# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

June 2011 , Volume 15 , Issue 4

Special issue on microfluidics and complex flows

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2011, 15(4): i-ii doi: 10.3934/dcdsb.2011.15.4i +[Abstract](1069) +[PDF](99.7KB)
Abstract:
The topic of microfluidics and complex flows is of paramount interest for various applications at microscales such as biology (blood flow), food industry, polymeric flows, micro film coating etc., among other Science and Engineering problems.

2011, 15(4): 917-934 doi: 10.3934/dcdsb.2011.15.917 +[Abstract](1597) +[PDF](470.3KB)
Abstract:
We introduce a multilayer model to solve three-dimensional sediment transport by wind-driven shallow water flows. The proposed multilayer model avoids the expensive Navier-Stokes equations and captures stratified horizontal flow velocities. Forcing terms are included in the system to model momentum exchanges between the considered layers. The topography frictions are included in the bottom layer and the wind shear stresses are acting on the top layer. To model the bedload transport we consider an Exner equation for morphological evolution accounting for the velocity field on the bottom layer. The coupled equations form a system of conservation laws with source terms. As a numerical solver, we apply a kinetic scheme using the finite volume discretization. Preliminary numerical results are presented to demonstrate the performance of the proposed multilayer model and to confirm its capability to provide efficient simulations for sediment transport by wind-driven shallow water flows. Comparison between results obtained using the multilayer model and those obtained using the single-layer model are also presented.
2011, 15(4): 935-956 doi: 10.3934/dcdsb.2011.15.935 +[Abstract](1612) +[PDF](872.0KB)
Abstract:
We propose a time discretization scheme for a class of ordinary differential equations arising in simulations of fluid/particle flows. The scheme is intended to work robustly in the lubrication regime when the distance between two particles immersed in the fluid or between a particle and the wall tends to zero. The idea consists in introducing a small threshold for the particle-wall distance below which the real trajectory of the particle is replaced by an approximated one where the distance is kept equal to the threshold value. The error of this approximation is estimated both theoretically and by numerical experiments. Our time marching scheme can be easily incorporated into a full simulation method where the velocity of the fluid is obtained by a numerical solution to Stokes or Navier-Stokes equations. We also provide a derivation of the asymptotic expansion for the lubrication force (used in our numerical experiments) acting on a disk immersed in a Newtonian fluid and approaching the wall. The method of this derivation is new and can be easily adapted to other cases.
2011, 15(4): 957-969 doi: 10.3934/dcdsb.2011.15.957 +[Abstract](1625) +[PDF](1067.0KB)
Abstract:
This work describes a three-dimensional (3D) numerical simulation of the flow field of the complete enclosed Central Nervous System (CNS) including the ventricular system, the spinal cord and spinal sub-arachnoid space (SAS). Previous works on the topic consider only parts of the complete system imposing artificial boundary conditions at internal cross sections. The computational domain was constructed from MRI data using Materialise Software Mimics. In this work pulsatile velocity inlets in the lateral ventricles, due to the cardiac cycle, were used to simulate the dynamic nature of the CSF, whilst pressure outlets were used to model the areas of CSF re-absorption. A porous medium formulation (Darcy flow) was considered in the SAS to account for the effect of the arachnoid trabeculae within these areas. The simulation was run using the commercial CFD code Fluent using the laminar solver and transient simulation. A maximum CSF velocity was found to be in the region 11.8 mm/s, with a peak pressure drop through the aqueduct of the order of 2.8 Pa corresponding to a calculated peak Reynolds number of 12. CSF pressure at the exits of Magendie and Luschke were found to vary over the cardiac cycle, with pressure at the exits of Luschke being higher than Magendie for large periods of the cycle. CSF was seen to enter the SAS as a laminar jet from the exits of Magendie and Luschke. By only considering the cardiac cycle a very slow CSF motion within the spinal SAS was observed with magnitudes significantly reduced after a depth of 50mm down the column from the exit of Magendie. This result suggests that pulsating wall motion in the region of the spinal cord, due to respiratory effects, needs to be considered in order to predict experimentally observed flow recirculation within the spinal sac. 287 words.
2011, 15(4): 971-990 doi: 10.3934/dcdsb.2011.15.971 +[Abstract](1694) +[PDF](527.1KB)
Abstract:
Microfiltration of particles is modelled by the motion of particles embedded in a Stokes flow near a porous membrane in which Darcy equations apply. Stokes flow also applies on the other side of the membrane. A pressure gradient is applied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces. This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is much larger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order. The method is applied to the example case of a spherical particle moving normal to a membrane. The solution, limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step the boundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.
2011, 15(4): 991-998 doi: 10.3934/dcdsb.2011.15.991 +[Abstract](1609) +[PDF](266.9KB)
