Binary Fluid Convection Research Articles

Effects of inclination angle and fluid parameters on binary fluid convection in a tilted rectangular cavity

Thermal convection of binary fluid mixtures with negative separation ratios in a tilted rectangular cavity heated from below is numerically investigated using a high-accuracy finite-difference method. The fluids under consideration are Newtonian fluids with the Prandtl number (Pr) and Lewis number (Le) in the ranges of 0.01≤Pr≤10 and 0.001≤Le≤1, respectively. Extensive direct numerical simulations are conducted to explore the dynamics of the system by varying the control parameters, namely, the relative Rayleigh number (r) and the inclination angle of the cavity (β) together with the separation ratio (ψ) in the ranges of 1.0≤r≤2.0, 0°≤β≤90°, and −0.6≤ψ≤0. It is found that the inclination angle leads to a distinct trend for heat and mass transfer: the Nusselt number first decreases for small β, then gradually increases for moderate β, and finally decreases again for large β when r varies from 1.4 to 2.0. The optimal inclination angle lies in the range of 54°≤β∗≤64°, where the heat transport peaks. The Sherwood number initially decreases for small β, then increases with increasing β, and finally remains constant for large β. We present the bifurcation diagram of the solutions for different separation ratios, focusing on the distinct variation trends of the inclination angle. The corresponding heat- and mass-transfer properties of the flow on different branches of the bifurcation diagram are investigated. Furthermore, we explore the effect of the relative Rayleigh number, Prandtl number and Lewis number as well as the separation ratio on heat and mass transfer for various inclination angles, and obtain the Nusselt number as a function of the separation ratio and inclination angle.

Convection instability of linear Oldroyd-B fluids in a vertical channel with non-Fourier heat flux model

Owing to the importance of non-Fourier heat flux model in several natural and engineering processes, the convection of binary viscoelastic fluid in a vertical channel with non-Fourier heat flux model is investigated. The linear Oldroyd-B constitutive equation is used to model viscoelasticity. The presence of the basic flow in the vertical y-direction makes the problem challenging compared with the case in Rayleigh–Bénard convection. We use the Chebyshev collocation method to explore the instability characteristics of the linear Oldroyd-B fluid under a wide variety of physical parameters. Results show that the non-Fourier effect and relaxation time contribute to destabilize the system for oscillatory convection. The retardation time can inhibit the instability of the convective system. In the absence of the non-Fourier effect, the vertical fluid layer cannot support oscillatory motions. Oscillatory motion is possible, and the neutral stability curve branches when the non-Fourier effect is taken into account in the fluid. In addition, a new interesting phenomenon can be found: under the coupling action of viscoelastic fluids and the non-Fourier effect, the neutral stability curve would change from single to two branches and then to a single branch with the increase in relaxation time.

Collisions of localized patterns in a nonvariational Swift-Hohenberg equation.

The cubic-quintic Swift-Hohenberg equation(SH35) has been proposed as an order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use numerical continuation, together with extensive direct numerical simulations (DNSs), to study SH35 with an additional nonvariational quadratic term to model the effects of breaking the midplane reflection symmetry. The nonvariational structure of the model leads to the propagation of asymmetric spatially localized structures (LSs). An asymptotic prediction for the drift velocity of such structures, derived in the limit of weak symmetry breaking, is validated numerically. Next, we present an extensive study of possible collision scenarios between identical and nonidentical traveling structures, varying a temperaturelike control parameter. These collisions are inelastic and result in stationary or traveling structures. Depending on system parameters and the types of structures colliding, the final state may be a simple bound state of the initial LSs, but it can also be longer or shorter than the sum of the two initial states as a result of nonlinear interactions. The Maxwell point of the variational system, where the free energy of the global pattern state equals that of the trivial state, is shown to have no bearing on which of these scenarios is realized. Instead, we argue that the stability properties of bound states are key. While individual LSs lie on a modified snakes-and-ladders structure in the nonvariational SH35, the multipulse bound states resulting from collisions lie on isolas in parameter space, disconnected from the trivial solution. In the gradient SH35, such isolas are always of figure-eight shape, but in the present nongradient case they are generically more complex, although the figure-eight shape is preserved in a small subset of cases. Some of these complex isolas are shown to terminate in T-point bifurcations. A reduced model is proposed to describe the interactions between the tails of the LSs. The model consists of two coupled ordinary differential equations (ODEs) capturing the oscillatory nature of SH35 profiles at the linear level. It contains three parameters: two interaction amplitudes and a phase, whose values are deduced from high-resolution DNSs using gradient descent optimization. For collisions leading to the formation of simple bound states, the reduced model reproduces the trajectories of LSs with high quantitative accuracy. When nonlinear interactions lead to the creation or deletion of wavelengths, the model performs less well. Finally, we propose an effective signature of a given interaction in terms of net attraction or repulsion relative to free propagation. It is found that interactions can be attractive or repulsive in the net, irrespective of whether the two closest interacting extrema are of the same or opposite signs. Our findings highlight the rich temporal dynamics described by this bistable nonvariational SH35, and show that the interactions in this system can be quantitatively captured, to a significant extent, by a highly reduced ODE model.

