scholarly journals Spatially Developing Modes: The Darcy–Bénard Problem Revisited

Physics ◽  
2021 ◽  
Vol 3 (3) ◽  
pp. 549-562
Author(s):  
Antonio Barletta
Keyword(s):  
Porous Layer ◽  
Neutral Stability ◽  
Classical Analysis ◽  
Small Perturbations ◽  
Spatial Instability ◽  
Isothermal Plane ◽  
Different Temperatures ◽  
Spatially Periodic ◽  
Time Periodic ◽  
Bénard System

In this paper, the instability resulting from small perturbations of the Darcy–Bénard system is explored. An analysis based on time–periodic and spatially developing Fourier modes is adopted. The system under examination is a horizontal porous layer saturated by a fluid. The two impermeable and isothermal plane boundaries are considered to have different temperatures, so that the porous layer is heated from below. The spatial instability for the system is defined by taking into account both the spatial growth rate of the perturbation modes and their propagation direction. A comparison with the neutral stability condition determined by using the classical spatially periodic and time–evolving Fourier modes is performed. Finally, the physical meaning of the concept of spatial instability is discussed. In contrast to the classical analysis, based on spatially periodic modes, the spatial instability analysis, involving time–periodic Fourier modes, is found to lead to the conclusion that instability occurs whenever the Rayleigh number is positive.

The development of asymmetry and period doubling for oscillatory flow in baffled channels

1996 ◽  
Vol 328 ◽  
pp. 19-48 ◽  
Author(s):  
E. P. L. Roberts ◽  
M. R. Mackley
Keyword(s):  
Reynolds Number ◽  
Oscillatory Flow ◽  
Three Dimensional ◽  
Period Doubling ◽  
Progressive Increase ◽  
Newtonian Flow ◽  
Chaotic Flow ◽  
Spatially Periodic ◽  
Dimensional Simulation ◽  
Time Periodic

We report experimental and numerical observations on the way initially symmetric and time-periodic fluid oscillations in baffled channels develop in complexity. Experiments are carried out in a spatially periodic baffled channel with a sinusoidal oscillatory flow. At modest Reynolds number the observed vortex structure is symmetric and time periodic. At higher values the flow progressively becomes three-dimensional, asymmetric and aperiodic. A two-dimensional simulation of incompressible Newtonian flow is able to follow the flow pattern at modest oscillatory Reynolds number. At higher values we report the development of both asymmetry and a period-doubling cascade leading to a chaotic flow regime. A bifurcation diagram is constructed that can describe the progressive increase in complexity of the flow.

Time-periodic spatially periodic planforms in Euclidean equivariant partial differential equations

1995 ◽  
Vol 352 (1698) ◽  
pp. 125-168 ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Pattern ◽  
Invariant Equilibrium ◽  
Periodic Patterns ◽  
Planar Lattice ◽  
Diffusive Convection ◽  
Spatially Periodic ◽  
Spatio Temporal ◽  
Boussinesq Fluid ◽  
Time Periodic

In Rayleigh-Bénard convection, the spatially uniform motionless state of a fluid loses stability as the Rayleigh number is increased beyond a critical value. In the simplest case of convection in a pure Boussinesq fluid, the instability is a symmetry-breaking steady-state bifurcation that leads to the formation of spatially periodic patterns. However, in many double-diffusive convection systems the heat-conduction solution actually loses stability via Hopf bifurcation. These hydrodynamic systems provide motivation for the present study of spatiotemporally periodic pattern formation in Euclidean equivariant systems. We call such patterns planforms . We classify, according to spatio-temporal symmetries and spatial periodicity, many of the time-periodic solutions that may be obtained through equivariant Hopf bifurcation from a group-invariant equilibrium. Instead of focusing on plan- forms periodic with respect to a specified planar lattice, as has been done in previous investigations, we consider all planforms that are spatially periodic with respect to some planar lattice. Our classification results rely only on the existence of Hopf bifurcation and planar Euclidean symmetry and not on the particular dif­ferential equation.

