Perturbation solution of the coupled Stokes-Darcy problem

A new methodology is proposed for the topology optimization of fluid in Stokes flow. The binary design variable and no-slip condition along the solid–fluid interface are regularized to allow for the use of continuous mathematical programming techniques. The regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by Darcy's law. Fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the Stokes and Darcy equations. The mixed formulation of the new Darcy–Stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems. Convergence to the no-slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid–fluid interface are not needed. The optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature. The advantages of the Darcy–Stokes approach include that it uses existing stabilization techniques to solve the finite element problem, it produces 0–1 (void–solid) topologies (i.e. there are no regions of artificial material), and that it can potentially be used to optimize the layout of a microscopically porous material. Copyright © 2005 John Wiley & Sons, Ltd.

This research investigates a number of problems, related to Stokes flow and to heat flow. The Stokes flow is inspired by glass flow in the process of making bottles or jars. The heat flow is related to a heat conduction model problem, and a problem about hot-spots formation in the microwave heating. We will discuss the first problem. There are two phases during the industrial process of making glass, viz. the pressing phase and the blowing phase. We consider some mathematical aspects of the pressing phase. The motion of glass at temperatures above 6000C can be described by the Navier-Stokes equations. Since glass is a highly viscous fluid, those equations can be simplified to the Stokes equations. We use two different methods to solve these equations, viz. perturbation and operator methods. The perturbation method is based on the geometry being slowly varying. As a result, we obtain the velocity analytically. This result has a good agreement with numerical results based on finite element modelling. Using the velocity obtained we derive the formula for the force on the plunger. Next, we consider the operator method. Using this method, the Stokes equations can be transformed into an operator equation on the boundary ??? with a tangent vector field a on the boundary ??? as unknown. Solving this operator equation shows, that the solutions of the Stokes equations can be parameterized by aH, the harmonic extension of a to the interior of the domain ??. As an application, we present some full explicit solutions of the Stokes equations for several domains such as the interior and exterior of the unit ball and of the unit disk, an infinite strip, a half space, and a wedge. In the second problem, we consider the heat conduction problem inside two types of geometry, viz. slowly and slightly varying geometry. Using this problem, we show the difference between those geometries. An example that involves the boundary layers at the ends is presented. Finally, we consider a simplified model of the microwave heating of a one-dimensional unit slab. This slab consists of three layers that have different thermal conductivities. We consider only the steady state problem with Dirichlet boundary conditions and continuity of heat flux across the layers. Using a fundamental-mode approximation of eigenfunction expansion, we investigate the effect of thermal conductivity on the formation of hot-spots where the temperature increases catastrophically as a function of d, the amplitude of the applied electric field. First, we consider a unit slab geometry. In this geometry, we find the critical value dcr, for which slight changes in d yields a sudden jump to another stable solution, now with a much higher temperature. Next, we consider a unit slab consisting of three layers of material with different thermal conductivity (µ). We assume the inner layer has the smallest value of µ. We find the temperature in this layer is much higher than that in other layers. Then, we consider only the inner layer. For a given value of d and changing values of µ, we get a temperature jump near some values of µ. This jump shows that there is a critical value of µ and signifies the formation of a hot-spot.

A hybrid high-order method for a coupled Stokes-Darcy problem on general meshes

The simulation of a stationary fluid flow past an obstacle by the lattice Boltzmann method (LBM) in two dimensions is discussed. The combination of second-order expressions for far-field boundary conditions and a suitable treatment of the no-slip boundary condition at the obstacle surface with the nested grid-refinement technique can be applied to the LBM, resulting in a highly efficient method for the treatment of exterior flows. Furthermore, via replacing the nested time stepping by local time stepping, the resolution process can be substantially accelerated. A highly accurate drag coefficient was used to make an error assessment for various no-slip boundary conditions imposed on the obstacle surface. The analysis showed that the equilibrium method for treating the no-slip conditions is superior to halfway bounce-back and full-way bounce-back no-slip conditions when the relaxation time tau=1 . Also a tau -dependence test was made to evaluate the performance of different methods in the treatment of the no-slip boundary conditions.

Universal image systems for non-periodic and periodic Stokes flows above a no-slip wall

A note on the relationship between the solutions of the stokes flow equations in a half-space with a no-slip boundary condition and with a slip boundary condition

Vertically bounded fingering double diffusive convection is numerically investigated, focusing on the influences of different velocity boundary conditions, i.e., the no-slip condition, which is inevitable in the lab-scale experimental researches, and the free-slip condition, which is an approximation for the interfaces in many natural environments, such as the oceans. For both boundary conditions the flow is dominated by fingers and the global responses follow the same scaling laws, with enhanced prefactors for the free-slip cases. Therefore, the laboratory experiments with the no-slip boundaries serve as a good model for the finger layers in the ocean. Moreover, in the free-slip case, although the tangential shear stress is eliminated at the boundaries, the local dissipation rate in the near-wall region may exceed the value found in the no-slip cases, which is caused by the stronger vertical motions of horizontally focused fingers and sheet structures near the free-slip boundaries. This counterintuitive result might be relevant for properly estimating and modeling the mixing and entrainment phenomena at free surfaces and interfaces widespread in oceans and geophysical flows.