Analyzing Slightly Inclined Cylindrical Binary Fluid Convection via Higher Order Dynamic Mode Decomposition

Analyzing Slightly Inclined Cylindrical Binary Fluid Convection via Higher Order Dynamic Mode Decomposition

Fermi-Pasta-Ulam-Tsingou recurrence of periodic anomalous waves in the complex Ginzburg-Landau and in the Lugiato-Lefever equations.

The complex Ginzburg-Landau (CGL) equation, an envelope model relevant in the description of several natural phenomena like binary-fluid convection and second-order phase transitions, and the Lugiato-Lefever (LL) equation, describing the dynamics of optical fields in pumped lossy cavities, can be viewed as nonintegrable generalizations of the nonlinear Schrödinger (NLS) equation, including diffusion, linear and nonlinear loss or gain terms, and external forcing. In this paper we treat the nonintegrable terms of both equations as small perturbations of the integrable focusing NLS equation, and we study the Cauchy problem of the CGL and LL equations corresponding to periodic initial perturbations of the unstable NLS background solution, in the simplest case of a single unstable mode. Using the approach developed in a recent paper by the authors with P. G. Grinevich [Phys. Rev. E 101, 032204 (2020)10.1103/PhysRevE.101.032204], based on the finite gap method and the theory of perturbations of soliton PDEs, we construct the proper analytic models describing quantitatively how the solution evolves, after a suitable transient, into a Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence of anomalous waves (AWs) described by slowly varying lower dimensional patterns (attractors) in the (x,t) plane, characterized by Δx=L/2 or Δx=0 in the case in which loss or gain, respectively, effects prevail, where Δx is the x-shift of the position of the AW during the recurrence and L is the period. We also obtain, in the CGL case, the analytic condition for which loss and gain exactly balance, stabilizing the ideal FPUT recurrence of periodic NLS AWs; such a stabilization is not possible in the LL case due to the external forcing. These processes are described, to leading order, in terms of elementary functions of the initial data in the CGL case, and in terms of elementary and special functions of the initial data in the LL case.

The effect of gas-phase transport on Marangoni convection in volatile binary fluids driven by a horizontal temperature gradient

Recent experimental and numerical studies of convection in confined layers of volatile binary liquids with a free surface subjected to a horizontal temperature gradient have observed a reversal in the direction of interfacial flow as the concentration of air in the vapor space above the liquid is decreased. These observations suggest that transport in the gas phase has a significant effect on the balance between thermocapillary and solutocapillary stresses, the competition between which determines the flow direction. In order to develop a quantitative description of the flow reversal, we use the two-sided (liquid/gas) transport model introduced previously to obtain approximate analytical solutions for the interfacial temperature and composition of the liquid, hence predict thermocapillary and solutocapillary stresses, and the flow direction. Therefore, our solutions provide useful guidelines for choosing the optimal binary coolants composition and operating conditions for thermal management applications. Despite the complex nature of this problem, we have found that the mass transport in the gas phase is effectively one-dimensional and independent of the flow in moderate to large aspect-ratio cavity for sufficiently low temperature gradients, which allows this problem to be simplified and solved analytically in a sequential manner. Our theoretical predictions agree well with the results of numerical simulations, which indicates that the analytical analysis captures the essential physics of the problem.

