Decay properties for the incompressible Navier-Stokes flows in a half space

In this article, we give a comprehensive characterization of $L^1$ -summability for the Navier-Stokes flows in the half space, which is a long-standing problem. The main difficulties are that $L^q-L^r$ estimates for the Stokes flow don't work in this end-point case: $q=r=1$ ; the projection operator $P: L^1\longrightarrow L^1_\sigma$ is not bounded any more; useful information on the pressure function is missing, which arises in the net force exerted by the fluid on the noncompact boundary. In order to achieve our aims, by making full use of the special structure of the half space, we decompose the pressure function into two parts. Then the knotty problem of handling the pressure term can be transformed into establishing a crucial and new weighted $L^1$ -estimate, which plays a fundamental role. In addition, we overcome the unboundedness of the projection $P$ by solving an elliptic problem with homogeneous Neumann boundary condition.

Quasilocal Smarr relation for an asymptotically flat spacetime

We investigate the thermodynamics of Einstein–Maxwell (-dilaton) theory for an asymptotically flat spacetime in a quasilocal frame. We firstly define a quasilocal thermodynamic potential via the Euclidean on-shell action and formulate a quasilocal Smarr relation from Euler’s theorem. Then we calculate the quasilocal energy and surface pressure by employing a Brown–York quasilocal method along with Mann–Marolf counterterm and find entropy from the quasilocal thermodynamic potential. These quasilocal variables are consistent with the Tolman temperature and the entropy in a quasilocal frame turns out to be same as the Bekenstein–Hawking entropy. As a result, we found that a surface pressure term and its conjugate variable, a quasilocal area, do not participate in a quasilocal thermodynamic potential, but should be present in a quasilocal Smarr relation and the quasilocal first law of black hole thermodynamics. For dyonic black hole solutions having dynamic dilaton field, a non-trivial dilaton contribution should occur in the quasilocal first law but not in the quasilocal Smarr relation.

A combined model approach for debris-flow impact forces

<p>For the design of mitigation measures knowledge of debris-flow impact forces, usually estimated based on hydrostatic, hydrodynamic, or combined approaches, is essential. As these approaches are based on Newtonian fluids, they must be adjusted by empirical correction factors to account for the solid-fluid nature of debris flows. The values for the correction factors shown in the literature vary over a wide range and several studies showed a clear dependence with the Froude regime of debris flows.</p><p>To better understand the correction factors and to be able to calculate them using parameters that describe the flow behaviour a total of 32 experiments were conducted in the course of the project “Debris flow impact forces on bridge super structures (DEFSUP)”, funded by the Austrian Science Fund (FWF). Two different material compositions, different water contents as well as a total impact and a bypassing of the measuring block were tested.</p><p>The experimental setup designed within the project consists of a 4 m long semi-circular channel with a diameter of 300 mm and an inclination of 20°. The material is released from a rectangular reservoir in a dam-break scenario and accelerated with zero roughness on a length of 1.2 m and transferred to the semi-circle profile. The subsequently introduced roughness with a grain diameter of 1-2 mm generates a stationary phenomenological debris flow until it hits the measuring setup. With a starting volume of 50 kg, flow heights between 8 and 12 cm and velocities from 0.8 to 2.2 m/s were achieved according to the material composition and different water content. With these different mixtures a Froude-range from 0.6 to 3.6 was covered. In addition, normal stresses and pore water pressures were measured at the exact same point.</p><p>A detailed analysis of the measured impact forces together with the above mentioned measured parameters showed that the hydrodynamic correction factor is a constant mainly corresponding to the liquification ratio of the debris-flow mixture. Hence, the hydrodynamic correction factor can be regarded as a drag coefficient and seems to depend mainly on the internal friction of the flowing medium. At low Froude numbers measured impact forces exceed even a full momentum transfer if the mean bulk density is used for the calculation. This indicates that the impact forces can no longer be described by the hydrodynamic approach alone. For this reason, an additional pressure term based on a hydrostatic approach is considered in the combined concept. This additional pressure term depends on the dynamics of flow (Froude number) and can be modelled via a dynamic earth pressure coefficient.</p><p>The findings from these experiments contribute to a better prediction of debris-flows impact forces in terms of their material composition and flow behaviour.</p>

Semi-martingale driven variational principles

Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler–Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.

A generalization of the Einstein–Maxwell equations

The proposed modifications of the Einstein–Maxwell equations include: (1) the addition of a scalar term to the electromagnetic side of the equation rather than to the gravitational side, (2) the introduction of a four-dimensional, nonlinear electromagnetic constitutive tensor, (3) the addition of curvature terms arising from the non-metric components of a general symmetric connection and (4) the addition of a non-isotropic pressure tensor. The scalar term is defined by the condition that a spherically symmetric particle be force-free and mathematically well behaved everywhere. The constitutive tensor introduces two structure fields: One contributes to the mass and the other contributes to the angular momentum. The additional curvature terms couple both to particle solutions and to localized electromagnetic and gravitational wave solutions. The pressure term is needed for the most general spherically symmetric, static metric. It results in a distinction between the Schwarzschild mass and the inertial mass.

Influence of the initial hydrostatic pressure on contact area coefficient under drainage condition

The expression of effective stress proposed by Terzaghi has always been questioned. Many correction formulas are modification of pore pressure term. The pore pressure factor is related to porosity, contact area and other factors. When the particles are in point contact, the expression of the effective stress is that proposed by Terzaghi, while for the surface contact particles, the actual effective stress increases the stress produced by pore pressure passing through the contact surface based on the Terzaghi effective stress. There are many factors that affect the development of contact area and pore pressure, therefore, a series of the drained triaxial tests were carried out on four groups of sand samples with different initial hydrostatic pressures to study the influence of different initial hydrostatic pressures on the effective stress due to the term of contact area (σα). The test results show that the shear strength is increases with the initial hydrostatic pressure under the same effective confining pressure, which indirectly indicates that the initial hydrostatic pressure increases the contact area stress.

Simulation of complex free surface flows

We study the Serre-Green-Naghdi system under a non-hydrostatic formulation, modelling incompressible free surface flows in shallow water regimes. This system, unlike the well-known (nonlinear) Saint-Venant equations, takes into account the effects of the non-hydrostatic pressure term as well as dispersive phenomena. Two numerical schemes are designed, based on a finite volume - finite difference type splitting scheme and iterative correction algorithms. The methods are compared by means of simulations concerning the propagation of solitary wave solutions. The model is also assessed with experimental data concerning the Favre secondary wave experiments [12].

Analytical solutions for unsteady forced convection pulsating flow in a microchannel in the presence of EDL effects

A forced convection flow driven in a microchannel by an applied pressure gradient that fluctuates with small amplitude harmonically in time in the presence of electrical double layer effects is investigated. An analytical expression for the electrostatic potential is obtained via Poisson’s equation. Based on this solution, we further obtain analytical solutions for velocity and temperature for both the cases Pr ≠ 1 and Pr = 1. Results show that they match each other as Pr → 1− and Pr → 1+. The explicit expression of the transient Nusselt number is derived. We notice that the Debye–Hückel parameter γ and the angular velocity Ω are key factors for flow behaviours. Our proposed study adds some new insights by including the time-dependent pressure term that is usually overlooked in previous works. It is expected that this work could help to understand the transportal mechanisms of forced convection flow in microfluidic equipment and instruments.