Onset of Darcy–Brinkman convection in a binary viscoelastic fluid saturated porous layer

The onset of Darcy–Brinkman double-diffusive convection in a binary viscoelastic fluid-saturated porous layer is studied using both linear and weakly nonlinear stability analyses. The Oldroyd-B model is employed to describe the rheological behavior of the fluid. An extended form of Darcy–Oldroyd law incorporating the Brinkman’s correction and time derivative is used to describe the fluid flow and the Oberbeck–Boussinesq approximation is invoked. The onset criterion for stationary and oscillatory convection is derived analytically. The effects of rheological parameters, Darcy number, normalized porosity, Lewis number, solute Rayleigh number, and Darcy–Prandtl number on the stability of the system is investigated. The results indicated that there is a competition among the processes of thermal, solute diffusions and viscoelasticity that causes the convection to set in through the oscillatory modes rather than the stationary. The Darcy–Prandtl number has a dual effect on the threshold of oscillatory convection. The nonlinear theory based on the method of truncated representation of Fourier series is used to find the transient heat and mass transfer. Some existing results are reproduced as the particular cases of present study.     

Gravity-elastic deformation and stability of fluid–fluid interface in porous media

A new generalized model is proposed to describe deformations of the mobile interface separating two immiscible weakly compressible fluids in a weakly deformable porous medium. It describes gravity non-equilibrium processes, including evolution of the gravitational instability, and represents a system of parabolic or anti-parabolic equations.     

Linear stability of a fluid channel with a porous layer in the center简

We perform a Poiseuille flow in a channel linear stability analysis of a inserted with one porous layer in the centre, and focus mainly on the effect of porous filling ratio. The spectral collocation technique is adopted to solve the coupled linear stability problem. We investigate the effect of permeability, σ, with fixed porous filling ratio ψ = 1/3 and then the effect of change in porous filling ratio. As shown in the paper, with increasing σ, almost each eigenvalue on the upper left branch has two subbranches at ψ = 1/3. The channel flow with one porous layer inserted at its middle （ψ = 1/3） is more stable than the structure of two porous layers at upper and bottom walls with the same parameters. By decreasing the filling ratio ψ, the modes on the upper left branch are almost in pairs and move in opposite directions, especially one of the two unstable modes moves back to a stable mode, while the other becomes more instable. It is concluded that there are at most two unstable modes with decreasing filling ratio ψ. By analyzing the relation between ψ and the maximum imaginary part of the streamwise phase speed, Cimax, we find that increasing Re has a destabilizing effect and the effect is more obvious for small Re, where ψ a remarkable drop in Cimax can be observed. The most unstable mode is more sensitive at small filling ratio ψ, and decreasing ψ can not always increase the linear stability. There is a maximum value of Cimax which appears at a small porous filling ratio when Re is larger than 2 000. And the value of filling ratio 0 corresponding to the maximum value of Cimax in the most unstable state is increased with in- creasing Re. There is a critical value of porous filling ratio （= 0.24） for Re = 500; the structure will become stable as ψ grows to surpass the threshold of 0.24; When porous filling ratio ψ increases from 0.4 to 0.6, there is hardly any changes in the values of Cimax. We have also observed that the critical Reynolds number is especially sensitive for small ψ where the fastest drop is observed, and there may be a wide range in which the porous filling ratio has less effect on the stability （ψ ranges from 0.2 to 0.6 at σ = 0.002）. At larger permeability, σ, the critical Reynolds number tends to converge no matter what the value of porous filling ratio is.     

On a Darcy–Stefan Problem arising in freezing and thawing of saturated porous media

A model with phase change for material convection in a saturated porous medium with a frozen region is formulated as a Darcy-Stefan problem. We propose a new generalized formulation for this Stefan-type problem with convection governed by Darcy's law. This approach, which is valid for irregular geometries with irregular subregions, has the advantage of not requiring the smoothness of the temperature, that restricted previous mathematical works to two-dimensional particular cases. We show existence of generalized solutions, passing to the limit in suitable approximated problems, which in principle can be solved numerically by the finite element method. Received October 20, 1998     

Effects of walls temperature variation on double diffusive natural convection of Al2O3–water nanofluid in an enclosure

The effect of wall temperature variations on double diffusive natural convection of Al₂O₃–water nanofluid in a differentially heated square enclosure with constant temperature hot and cold vertical walls is studied numerically. Transport mechanisms of nanoparticles including Brownian diffusion and thermophoresis that cause heterogeneity are considered in non-homogeneous model. The hot and cold wall temperatures are varied, but the temperature difference between them is always maintained 5 °C. The thermophysical properties such as thermal conductivity, viscosity and density and thermophoresis diffusion and Brownian motion coefficients are considered variable with temperature and volume fraction of nanoparticles. The governing equations are discretized using the control volume method. The results show that nanoparticle transport mechanisms affect buoyancy force and cause formation of small vortexes near the top and bottom walls of the cavity and reduce the heat transfer. By increasing the temperature of the walls the effect of transport mechanisms decreases and due to enhanced convection the heat transfer rate increases.     