Effect of small inclination on binary convection in elongated rectangular cells.

We analyze the effect of a small inclination on the well-studied problem of two-dimensional binary fluid convection in a horizontally extended closed rectangular box with a negative separation ratio, heated from below. The horizontal component of gravity generates a shear flow that replaces the motionless conduction state when inclination is not present. This large-scale flow interacts with the convective currents resulting from the vertical component of gravity. For very small inclinations the primary bifurcation of this flow is a Hopf bifurcation that gives rise to chevrons and blinking states similar to those obtained with no inclination. For larger but still small inclinations this bifurcation disappears and is superseded by a fold bifurcation of the large-scale flow. The convecton branches, i.e., branches of spatially localized states consisting of counterrotating rolls, are strongly affected, with the snaking bifurcation diagram present in the noninclined system destroyed already at small inclinations. For slightly larger but still small inclinations we obtain small-amplitude localized states consisting of corotating rolls that evolve continuously when the primary large-scale flow is continued in the Rayleigh number. These localized states lie on a solution branch with very complex behavior strongly dependent on the values of the system parameters. In addition, several disconnected branches connecting solutions in the form of corotating rolls, counterrotating rolls, and mixed corotating and counterrotating states are also obtained.

Complex convective structures in three-dimensional binary fluid convection in a porous medium

Three-dimensional convection in a binary mixture in a porous medium heated from below is studied. For negative separation ratios steady spatially localized convection patterns are expected. Such patterns, localized in two dimensions, are computed in square domains with periodic boundary conditions and three different aspect ratios. Numerical continuation is used to examine their growth as the Rayleigh number varies and to study their interaction with their images under periodic replication once they reach the size of the domain. In relatively small domains with six critical wavelengths on a side three different structures are found, two with four arms extended either along the principal axes or the diagonals and one with eight arms. As these states are followed as a function of the Rayleigh number all succeed in growing so as to ultimately fill the domain thereby generating an extended convection pattern. In contrast, in larger domains, with 12 or 18 critical wavelengths on a side, the solution branches undergo much more complex behavior but appear to fail to generate spatially extended states. In each case both D4- and D2-symmetric structures are computed.

Three-dimensional numerical study of natural convection of supercritical binary fluid

Three-dimensional numerical study of natural convection of supercritical binary fluid

一种新的混合流体对流竖向镜面对称对传波斑图

The 2D hydrodynamic equations for binary fluid convection were numerically simulated with the SIMPLE method. For separation ratio ψ=－0.6 of the binary fluid mixture and aspect ratio Γ=20 of the rectangular cell, a newtype counterpropagating wave pattern of vertical mirror symmetry was found for the first time and its dynamics was preliminarily studied. At the center of the counterpropagating wave pattern of vertical mirror symmetry was a standing wave, of which the wavelength extended with time. As the wavelength increased to a certain critical value, a roll split into 2 rolls, and a new roll with a 180° phase difference formed between them 2. The roll located at the center line only has phase mutation and wavelength contraction or extension, without moving toward the left or right. The convective rolls propagating toward the left or right exist on both sides of the center line. The 2 phase mutations of the standing wave form a period, and the standing wave period increases with reduced Rayleigh number Ra_r. This type of convective structure exists in the range of Ra_r∈(3.6,4.3].The convection system produces the traveling wave pattern with defect for Ra_r≤3.6. The system shifts to the traveling wave pattern for Ra_r>4.3.The work shows that the counterpropagating wave pattern of vertical mirror symmetry is a stable flow pattern between the traveling wave pattern with defect and the traveling wave pattern.

Onset and nonlinear regimes of convection of binary fluid with negative separation ratio in square cavity heated from above

The present paper is concerned with studying the onset and nonlinear regimes of convection in a binary liquid mixture with negative separation ratio in square cavity heated from above. The problem of temporal evolution of small perturbations of the non-stationary base state is solved in the framework of the linear approach. A direct numerical simulation based on the solution of full unsteady equations is carried out. The results of calculations show that the wavelength of the most dangerous perturbations and the time of the onset of instability decrease with the growth of solutal Rayleigh number (level of gravity) according to the power laws. The data on the structure of the critical perturbations and nonlinear regimes established after the loss of stability are obtained. The results of direct numerical simulation are compared with the dependencies of the onset time of instability and wave number of the most dangerous perturbations on the solutal Rayleigh number, obtained by solving the linear problem of temporal evolution of small perturbations; good agreement is found.