scholarly journals On the stability of carbon sequestration in an anisotropic horizontal porous layer with a first-order chemical reaction

2019 ◽  
Vol 475 (2226) ◽  
pp. 20180365 ◽  
Author(s):  
K. Gautam ◽  
P. A. L. Narayana
Keyword(s):  
Chemical Reaction ◽  
Porous Layer ◽  
Nonlinear Stability ◽  
Anisotropy Ratio ◽  
Neutral Stability ◽  
First Order ◽  
Order Chemical Reaction ◽  
Linear And Nonlinear ◽  
First Order Chemical Reaction ◽  
The Stability

Carbon dioxide (CO 2 ) sequestration in deep saline aquifers is considered to be one of the most promising solutions to reduce the amount of greenhouse gases in the atmosphere. As the concentration of dissolved CO 2 increases in unsaturated brine, the density increases and the system may ultimately become unstable, and it may initiate convection. In this article, we study the stability of convection in an anisotropic horizontal porous layer, where the solute is assumed to decay via a first-order chemical reaction. We perform linear and nonlinear stability analyses based on the steady-state concentration field to assess neutral stability curves as a function of the anisotropy ratio, Damköhler number and Rayleigh number. We show that anisotropy in permeability and solutal diffusivity play an important role in convective instability. It is shown that when solutal horizontal diffusivity is larger than the vertical diffusivity, varying the ratio of vertical to horizontal permeabilities does not significantly affect the behaviour of instability. It is also noted that, when horizontal permeability is higher than the vertical permeability, varying the ratio of vertical to horizontal solutal diffusivity does have a substantial effect on the instability of the system when the reaction rate is dominated by the diffusion rate. We used the Chebyshev-tau method coupled with the QZ algorithm to solve the eigenvalue problem obtained from both the linear and nonlinear stability theories.

scholarly journals Stability of a Buoyant Oldroyd-B Flow Saturating a Vertical Porous Layer with Open Boundaries

Fluids ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 375
Author(s):  
Stefano Lazzari ◽  
Michele Celli ◽  
Antonio Barletta
Keyword(s):  
Rayleigh Number ◽  
Porous Layer ◽  
Critical Rayleigh Number ◽  
Stationary Flow ◽  
Vertical Axis ◽  
Seepage Flow ◽  
Neutral Stability ◽  
Linear Viscoelastic ◽  
Infinite Height ◽  
Limiting Case

The performance of several engineering applications are strictly connected to the rheology of the working fluids and the Oldroyd-B model is widely employed to describe a linear viscoelastic behaviour. In the present paper, a buoyant Oldroyd-B flow in a vertical porous layer with permeable and isothermal boundaries is investigated. Seepage flow is modelled through an extended version of Darcy’s law which accounts for the Oldroyd-B rheology. The basic stationary flow is parallel to the vertical axis and describes a single-cell pattern where the cell has an infinite height. A linear stability analysis of such a basic flow is carried out to determine the onset conditions for a multicellular pattern. This analysis is performed numerically by employing the shooting method. The neutral stability curves and the values of the critical Rayleigh number are evaluated for different retardation time and relaxation time characteristics of the fluid. The study highlights the extent to which the viscoelasticity has a destabilising effect on the buoyant flow. For the limiting case of a Newtonian fluid, the known results available in the literature are recovered, namely a critical value of the Darcy–Rayleigh number equal to 197.081 and a corresponding critical wavenumber of 1.05950.

Thermal Convection of a Non-Fourier Fluid in a Vertical Slot

2016 ◽  
Vol 138 (5) ◽  
Author(s):  
Mohammad Niknami ◽  
Roger E. Khayat
Keyword(s):  
Single Phase ◽  
Neutral Stability ◽  
Retardation Time ◽  
Oscillatory Convection ◽  
Stationary Convection ◽  
Ultrafast Processes ◽  
Vertical Slot ◽  
Different Temperatures ◽  
Prandtl Numbers ◽  
Critical State Parameters

