Linear stability of stratified two-phase flows in horizontal channels to arbitrary wavenumber disturbances is studied. The problem is reduced to Orr-Sommerfeld equations for the stream function disturbances, defined in each sublayer and coupled via boundary conditions that account also for possible interface deformation and capillary forces. Applying the Chebyshev collocation method, the equations and interface boundary conditions are reduced to the generalized eigenvalue problems solved by standard means of numerical linear algebra for the entire spectrum of eigenvalues and the associated eigenvectors. Some additional conclusions concerning the instability nature are derived from the most unstable perturbation patterns. The results are summarized in the form of stability maps showing the operational conditions at which a stratified-smooth flow pattern is stable. It is found that for gas-liquid and liquid-liquid systems the stratified flow with a smooth interface is stable only in confined zone of relatively low flow rates, which is in agreement with experiments, but is not predicted by long-wave analysis. Depending on the flow conditions, the critical perturbations can originate mainly at the interface (so-called "interfacial modes of instability") or in the bulk of one of the phases (i.e., "shear modes"). The present analysis revealed that there is no definite correlation between the type of instability and the perturbation wavelength

Factorization of the incompressible Stokes operator linking pressure and velocity is revisited.The main purpose is to use the inverse of the Stokes operator with a large time step as a preconditioner for Newton and Arnoldi iterations applied to computation of steady threedimensional flows and to study of their stability. It is shown that the Stokes operator can be inversed within an acceptable computational effort. This inverse includes fast direct inverses of several Helmholtz operators and iterative inverse of the pressure matrix. It is shown, additionally, that fast direct solvers can be attractive for the inverse of the Helmholtz and Laplace operators on fine grids and at large Reynolds numbers, as well as for other problems where convergence of iterative methods slows down. Implementation of the Stokes operator inverse to time-stepping-based formulation of the Newton and Arnoldi iterations is discussed. PACS 47.11.Df Reynolds numbers the eigenvalue decomposition based direct solver [ 1] becomes more efficient than iterative solvers. In particular, since computational requirements of the direct solver do not depend on the time step and Reynolds number, all the time steps are completed within the same CPU time, which is an attractive feature by itself.Second, we are interested in application of time-stepping algorithms to steady state Newton solvers and Arnoldi eigensolvers preconditioned by an inverse Stokes operator [2] below. To become an effective preconditioner the Stokes operator must be evaluated with a large time step.The latter becomes especially difficult when three-dimensional flows are studied on fine grids making most of traditional iterative methods to disconverge. In particular, we are interested in coupled incompressible pressure-velocity solvers, which are more computationally demanding than segregated ones, but possess important advantages: more stable time integration, correct calculation of pressure at each time step, and a possibility to proceed without pressure boundary conditions. Applied as preconditioners to Newton and Arnoldi solvers the coupled methods are expected to perform well if the Stokes operator with a large time step can be efficiently inversed.Considering 2D stability problems one can apply a direct sparse solver to inverse the 2D Stokes operator [4], however this becomes too memory demanding for fine three-dimensional grids. A similar approach with the same restrictions in 3D cases was implemented in [5] for explicitly calculated Jacobian matrices. At the same time, our recent pressure-velocity coupled multigrid solver [6], which performs well at small time steps fails to converge at large steps needed for 3D stability studies [4]. Based on the above experience, in this paper we recall the well-known factorization of the Stokes operator, which we use for computation of its inverse applying fast direct methods where possible. Using the finite volume method, we arrive to an analog of the Uzawa scheme [7], in which only one matrix, called "pressure matrix" has to be inversed ...

We have carried out direct numerical simulations (DNS) of the fluctuating Navier-Stokes equation together with the particle equations governing the motion of a nanosized particle or nanoparticle (NP) in a cylindrical tube. The effects of the confining boundary, its curvature, particle size, and particle density variations have all been investigated. To reveal how the nature of the temporal correlations (hydrodynamic memory) in the inertial regime is altered by the full hydrodynamic interaction due to the confining boundaries, we have employed the Arbitrary Lagrangian-Eulerian (ALE) method to determine the dynamical relaxation of a spherical NP located at various positions in the medium over a wide span of time scales compared to the fluid viscous relaxation time τv = a2/v, where a is the spherical particle radius and v is the kinematic viscosity. The results show that, as compared to the behavior of a particle in regions away from the confining boundary, the velocity autocorrelation function (VACF) for a particle in the lubrication layer initially decays exponentially with a Stokes drag enhanced by a factor that is proportional to the ratio of the particle radius to the gap thickness between the particle and the wall. Independent of the particle location, beyond time scales greater than a2/v, the decay is always algebraic followed by a second exponential decay (attributed to the wall curvature) that is associated with a second time scale D2/v, where D is the vessel diameter.

The three-dimensional linearized optimal energy growth mechanism, in plane parallel shear flows, is re-examined in terms of the role of vortex stretching and the interplay between the spanwise vorticity and the planar divergent components. For high Reynolds numbers the structure of the optimal perturbations in Couette, Poiseuille and mixing-layer shear profiles is robust and resembles localized plane waves in regions where the background shear is large. The waves are tilted with the shear when the spanwise vorticity and the planar divergence fields are in (out of) phase when the background shear is positive (negative). A minimal model is derived to explain how this configuration enables simultaneous growth of the two fields, and how this mutual amplification affects the optimal energy growth. This perspective provides an understanding of the three-dimensional growth solely from the two-dimensional dynamics on the shear plane.

Non-modal transient growth of disturbances in an isothermal viscous mixing layer flow is studied for the Reynolds numbers varying from 100 up to 5000 at different streamwise and spanwise wavenumbers. It is found that the largest non-modal growth takes place at the wavenumbers for which the mixing layer flow is stable. In linearly unstable configurations the non-modal growth can only slightly exceed the exponential growth at short times. Contrarily to the fastest exponential growth, which is two-dimensional, the most profound non-modal growth is attained by oblique three-dimensional oblique waves propagating at an angle with respect to the base flow. By comparing results of several mathematical approaches, it is concluded that within the considered mixing layer model with the tanh base velocity profile, the non-modal optimal disturbances growth results from the discrete part of the spectrum only. Finally, full three-dimensional DNS with the optimally perturbed base flow confirms the presence of the structures determined by the transient growth analysis. The time evolution of optimal perturbations is presented and exhibit growth and decay of flow structures that sometimes become similar to those observed at late stages of time evolution of the KelvinHelmholtz billows. It is shown that non-modal optimal disturbances yield a strong mixing without a transition to turbulence.2