Abstract:
In this paper, the derivation of the convected derivatives for the heat flux and stress tensor is revisited. A kinematic approach is adopted based on material invariance. These upper-convected derivatives are used in the literature to generalize Newton's law of viscosity and Fourier's heat law of heat. The former constitutive law represents the behaviour of a viscoelastic fluid of the Boger type obeying the Oldroyd-B model, and the latter represents fluids obeying the Maxwell-Cattaneo's heat equation. The invariance of the derivatives under orthogonal transformation is also shown. Although the presentation here is limited to the derivatives of vector and second-rank tensor fluxes, the formulation can be generalized to generate the convected derivative of a tensor flux of arbitrary rank. Finally, the connection with micro- or nano-channel flow is noted.
2011, 15(4): 999-1018 doi: 10.3934/dcdsb.2011.15.999 +[Abstract](1636) +[PDF](551.8KB)
Abstract:
In this paper we use a layer potential method to obtain an existence and solvability result in Sobolev spaces for a Dirichlet-transmission problem given in terms of general Brinkman operators, when the solution is defined in two adjacent Lipschitz or $C^1$ domains on a Riemannian manifold and satisfies prescribed transmission conditions at the interface between these domains, and an additional Dirichlet condition on an external boundary.
2011, 15(4): 1019-1044 doi: 10.3934/dcdsb.2011.15.1019 +[Abstract](1640) +[PDF](821.2KB)
Abstract:
In order to employ continuum models in the analysis of the flow behaviour of a viscous Newtonian fluid at micro scale devices, it is necessary to consider at the wall surfaces appropriate slip boundary conditions instead of the classical non-slip condition. To account for the slip condition at the nano-scale, we used the Navier's type boundary condition that relates the tangential fluid velocity at the boundaries to the tangential shear rate. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the Stokes velocity field, which directly incorporates into the integral equations the local tangential shear rate at the wall surfaces, is presented. This formulation is used to numerically simulate concentric and eccentric rotating Couette mixers and a Single rotor mixer, including the effect of thermal creep in cases of rarefied gases. The numerical results obtained for the Couette mixers, concentric and eccentric, are validated again the corresponding analytical solutions, showing excellent agreements.
2011, 15(4): 1045-1064 doi: 10.3934/dcdsb.2011.15.1045 +[Abstract](1270) +[PDF](293.8KB)
Abstract:
This paper examines the slow viscous migration of a collection of $N\geq 1$ spherical bubbles immersed in a bounded Newtonian liquid under the action of prescribed uniform gravity field and/or arbitrary ambient Stokes flow. The liquid domain is either open or closed with fixed boundary(ies) where the ambient Stokes flow vanishes. The incurred translational velocity of each bubble is obtained by resorting to a well-posed boundary formulation which requires to invert $3N$ boundary-integral equations. Depending upon the selected Green tensor, these integral equations holds on the cluster's surface plus the boundaries or solely on the cluster's surface. The advocated numerical strategy resorts to quadratic triangular curvilinear boundary elements on each encoutered surface and enables one to accurately compute at a reasonable cpu time cost each bubble velocity. A special attention is paid, both theoretically and numerically, to the case of $N-$bubble cluster located near a solid and motionless plane wall with numerical results given and discussed for a few clusters subject to gravity effects and ambient linear or quadratic shear flows.
2011, 15(4): 1065-1076 doi: 10.3934/dcdsb.2011.15.1065 +[Abstract](1380) +[PDF](410.2KB)
Abstract:
Two dimensional numerical simulations of sets of vesicles in a Poiseuille flow are presented. Vesicles are a simple model to describe the dynamics of red cells in blood flow. At the scale of vesicles, the hydrodynamics is well described by the Stokes equation, whose linearity allows the use of Green's functions via the boundary integral method. This is coupled with the fast multipole method to acheive optimal scaling with respect to the number of discretization points. Results are presented for sets of different number of vesicles, showing their spatial organization. Vesicles assume a parachute-like shape and align one to the other in the centre of the parabolic profile. The relative distances depend on the total number of vesicles and on the position in the set.
2011, 15(4): 1077-1093 doi: 10.3934/dcdsb.2011.15.1077 +[Abstract](1367) +[PDF](1029.3KB)
Abstract:
By using an accurate indirect boundary element method, the break-up of a low-viscosity-ratio isolated drop is investigated numerically in a contraction flow at vanishing Reynolds numbers. A practical mathematical method is constructed to detect the asymptotic behavior of the maximum curvature at the point of pinch-off and is used to predict an impending breakup and the breakup time. The 3-D numerical simulation presented here can accurately capture not only the primary breakup of a low viscosity drop as it moves through a constricted geometry, but also secondary breakups and the presence of a set of satellite drops. The results agree qualitatively with laboratory experiments and two-dimensional simulations, but provide more details, aiding the understanding of the process of low-viscosity-ratio drop breakup in an arbitrarily shaped confined outer flow.

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