Roughness-Induced Effect at Main order on the Reynolds Approximation

Usually the Stokes equations that govern a flow in a smooth thin domain (with thickness of order $\varepsilon$) are related to the Reynolds equation for the pressure $p_{\mathrm{smooth}}$. In this paper, we show that for a rough thin domain (with rugosities of order $\varepsilon^2$) the flow is governed by a modified Reynolds equation for a pressure $p_{\mathrm{rough}}$. Moreover, we find the relation $p_{\mathrm{rough}}=K\,p_{\mathrm{smooth}}$, where K is an explicit coefficient depending only on the form of the rugosities and on the viscosity of the fluid. In some sense, we see that the flow may be accelerated using adequate rugosity profiles on the bottom. The limit system is mathematically justified through a variant of the notion of two-scale convergence, the originality and difficulty being the anisotropy in the height profile.

Navier–Stokes equations in 3D thin domains with Navier friction boundary condition

In this paper, we investigate a convection–diffusion–reaction problem in a thin domain endowed with the Robin-type boundary condition describing the reaction catalyzed by the upper wall. Motivated by the microfluidic applications, we allow the oscillating behavior of the upper boundary and analyze the resonant case where the amplitude and period of the oscillation have the same small order as the domain's thickness. Depending on the magnitude of the reaction mechanism, we rigorously derive three different asymptotic models via the unfolding operator method. In particular, we identify the critical case in which the effects of the domain's geometry and all physically relevant processes become balanced.

Some Results on the Navier–Stokes Equations in Thin 3D Domains

Multiscale dimension reduction for flow and transport problems in thin domain with reactive boundaries

Mud diapirism is a common phenomenon of accretionary convergent margins but less common in erosive margins. Fluid venting associated with mud diapirism is of importance for the dewatering of the forearc and the resulting devolatilisation of the entire subduction zone. The margin offshore Costa Rica is today interpreted as erosive and subdivided into two major structural domains on grounds of the roughness of the downgoing plate: a smooth domain in the north where normal oceanic crust originating at the East Pacific Rise, and a rough southern domain where the margin is uplifted and fractured by the collision of the Cocos Ridge and numerous adjacent volcanic seamounts. These structural differences are reflected in differences in the output at the volcanic arc, dewatering mechanisms, and the abundance and geometry of mud mounds in the forearc. Diapiric mud mound occurrences in the smooth domain are most abundant in the middle and upper slope and apparently do not correlate with the maximum of compactional water release of the incoming sedimentary sequence. We invoke rapid changes in sedimentation rate and addition of accommodation space due to extensional faulting of the wedge to explain the observed mound distribution.

On the regularity of the Navier–Stokes equation in a thin periodic domain

We use the method of the topological degree, the theory of fractional powers of positive operators, and the Grisvard formula together with results proved by G. Raugel and G. R. Sell to study the periodic solutions of the incompressible Navier–Stokes equations in a thin three-dimensional domain.

We consider the flow of two Newtonian, incompressible and nonmiscible fluids in a 2D thin domain. Starting from the Stokes equations, we derive a generalized Buckley–Leverett equation for the first fluid saturation. We study the asymptotic behavior of the flow when the thickness of the gap tends to zero. Assuming that the fluids interface, which is a free boundary, is described by curves of uniformly bounded variation, we prove that the limit problem obeys a generalized Reynolds law. Moreover, when the two sides of the gap are fixed, the saturation of the limit problem is solution of the classical Buckley–Leverett equation.

We consider the Navier-Stokes equations in a three-dimensional curved thin domain around a given closed surface under Navier's slip boundary conditions. When the thickness of the thin domain is sufficiently small, we establish the global existence of a strong solution for large data. We also show several estimates for the strong solution with constants explicitly depending on the thickness of the thin domain. The proofs of these results are based on a standard energy method and a good product estimate for the convection and viscous terms following from a detailed study of average operators in the thin direction. We use the average operators to decompose a three-dimensional vector field on the thin domain into the almost two-dimensional average part and the residual part, and derive good estimates for them which play an important role in the proof of the product estimate.

Let Ω ⊂ ℝNbe a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W1,p(Ω) ↪ Lq(∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem [Formula: see text] for functions that verify u|A= 0. It is known that there exists an optimal hole that minimizes the best constant SAamong subsets of Ω of the prescribed volume.In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain.

Understanding fluid flow through a rough‐walled fracture is important in many problems such as petroleum and geothermal reservoir exploitation, geological storage of CO2, and sitting of radioactive waste repositories. In order to advance the understanding of fracture flow, we conducted the first direct measurement of flow velocity across rough‐walled fractures at Reynolds number (Re) of 0.014 to 0.086. The results were used for an order of magnitude analysis to evaluate assumptions underlying the Stokes and the Reynolds equations, which are derived from simplifying the Navier–Stokes equations. Even at very rough subregions, viscous forces were at least 2 orders of magnitude greater than inertial forces, indicating that the Stokes equations are valid for Re < 0.1. However, the assumption made in the derivation of the Reynolds equation that ∂2ux/∂z2 is dominant over other viscous terms was not satisfied even at moderate roughness for Re < 0.1. The Reynolds equation overestimated flow rate.

Chapter 5 - Isolation of Rough and Smooth Membrane Domains of the Endoplasmic Reticulum from Rat Liver

In ice sheet and glacier modelling, the Finite Element Method is rapidly gaining popularity. However, constructing and updating meshes for ice sheets and glaciers is a non-trivial and computationally demanding task due to their thin, irregular, and time dependent geometry. In this paper we introduce a novel approach to ice dynamics computations based on the unfitted Finite Element Method CutFEM, which lets the domain boundary cut through elements. By employing CutFEM, complex meshing and remeshing is avoided as the glacier can be immersed in a simple background mesh without loss of accuracy. The ice is modelled as a non-Newtonian, shear-thinning fluid obeying the p-Stokes (full Stokes) equations with the ice atmosphere interface as a moving free surface. A Navier slip boundary condition applies at the glacier base allowing both bedrock and subglacial lakes to be represented. Within the CutFEM framework we develop a strategy for handling non-linear viscosities and thin domains and show how glacier deformation can be modelled using a level set function. In numerical experiments we show that the expected order of accuracy is achieved and that the method is robust with respect to penalty parameters. As an application we compute the velocity field of the Swiss mountain glacier Haut Glacier d'Arolla in 2D with and without an underlying subglacial lake, and simulate the glacier deformation from year 1930 to 1932, with and without surface accumulation and basal melt.

Transitional endoplasmic reticulum (tER) consists of confluent rough and smooth endoplasmic reticulum (ER) domains. In a cell-free incubation system, low-density microsomes (1.17 g cc(-1)) isolated from rat liver homogenates reconstitute tER by Mg(2+)GTP- and Mg(2+)ATP-hydrolysis-dependent membrane fusion. The ATPases associated with different cellular activities protein p97 has been identified as the relevant ATPase. The ATP depletion by hexokinase or treatment with either N-ethylmaleimide or anti-p97 prevented assembly of the smooth ER domain of tER. High-salt washing of low-density microsomes inhibited assembly of the smooth ER domain of tER, whereas the readdition of purified p97 with associated p47 promoted reconstitution. The t-SNARE syntaxin 5 was observed within the smooth ER domain of tER, and antisyntaxin 5 abrogated formation of this same membrane compartment. Thus, p97 and syntaxin 5 regulate assembly of the smooth ER domain of tER and hence one of the earliest membrane differentiated components of the secretory pathway.