Pore-scale modeling of viscoelastic flow in porous media using a Bautista–Manero fluid

This article examines the extensional flow and viscosity and the converging–diverging geometry as the basis of the peculiar viscoelastic behavior in porous media. The modified Bautista–Manero model, which successfully describes elasticity, thixotropic time dependency and shear-thinning, was used for modeling the flow of viscoelastic materials which also show thixotropic attributes. An algorithm, originally proposed by Philippe Tardy, that employs this model to simulate steady-state time-dependent flow was implemented in a non-Newtonian flow simulation code using pore-scale modeling. The simulation results using two topologically-complex networks confirmed the importance of the extensional flow and converging–diverging geometry on the behavior of non-Newtonian fluids in porous media. The analysis also identified a number of correct trends (qualitative and quantitative) and revealed the effect of various fluid and flow parameters on the flow process. The impact of some numerical parameters was also assessed and verified.     

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

We investigate the resonant double Hopf bifurcation in a diffusive complex Ginzburg–Landau model with delayed feedback and phase shift. The conditions for the existence of resonant double Hopf bifurcation are obtained by analyzing the roots’ distribution of the characteristic equation, and a general formula to determine the bifurcation point is given. For the cases of 1:2 and 1:3 resonance, we choose time delay, feedback strength and phase shift as bifurcation parameters and derive the normal forms which are proved to be the same as those in non-resonant cases. The impact of cubic terms on the unfolding types is discussed after obtaining the normal form till 3rd order. By fixing phase shift, we find that varying time delay and feedback strength simultaneously can induce the coexistence of two different periodic solutions, the existence of quasi-periodic solutions and strange attractors. Also, the effects on the existence of transient quasi-periodic solution exerted by the phase shift are illustrated.

     

Average and extremal properties of heat transfer and shear stress on a wall surface in Rayleigh–Bénard convection

In this study surface-averaged and extremal properties of heat transfer and shear stress on the upper wall surface of Rayleigh–Bénard convection are numerically examined. The Prandtl number was raised up to 10³, and the Rayleigh number was changed between 10⁴ and 10⁷. As a result, average Nusselt number Nu and shear rate τ/Pr depends on Pr, Ra, and the entire numerical results are distributed between two correlation equations corresponding to small and large Pr. The small and large Pr equations are closely related to steady and unsteady flow regimes, respectively. Nevertheless, a single relation τ/Pr ~ Nu ^3.0 exists to explain the entire results. Similarly the change of local maximal properties Nu _max and τ _max/Pr depends on Pr, Ra, and these values are also distributed between two correlation equations corresponding to small and large Pr cases. Despite such complicated dependence we can obtain a correlation equation as a form of τ _max/Pr ~ Nu _max^2.6, which has not been obtained theoretically.     

Migration of a solid particle in the vicinity of a plane fluid–fluid interface

The slow migration of a small and solid particle in the vicinity of a gas–liquid, fluid–fluid or solid–fluid plane boundary when subject to a gravity or an external flow field is addressed. By contrast with previous works, the advocated approach holds for arbitrarily shaped particles and arbitrary external Stokes flow fields complying with the conditions on the boundary. It appeals to a few theoretically established and numerically solved boundary-integral equations on the particle’s surface. This integral formulation of the problem allows us to provide asymptotic approximations for a distant boundary and also, implementing a boundary element technique, accurate numerical results for arbitrary locations of the boundary. The results obtained for spheroids, both settling or immersed in external pure shear and straining flows, reveal that the rigid-body motion experienced by a particle deeply depends upon its shape and also upon the boundary location and properties.     

MHD mixed convection–radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour’s Effects

This work is focused on the numerical modeling of steady, laminar, heat and mass transfer by MHD mixed convection from a semi-infinite, isothermal, vertical and permeable surface immersed in a uniform porous medium in the presence of thermal radiation and Dufour and Soret effects. A mixed convection parameter for the entire range of free-forced-mixed convection is employed and the governing equations are transformed into non-similar equations. These equations are solved numerically by an efficient, implicit, iterative, finite-difference scheme. The obtained results are checked against previously published work on special cases of the problem and are found to be in excellent agreement. A parametric study illustrating the influence of the thermal radiation coefficient, magnetic field, porous medium inertia parameter, concentration to thermal buoyancy ratio, and the Dufour and Soret numbers on the fluid velocity, temperature and concentration as well as the local Nusselt and the Sherwood numbers is conducted. The obtained results are shown graphically and the physical aspects of the problem are discussed.     

Dynamics and pattern formation of a diffusive predator–prey model in the presence of toxicity

Nonlinear Dynamics - In this paper, we develop a diffusivepredator–prey model with toxins under the homogeneous Neumann boundary condition. First, the persistence property and global...     