Abstract. We report laboratory determinations of the shear resistance to sliding melting ice with entrained particles over a hard, impermeable surface. With higher particle concentrations and larger particle sizes, Coulomb friction at particle-bed contacts dominates and the shear stress increases linearly with normal load. We term this the sandy regime. When either particle concentration or particle size is reduced below a threshold, the dependence of shear resistance on normal load is no longer statistically significant. We term this regime slippery. We use force and mass balance considerations to examine the flow of melt water beneath the simulated basal ice. At high particle concentrations, the transition from sandy to slippery behavior occurs when the particle size is comparable to the thickness of the melt film that separates the sliding ice from its bed. For larger particle sizes, a transition from sandy to slippery behavior occurs when the particle concentration drops sufficiently that the normal load is no longer transferred completely to the particle-bed contacts. We estimate that the melt films separating the particles from the ice are approximately 0.1 µm thick at this transition. Our laboratory results suggest the potential for abrupt transitions in the shear resistance beneath hard-bedded glaciers with changes in either the thickness of melt layers or the particle loading.

Copper nanoparticles deposited inside the pores of anodized aluminium oxide using atomic layer deposition

We have examined the effect of the no-slip boundary condition on the steady streaming around a radially pulsating bubble, located at the velocity node of a standing sound wave. The no-slip condition can take place on the bubble surface under many circumstances, especially if the surface is contaminated or otherwise coated. We applied the singular perturbation analysis with the oscillation amplitude, ε=U0/(aω)<<1, as a small parameter and took the frequency parameter, M2=iωa2/ν, to be large. Here, a, U0, ω and ν are length scale, velocity scale, frequency, and kinematic viscosity, respectively. We further assumed that the bubble stays spherical in shape, the wavelength is significantly larger than the bubble radius, and no fluid transport takes place from the bubble to the surroundings. Additionally, the lateral and the radial oscillations, while at the same frequency, have a phase difference φ, and the amplitudes of both the oscillations are small compared to the bubble radius. We found that the streaming happened at a lower order than the non-pulsating case and it is more intense than the case of the shear free boundary condition. Furthermore, the phase difference was found to play a significant role in the streaming flow direction and its intensity.

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Flow singularity and slip velocity in plane extrudate swell computations

The shear motion in Newtonian fluids, that is, the fluid vorticity, represents an intrinsic loss mechanism governed by a diffusion equation. Its description involves the trace-free part of the fluid viscous stress tensor. This part is missing in the Biot theory of poroelasticity. As a result, the fluid vorticity is not captured and only one shear wave (S-wave) is predicted. The missing fluid vorticity has implications for the propagation of S-waves across discontinuities. This becomes most apparent in the problem of S-wave propagation across the welded contact of an elastic solid with a porous medium. At such a contact, the no-slip condition between the elastic solid and the constituent parts of the porous medium, the solid frame and the pore fluid, must hold. This requirement translates into a vanishing relative motion of the fluid with respect to the solid frame, that is, the filtration field, at the contact. Nevertheless, our analysis indicates that for the Biot theory, in the low-frequency regime, a nonzero, although insignificantly small, filtration field exists at the contact. However, more importantly, the filtration field is noticeable when the transition to the high-frequency regime occurs. This constitutes a disagreement with the requirement of a no-slip boundary condition and renders the prediction unphysical. This shortcoming is circumvented by including the fluid viscous stress tensor into the poroelastic constitutive relations, as stipulated by the de la Cruz-Spanos poroelasticity theory. Then, a second S-wave is predicted that manifests as the fluid vorticity at the macroscale. This process is distinct from the fast S-wave, the other predicted S-wave akin to the Biot S-wave. We find that the generation of this process at the contact induces a filtration field equal and opposite to that associated with the fast S-wave. Therefore, the no-slip condition is satisfied and the S-wave reflection/transmission across a discontinuity becomes physically meaningful.

In this work, we show how to impose no-slip boundary conditions for an H ( curl , Ω ) -based formulation for incompressible Stokes flow, which is used in structure-preserving discretizations of Navier-Stokes and magnetohydrodynamics equations. At first glance, it seems straightforward to apply no-slip boundary conditions: the tangential part is an essential boundary condition on H ( curl , Ω ) and the normal component can be naturally enforced through integration-by-parts of the divergence term. However, we show that this can lead to an ill-posed discretization and propose a Nitsche-based finite element method instead. We analyze the discrete system, establishing stability and deriving a priori error estimates. Numerical experiments validate our analysis and demonstrate optimal convergence rates for the velocity field.

Transient failure of a borehole excavated in a poroelastic continuum