. In the context of temporal flow stability, the magnitude of the pressure gradient is pivotal in determining the streamwise extent of the Tollmien–Schlichting (TS) waves. In highly accelerated laminar flows, these waves experience considerable elongation. Conversely, under the influence of a strong adverse pressure gradient, the characteristic streamwise length of the smallest unstable wavelength, which is necessary for destabilization via TS waves, is significantly reduced. Furthermore, flows subjected to acceleration (β > 0) exhibit a higher propensity to transition towards a more stable state during the initial transient phase. For instance, the time response required to reach the steady-state critical Reynolds number was approximately 1τ for β = 0.18 (FPG) and τ = 6.8 for β = −0.18 (APG). 

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4. Nonlinear inviscid damping near monotonic shear flows

   https://doi.org/10.4310/ACTA.2023.v230.n2.a2  

   Ionescu, Alexandru D.; Jia, Hao (July 2023, Acta Mathematica)

   Tobias Ekholm (Ed.)

   We prove nonlinear asymptotic stability of a large class of monotonic shear flows among solutions of the 2D Euler equations in the channel $$\mathbb{T}\times[0,1]$$. More precisely, we consider shear flows $(b(y),0)$ given by a function $$b$$ which is Gevrey smooth, strictly increasing, and linear outside a compact subset of the interval $(0,1)$ (to avoid boundary contributions which are incompatible with inviscid damping). We also assume that the associated linearized operator satisfies a suitable spectral condition, which is needed to prove linear inviscid damping. Under these assumptions, we show that if $$u$$ is a solution which is a small and Gevrey smooth perturbation of such a shear flow $(b(y),0)$ at time $t=0$, then the velocity field $$u$$ converges strongly to a nearby shear flow as the time goes to infinity. This is the first nonlinear asymptotic stability result for Euler equations around general steady solutions for which the linearized flow cannot be explicitly solved. 

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5. Deep learning closure models for large-eddy simulation of flows around bluff bodies

   https://doi.org/10.1017/jfm.2023.446  

   Sirignano, Justin; MacArt, Jonathan F. (July 2023, Journal of Fluid Mechanics)

   Near-wall flow simulation remains a central challenge in aerodynamics modelling: Reynolds-averaged Navier–Stokes predictions of separated flows are often inaccurate, and large-eddy simulation (LES) can require prohibitively small near-wall mesh sizes. A deep learning (DL) closure model for LES is developed by introducing untrained neural networks into the governing equations and training in situ for incompressible flows around rectangular prisms at moderate Reynolds numbers. The DL-LES models are trained using adjoint partial differential equation (PDE) optimization methods to match, as closely as possible, direct numerical simulation (DNS) data. They are then evaluated out-of-sample – for aspect ratios, Reynolds numbers and bluff-body geometries not included in the training data – and compared with standard LES models. The DL-LES models outperform these models and are able to achieve accurate LES predictions on a relatively coarse mesh (downsampled from the DNS mesh by factors of four or eight in each Cartesian direction). We study the accuracy of the DL-LES model for predicting the drag coefficient, near-wall and far-field mean flow, and resolved Reynolds stress. A crucial challenge is that the LES quantities of interest are the steady-state flow statistics; for example, a time-averaged velocity component $$\langle {u}_i\rangle (x) = \lim _{t \rightarrow \infty } ({1}/{t}) \int _0^t u_i(s,x)\, {\rm d}s$$ . Calculating the steady-state flow statistics therefore requires simulating the DL-LES equations over a large number of flow times through the domain. It is a non-trivial question whether an unsteady PDE model with a functional form defined by a deep neural network can remain stable and accurate on $$t \in [0, \infty )$$ , especially when trained over comparatively short time intervals. Our results demonstrate that the DL-LES models are accurate and stable over long time horizons, which enables the estimation of the steady-state mean velocity, fluctuations and drag coefficient of turbulent flows around bluff bodies relevant to aerodynamics applications. 

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