On the stability of the simple shear flow of a Johnson–Segalman fluid

We solve the time-dependent simple shear flow of a Johnson–Segalman fluid with added Newtonian viscosity. We focus on the case where the steady-state shear stress/shear rate curve is not monotonic. We show that, in addition to the standard smooth linear solution for the velocity, there exists, in a certain range of the velocity of the moving plate, an uncountable infinity of steady-state solutions in which the velocity is piecewise linear, the shear stress is constant and the other stress components are characterized by jump discontinuities. The stability of the steady-state solutions is investigated numerically. In agreement with linear stability analysis, it is shown that steady-state solutions are unstable only if the slope of a linear velocity segment is in the negative-slope regime of the shear stress/shear rate curve. The time-dependent solutions are always bounded and converge to a stable steady state. The number of the discontinuity points and the final value of the shear stress depend on the initial perturbation. No regimes of self-sustained oscillations have been found.     

Simulation of conjugate radiation–forced convection heat transfer in a porous medium using the lattice Boltzmann method

In this paper, a lattice Boltzmann method is employed to simulate the conjugate radiation–forced convection heat transfer in a porous medium. The absorbing, emitting, and scattering phenomena are fully included in the model. The effects of different parameters of a silicon carbide porous medium including porosity, pore size, conduction–radiation ratio, extinction coefficient and kinematic viscosity ratio on the temperature and velocity distributions are investigated. The convergence times of modified and regular LBMs for this problem are 15 s and 94 s, respectively, indicating a considerable reduction in the solution time through using the modified LBM. Further, the thermal plume formed behind the porous cylinder elongates as the porosity and pore size increase. This result reveals that the thermal penetration of the porous cylinder increases with increasing the porosity and pore size. Finally, the mean temperature at the channel output increases by about 22% as the extinction coefficient of fluid increases in the range of 0–0.03.

     

The solid–fluid transition in a yield stress shear thinning physical gel

We present an experimental investigation of the solid–fluid transition in a yield stress shear thinning physical gel (Carbopol^® 940) under shear. Upon a gradual increase of the external forcing, we observe three distinct deformation regimes: an elastic solid-like regime (characterized by a linear stress–strain dependence), a solid–fluid phase coexistence regime (characterized by a competition between destruction and reformation of the gel), and a purely viscous regime (characterized by a power law stress-rate of strain dependence). The competition between destruction and reformation of the gel is investigated via both systematic measurements of the dynamic elastic moduli (as a function of stress, the amplitude, and temperature) and unsteady flow ramps. The transition from solid behavior to fluid behavior displays a clear hysteresis upon increasing and decreasing values of the external forcing. We find that the deformation power corresponding to the hysteresis region scales linearly with the rate at which the material is being forced (the degree of flow unsteadiness). In the asymptotic limit of small forcing rates, our results agree well with previous steady state investigations of the yielding transition. Based on these experimental findings, we suggest an analogy between the solid–fluid transition and a first-order phase transition, e.g., the magnetization of a ferro-magnet where irreversibility and hysteresis emerge as a consequence of a phase coexistence regime. In order to get further insight into the solid–fluid transition, our experimental findings are complemented by a simple kinetic model that qualitatively describes the structural hysteresis observed in our rheological experiments. The model is fairly well validated against oscillatory flow data by a partial reconstruction of the Pipkin space of the material’s response and its nonlinear spectral behavior.     

Electro-chemo-mechanical couplings in saturated porous media: elastic–plastic behaviour of heteroionic expansive clays

Chemically active saturated clays containing several cations are considered in a two-phase framework. The solid phase contains the negatively charged clay particles, absorbed water and ions. The fluid phase, or pore water, contains free water and ions. Electroneutrality is ensured in both phases, which gives rise to electrical fields. Water and ions can transfer between the two phases. In addition, a part of free water diffuses through the porous medium. A global understanding of all phenomena, deformation, transfer, diffusion and electroneutrality, is provided. Emphasis is laid on the electro-chemo-mechanical constitutive equations in an elastic–plastic setting. Elastic chemo-mechanical coupling is introduced through a potential, in such a way that the tangent elastic stiffness is symmetric. Material parameters needed to estimate the coupling are calibrated from specific experiments available in the recent literature. The elastic–plastic behaviour aims at reproducing qualitatively and quantitatively typical experimental phenomena observed on natural clays during chemical and mixed chemo-mechanical loadings, including chemical consolidation and swelling already described in Int. J. Solids Structures (39 (10), 2773–2806) in the simpler context of Na-Montmorillonite clays. Crucially, the successive exposure of a clay to pore solutions with chemical content dominated by a cation already present in the clay or quasi-absent leads to dramatically different volume changes, in agreement with experimental data.     

Global stability and Hopf-bifurcation in a zooplankton–phytoplankton model

We deal with a predator–prey model, representing a resource (phytoplankton) and two predators (zooplankton) system with toxin-producing delay. The response function is assumed here to be concave in nature. Firstly, the stability criterion of the model is analyzed both from a local and a global point of view. Our results imply that the toxin’s intrinsic characteristics, such as toxic liberation rate and toxin-producing delay, will not change the stability of the system irreversibly. Secondly, Hopf bifurcation of both systems with delay and without delay can occur via system parameters pertaining to the toxin. Our results indicate that the toxin produced by phytoplankton may be used as a bio-control agent for the Harmful Algal Bloom problems. Furthermore, the explicit algorithm for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are obtained using the normal form method and center manifold theorem. Finally, some numerical simulations are carried out to illustrate the results.