Generalized fundamental solutions for unsteady viscous flows

Chwang, Allen T.

doi:10.1103/physreve.63.051201

Cited by 18 publications

(15 citation statements)

References 23 publications

(16 reference statements)

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“…Efforts also have been made to solve unsteady Stokes flows by the singularity method [25,35,36] and inverse two-dimensional Stokes problems [37]. The MFS has also been applied to Brinkman flow including the case of Brinkman flow with Robin BC at the interface with another fluid [38].…”

Section: Introductionmentioning

confidence: 99%

Method of fundamental solutions for partial‐slip fibrous filtration flows

Zhao

Povitsky

2008

Numerical Methods in Fluids

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SUMMARYIn this study a Stokeslet-based method of fundamental solutions (MFS) for two-dimensional low Reynolds number partial-slip flows has been developed. First, the flow past an infinitely long cylinder is selected as a benchmark. The numerical accuracy is investigated in terms of the location and the number of the Stokeslets. The benchmark study shows that the numerical accuracy increases when the Stokeslets are submerged deeper beneath the cylinder surface, as long as the formed linear system remains numerically solvable. The maximum submergence depth increases with the decrease in the number of Stokeslets. As a result, the numerical accuracy does not deteriorate with the dramatic decrease in the number of Stokeslets. A relatively small number of Stokeslets with a substantial submergence depth is thus chosen for modeling fibrous filtration flows. The developed methodology is further examined by application to Taylor-Couette flows. A good agreement between the numerical and analytical results is observed for no-slip and partial-slip boundary conditions. Next, the flow about a representative set of infinitely long cylindrical fibers confined between two planar walls is considered to represent the fibrous filter flow. The obtained flowfield and pressure drop agree very well with the experimental data for this setup of fibers. The developed MFS with submerged Stokeslets is then applied to partial-slip flows about fibers to investigate the slip effect at fiber-fluid interface on the pressure drop. The numerical results compare qualitatively with the analytical solution available for the limit case of infinite number of fibers.

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Section: Introductionmentioning

confidence: 99%

Method of fundamental solutions for partial‐slip fibrous filtration flows

Zhao

Povitsky

2008

Numerical Methods in Fluids

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“…domains, or the so-called Stokeslet and Oseenlet, have been extensively examined [1,[4][5][6][7]. Recently, Venkatalaxmi et al [6] showed that the solution for an unsteady Stokeslet can be expressed in a general form that involves two scalar functions.…”

mentioning

confidence: 98%

“…Recently, Venkatalaxmi et al [6] showed that the solution for an unsteady Stokeslet can be expressed in a general form that involves two scalar functions. Shu and Chwang [7] derived analytical solutions for the generalized unsteady Stokeslet and Oseenlet.…”

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confidence: 99%

Interfacial capillary–gravity waves due to a fundamental singularity in a system of two semi-infinite fluids

2007

J Eng Math

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The interfacial capillary-gravity waves due to a transient fundamental singularity immersed in a system of two semi-infinite immiscible fluids of different densities are investigated analytically for two-and threedimensional cases. The two-fluid system, which consists of an inviscid fluid overlying a viscous fluid, is assumed to be incompressible and initially quiescent. The two fluids are each homogeneous, and separated by a sharp and stable interface. The Laplace equation is taken as the governing equation for the inviscid flow, while the linearized unsteady Navier-Stokes equations are used for the viscous flow. With surface tension taken into consideration, the kinematic and dynamic conditions on the interface are linearized for small-amplitude waves. The singularity is modeled as a simple mass source when immersed in the inviscid fluid above the interface, or as a vertical point force when immersed in the viscous fluid beneath the interface. Based on the integral solutions for the interfacial waves, the asymptotic wave profiles are derived for large times with a fixed distance-to-time ratio by means of the generalized method of stationary phase. It is found that there exists a minimum group velocity, and the wave system observed will depend on the moving speed of the observer. Two schemes of expansion of the phase function are proposed for the two cases when the moving speed of an observer is larger than, or close to the minimum group velocity. Explicit analytical solutions are presented for the long gravity-dominant and the short capillary-dominant wave systems, incorporating the effects of density ratio, surface tension, viscosity and immersion depth of the singularity.

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“…Price and Tan [15] provided a convolution-integral formulation for a transient Oseenlet. Chan and Chwang [10] and Shu and Chwang [11] derived a series of generalized 3D unsteady Oseenlets.…”

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confidence: 99%

Generation of free-surface gravity waves by an unsteady Stokeslet

2008

Arch Appl Mech

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The interaction of unsteady Stokeslets with the free surface of an initially quiescent incompressible fluid of infinite depth is investigated analytically for two-and three-dimensional cases. The disturbed flows are generated by an unsteady singular force moving perpendicularly downwards away from the surface. The analysis is based on the assumption that the motion satisfies the linearized unsteady Navier-Stokes equations with linear kinematic and dynamic boundary conditions. Firstly, the asymptotic representation for the transient free-surface waves due to an instantaneous Stokeslet is derived for a large time with a fixed distance-to-time ratio. As is well known, the corresponding inviscid waves predicted by the potential theory do not decay to zero as the time goes to infinity. In the present study, the transient waves predicted by the viscous theory eventually vanish due to the presence of viscosity, which is consistent with reality from the physical point of view. Secondly, the asymptotic solutions are obtained for the unsteady free-surface waves due to a harmonically oscillating Stokeslet. It is found that the unsteady waves can be decomposed into steady-state and transient responses. The steady state can be attained as time approaches infinity. It is shown that the viscosity of the fluid plays an important role in the evolution of the singularity-induced waves.

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Generalized fundamental solutions for unsteady viscous flows

Cited by 18 publications

References 23 publications

Method of fundamental solutions for partial‐slip fibrous filtration flows

Method of fundamental solutions for partial‐slip fibrous filtration flows

Interfacial capillary–gravity waves due to a fundamental singularity in a system of two semi-infinite fluids

Generation of free-surface gravity waves by an unsteady Stokeslet

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