Stability analysis of Electro-thermo convection of binary fluid with chemical reaction in a horizontal porous layer

In the present article, we illustrate the onset of electro-thermo convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces. The state of convection is considered, when the solubility of dissolved components depends on temperature. We use linear stability analysis to investigate how the vertical electric field and dissolution or precipitation of the component affects the onset of convection. Darcy-Brinkman’s law and Boussinesq approximation are employed with the equation of state taken to be linear with respect to temperature and concentration. We present a comparative study for four different bonding surfaces in linear case and weakly non-linear study in free-free (F/F) case for different controlling parameters. From the linear stability analysis, we find that the larger value of AC electric Rayleigh number (Rea) and Damköler number enhance (χ) the onset of convection whereas the larger value of inverse Darcy number delay (Da-1) the onset of convection. The stability criteria for different bounding surfaces are given as F/R>R/R>R/F>F/F (where F represents free and R stands for rigid bounding surfaces). The effect of parameters is qualitatively same for all surfaces but differs quantitatively. We are getting same kind of results for limiting cases such as pure electro convection, pure double diffusive electro-convection and pure thermal double diffusive convection. From weakly non-linear stability analysis, we show heat and mass transfer effect for unsteady and steady cases for same parameters. With increasing value of Rea and χ, enhance the unsteady and steady convection whereas reverse is obtained with increasing Da-1. We also draw streamlines, isotherms and isohalines in unsteady case for different times (0.001,0.03,0.06,0.1) as well as in steady case for different Rayleigh number (at critical Rayleigh number and more than critical). These plots represent state of conduction and convection.

Dufour-Soret Driven Double-Diffusive Convection in Nonreactive Binary Fluids with Temperature Dependent Viscosity

The aim of the present paper is to investigate the effects of temperature dependent viscosity on the stability of Dufour-Soret driven double-diffusive convection in nonreactive binary fluids, for general boundary conditions. Some analytical results concerning the stability of oscillatory motions and otherwise the complex growth rate of oscillatory motions are derived for this general problem. The expressions for Rayleigh numbers, when instability sets in as stationary modes, for each combination of rigid (slip free) and dynamically free (stress free) boundary conditions are derived numerically using Galerkin's method. The effects of linear and exponential temperature dependent viscosity on the onset stationary convection are studied and computed numerically. Various consequences of the obtained results for different convective problems have been worked out, as special cases.

A Skeleton of Collision Dynamics: Hierarchical Network Structure among Even-Symmetric Steady Pulses in Binary Fluid Convection

This paper presents a detailed analysis of the stability and network structure of thermal convection patterns of mixtures containing two miscible fluids. The stability of steady spatially localized solutions consisting of an even number of convection cells (even-SP) is investigated in detail because even-SPs emerge as transient states resulting from collisions between counterpropagating periodic traveling pulses, while odd-SPs do not. The even-SP branch is examined to analyze the distribution of the eigenvalues and the corresponding eigenfunctions to understand the stable/unstable direction in the vicinity of each solution and to investigate how they are connected to each other. Further, we construct a hierarchical network diagram consisting of even-SP solutions as nodes and stable and unstable manifolds connecting between them as edges, which shows a skeleton of transition processes of an arbitrary initial condition and asymptotic states.