The instability of steady natural convection of a non-Fourier fluid of the single-phase lagging (SPL) type between two vertical surfaces maintained at different temperatures is studied. SPL fluids possess a relaxation time, which reflects the delay in the response of the heat flux and the temperature gradient. The SPL model is particularly relevant to low-temperature liquids, ultrafast processes, and nanofluids (with a retardation time added in this case). Linear stability analysis is employed to obtain the critical state parameters, such as critical Grashof numbers. For intermediate Prandtl numbers (Pr = 7.5), when non-Fourier level exceeds a certain value, the neutral stability curve comprises a Fourier branch and an oscillatory branch. In this case, oscillatory convection increasingly becomes the mode of preference, compared to both conduction and stationary convection. Critical Grashof number decreases for fluids with higher non-Fourier levels.

scholarly journals Time-periodic measures, random periodic orbits, and the linear response for dissipative non-autonomous stochastic differential equations

2021 ◽  
Vol 8 (3) ◽  
Author(s):  
Michał Branicki ◽  
Kenneth Uda
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Periodic Orbits ◽  
Linear Response ◽  
Sufficient Conditions ◽  
Probability Measures ◽  
Climate Modelling ◽  
Small Perturbations ◽  
Asymptotic Probability ◽  
Time Periodic

AbstractWe consider a class of dissipative stochastic differential equations (SDE’s) with time-periodic coefficients in finite dimension, and the response of time-asymptotic probability measures induced by such SDE’s to sufficiently regular, small perturbations of the underlying dynamics. Understanding such a response provides a systematic way to study changes of statistical observables in response to perturbations, and it is often very useful for sensitivity analysis, uncertainty quantification, and improving probabilistic predictions of nonlinear dynamical systems, especially in high dimensions. Here, we are concerned with the linear response to small perturbations in the case when the time-asymptotic probability measures are time-periodic. First, we establish sufficient conditions for the existence of stable random time-periodic orbits generated by the underlying SDE. Ergodicity of time-periodic probability measures supported on these random periodic orbits is subsequently discussed. Then, we derive the so-called fluctuation–dissipation relations which allow to describe the linear response of statistical observables to small perturbations away from the time-periodic ergodic regime in a manner which only exploits the unperturbed dynamics. The results are formulated in an abstract setting, but they apply to problems ranging from aspects of climate modelling, to molecular dynamics, to the study of approximation capacity of neural networks and robustness of their estimates.

Darcy–Forchheimer Flow With Viscous Dissipation in a Horizontal Porous Layer: Onset of Convective Instabilities

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
A. Barletta ◽  
M. Celli ◽  
D. A. S. Rees
Keyword(s):  
Boundary Condition ◽  
Viscous Dissipation ◽  
Porous Layer ◽  
Linear Equations ◽  
Orientation Angle ◽  
Base Flow ◽  
Neutral Stability ◽  
Governing Parameter ◽  
Bottom Boundary ◽  
Forchheimer Flow

Parallel Darcy–Forchheimer flow in a horizontal porous layer with an isothermal top boundary and a bottom boundary, which is subject to a third kind boundary condition, is discussed by taking into account the effect of viscous dissipation. This effect causes a nonlinear temperature profile within the layer. The linear stability of this nonisothermal base flow is then investigated with respect to the onset of convective rolls. The third kind boundary condition on the bottom boundary plane may imply adiabatic/isothermal conditions on this plane when the Biot number is either zero (adiabatic) or infinite (isothermal). The solution of the linear equations for the perturbation waves is determined by using a fourth order Runge–Kutta scheme in conjunction with a shooting technique. The neutral stability curve and the critical value of the governing parameter R=GePe2 are obtained, where Ge is the Gebhart number and Pe is the Péclet number. Different values of the orientation angle between the direction of the basic flow and the propagation axis of the disturbances are also considered.

scholarly journals Time-periodic convective patterns in a horizontal porous layer with through-flow

2000 ◽  
Vol 58 (2) ◽  
pp. 265-281 ◽  
Author(s):  
Françoise Dufour ◽  
Marie-Christine Néel
Keyword(s):  
Porous Layer ◽  
Through Flow ◽  
Time Periodic
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