Onset of double-diffusive convection in a saturated porous layer with time-periodic surface heating

McKay, G.

doi:10.1007/s001610050091

Cited by 6 publications

(2 citation statements)

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“…Once nonlinearities are taken into account, however, the response remains bounded, but the system can still exhibit large-amplitude motion when forced at certain resonant frequencies. Analyses of parametric resonance have appeared for various fluid flows involving surface or interfacial waves wherein the system is forced through gravitational modulation [11,14,17] or time-dependent heating [15]. However, to our knowledge there has been no analysis performed on fluid-structure problems that captures the two-way interaction between a fluid and an immersed, flexible structure.…”

mentioning

confidence: 99%

Parametric Resonance in Immersed Elastic Boundaries

Cortez¹,

Peskin²,

Varela³

2004

SIAM J. Appl. Math.

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Abstract. In this paper, we investigate the stability of a fluid-structure interaction problem in which a flexible elastic membrane immersed in a fluid is excited via periodic variations in the elastic stiffness parameter. This model can be viewed as a prototype for active biological tissues such as the basilar membrane in the inner ear, or heart muscle fibers immersed in blood. Problems such as this, in which the system is subjected to internal forcing through a parameter, can give rise to "parametric resonance." We formulate the equations of motion in two dimensions using the immersed boundary formulation. Assuming small amplitude motions, we can apply Floquet theory to the linearized equations and derive an eigenvalue problem whose solution defines the marginal stability boundaries in parameter space. The eigenvalue equation is solved numerically to determine values of fiber stiffness and fluid viscosity for which the problem is linearly unstable. We present direct numerical simulations of the fluid-structure interaction problem (using the immersed boundary method) that verify the existence of the parametric resonances suggested by our analysis. 1. Introduction. Fluid-structure interaction problems abound in industrial applications as well as in many natural phenomena. Owing to the potentially complex, time-varying geometry and the nonlinear interactions that arise between fluid and solid, such problems represent a major challenge to both mathematical modelers and computational scientists.This paper deals with a model for fluid-structure interaction known as the immersed boundary (IB) formulation [20], which has proven particularly effective for dealing with complex problems in biological fluid mechanics. In this formulation, the immersed structure is treated as an elastic surface or interwoven mesh of elastic fibers that exert a singular force on a surrounding viscous fluid, while also moving with the velocity of adjacent fluid particles. The associated immersed boundary method has been used to solve a wide range of problems such as blood flow in the heart and arteries [2,21], swimming microorganisms [8], biofilms [7], and insect flight [16]. More recently, the IB method has also been extended to a much wider range of nonbiological problems involving flow past solid cylinders [12], flapping filaments [29], parachutes [10], and suspensions of flexible particles [25].

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mentioning

confidence: 99%

Parametric Resonance in Immersed Elastic Boundaries

Cortez¹,

Peskin²,

Varela³

2004

SIAM J. Appl. Math.

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“…[279][280][281][282] and chapter 10 of the book by [414]. In particular, [279,280] investigates the geophysical application to patterned ground formation due to a time-periodic surface flux generated by the Sun.…”

Section: Ltne Thermovibrational Convectionmentioning

confidence: 99%

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Straughan

2015

Advances in Mechanics and Mathematics

113

111

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In this chapter we describe work of [366] and of [393] on thermal convection in a Darcy porous material in a vertical layer subject to different temperatures on the vertical walls.For the case of a single temperature the problem where the porous medium occupies a vertical layer subject to differential heating from one side to the other has attracted much attention. This problem has much application to insulation, such as in the area of building design, and is of importance in window double glazing. For the single temperature situation the first proof that a vertical porous slab of Darcy type which is held at fixed but different temperatures on the vertical walls is stable to perturbations from the equilibrium state is due to [156]. His analysis is based on the linear theory and analyses the two-dimensional problem.[409] further analysed this problem but in three-dimensions and he treated the completely nonlinear situation by employing energy-like integral methods. In particular, [409] produced a threshold for the Rayleigh number which guarantees global nonlinear stability, i.e. regardless of how large the initial perturbation may be. He also employed a generalized energy method to demonstrate that the result of [156] is true in the fully three-dimensional case although the initial data must be restricted. Articles dealing with other interesting aspects of convection in a vertical porous slab or pipe involving effects like double diffusion, and LTNE, include the papers by [32, 33, 36, 46, 211, 220, 221, 329, 330, 362] and [460]. The methods of energy stability techniques applied specifically to convection in a vertical porous slab are addressed in [144, 229] and [356], and these are discussed in detail in the book by [414, pp. 126-134].The analogous problem in LTNE to that of [156] was first addressed by [366]. Specifically [366] dealt with the [156] problem of thermal convection in a vertical porous medium, but he employed the theory of local thermal non-equilibrium, allowing for different fluid and solid temperatures. The interesting work of [366] demonstrates that the vertical configuration is always stable according to linear theory, even with the much more complicated LTNE theory. [393] continued the problem of [366] but they analysed the complete nonlinear three-dimensional situation.

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Double-Diffusive Convection

Nield

Bejan

2012

Convection in Porous Media

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Onset of double-diffusive convection in a saturated porous layer with time-periodic surface heating

Cited by 6 publications

References 19 publications

Parametric Resonance in Immersed Elastic Boundaries

Parametric Resonance in Immersed Elastic Boundaries

Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Double-Diffusive Convection

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