Numerical investigation of binary fluid convection with a weak negative separation ratio in finite containers

By using a high-order compact finite difference method to solve the full hydrodynamic field equations, convection in binary fluid mixtures with a weak negative separation ratio of −0.1 in rectangular containers heated from below is numerically investigated. We consider the problem with the Prandtl number Pr ranging from 0.01 to 10 and the Lewis number Le from 0.0005 to 1. Several convective structures such as traveling wave, localized traveling wave, and undulation traveling wave convection as well as stationary overturning convection (SOC) are obtained. For the separation ratio considered, localized traveling wave state exists in a range of Rayleigh numbers spanning the critical point (the critical Rayleigh number at the onset of convection), and their length of the convective region is uniquely selected for a given parameter set. A bifurcation diagram of solution is drawn and the transitions between various traveling waves and the steady states on their upper branches are discussed. The effects of the fluid parameters and the aspect ratio of the container on the onset of convection and their saturated structures are studied in detail. Finally, several types of initial temperature fields are used to start simulations and five different stable SOC states with different mean wavenumbers are found. The corresponding heat and mass transfer properties of these stable SOC states are also investigated.

Three-dimensional spatially localized binary-fluid convection in a porous medium

AbstractThree-dimensional convection in a binary mixture in a porous medium heated from below is studied. For negative separation ratios steady spatially localized convection patterns are expected. Such patterns, spatially localized in two dimensions, are computed and numerical continuation is used to examine their growth and proliferation as parameters are varied. The patterns studied have the form of a core region with four extended side-branches and can be stable. A physical mechanism behind the formation of these unusual structures is suggested.

Travelling convectons in binary fluid convection

AbstractBinary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. With no-slip, fixed-temperature, no-mass-flux boundary conditions at the top and bottom stationary odd- and even-parity convectons fall on a pair of intertwined branches connected by branches of travelling asymmetric states. In appropriate parameter regimes the stationary convectons may be stable. When the boundary condition on the top is changed to Newton’s law of cooling the odd-parity convectons start to drift and the branch of odd-parity convectons breaks up and reconnects with the branches of asymmetric states. We explore the dependence of these changes and of the resulting drift speed on the associated Biot number using numerical continuation, and compare and contrast the results with a related study of the Swift–Hohenberg equation by Houghton & Knobloch (Phys. Rev.E, vol. 84, 2011, art. 016204). We use the results to identify stable drifting convectons and employ direct numerical simulations to study collisions between them. The collisions are highly inelastic, and result in convectons whose length exceeds the sum of the lengths of the colliding convectons.

Linear and nonlinear stability analysis of binary viscoelastic fluid convection

The linear and weakly nonlinear stability analysis of the quiescent state in a viscoelastic fluid subject to vertical solute concentration and temperature gradients is investigated. The non-Newtonian behavior of the viscoelastic fluid is characterized using the Oldroyd model. Analytical expressions for the critical Rayleigh numbers and corresponding wave numbers for the onset of stationary or oscillatory convection subject to cross diffusion effects is determined. A stability diagram clearly demarcates non-overlapping regions of finger and diffusive instabilities. A Lorenz system is obtained in the case of the weakly nonlinear stability analysis. The effect of Dufour and Soret parameters on the heat and mass transports are determined and discussed. Due to consideration of dilute concentrations of the second diffusing component the route to chaos in binary viscoelastic fluid systems is similar to that of single-component (thermal) viscoelastic fluid systems.

Spontaneous formation of travelling localized structures and their asymptotic behaviour in binary fluid convection

AbstractWe study spontaneous pattern formation and its asymptotic behaviour in binary fluid flow driven by a temperature gradient. When the conductive state is unstable and the size of the domain is large enough, finitely many spatially localized time-periodic travelling pulses (PTPs), each containing a certain number of convection cells, are generated spontaneously in the conductive state and are finally arranged at non-uniform intervals while moving in the same direction. We found that the role of PTP solutions and their strong interactions (collision) are important in characterizing the asymptotic state. Detailed investigations of pulse–pulse interactions showed the differences in asymptotic behaviour between that in a finite but large domain and in an infinite domain.

Linear and nonlinear stability analysis of binary Maxwell fluid convection in a porous medium

The stability analysis of the quiescent state in a Maxwell fluid-saturated densely packed porous medium subject to vertical concentration and temperature gradients is presented. A single phase model with local thermal equilibrium between the porous matrix and the Maxwell fluid is assumed. The critical Darcy–Rayleigh numbers and the corresponding wave numbers for the onset of stationary and oscillatory convection are determined. A Lorenz like system is obtained for weakly nonlinear stability analysis.