In this paper, we are interested in the linear and the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a laminar smooth density profile ρ 0 px 2 q being increasing in an infinite slab 2πLT ˆp´1, 1q (L ą 0, T is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ordinary differential equation of the vertical component of perturbed velocity on the finite interval p´1, 1q,with the boundary conditionswhere g ą 0 is the gravity constant, λ is the growth rate in time, k is the wave number transverse to the density profile and two Navier-slip coefficients ξ ˘are nonnegative constants. For each k P L ´1Zzt0u, we define a threshold of viscosity coefficient µcpk, Ξq for linear instability. So that, in the ksupercritical regime, i.e. µ ą µcpk, Ξq, we provide a spectral analysis adapting the operator method of Lafitte-Nguyễn in [12] and then prove that there are infinite solutions of (0.1)-(0.2). Based on infinitely unstable modes of the linearized problem, we consider a wide class of initial data to the nonlinear perturbation problem, extending Grenier's framework [6], to prove nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient, namely µ ą 3 sup kPL ´1Zzt0u µcpk, Ξq. ContentsData availability statement 2

In this paper, we show that a Dissipative Particle Dynamics (DPD) model of a viscous Newtonian fluid may actually produce a linear viscoelastic fluid. We demonstrate that a single set of DPD particles can be used to model a linear viscoelastic fluid with its physical parameters, namely the dynamical viscosity and the relaxation time in its memory kernel, determined from the DPD system at equilibrium. The emphasis of this study is placed on (i) the estimation of the linear viscoelastic effect from the standard parameter choice; and (ii) the investigation of the dependence of the DPD transport properties on the length and time scales, which are introduced from the physical phenomenon under examination. Transverse-current auto-correlation functions (TCAF) in Fourier space are employed to study the effects of the length scale, while analytic expressions of the shear stress in a simple small amplitude oscillatory shear flow are utilised to study the effects of the time scale. A direct mechanism for imposing the particle diffusion time and fluid viscosity in the hydrodynamic limit on the DPD system is also proposed.

In this study a Dissipative Particle Dynamics (DPD) method is employed with its input parameters directly determined from the fluid properties, such as the fluid mass density, water compressibility and viscosity. The investigation of thermal fluctuation scaling requires constant fluid properties, and this proposed DPD version meets this requirement. Its numerical verifications in simple or complex fluids under viscometric or non-viscometric flows indicate that (i) the level of thermal fluctuations in the DPD model for both types of fluids is consistently reduced with increasing in coarse-graining level; and (ii) viscometric or non-viscometric flows of a model fluid at different coarse-graining levels have a similar behaviour. Furthermore, to reduce the compressibility effect of the DPD fluid in simulating incompressible flows, a new simple treatment is presented and shown to be very effective.

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