Thermal Convection Of Viscoelastic Fluids Research Articles

Analytical and numerical analysis of bifurcations in thermal convection of viscoelastic fluids saturating a porous square box

We report theoretical and numerical results on bifurcations in thermal instability for a viscoelastic fluid saturating a porous square cavity heated from below. The modified Darcy law based on the Oldroyd-B model was used for modeling the momentum equation. In addition to Rayleigh number ℜ, two more dimensionless parameters are introduced, namely, the relaxation time λ1 and the retardation time λ2. Temporal stability analysis showed that the first bifurcation from the conductive state may be either oscillatory for sufficiently elastic fluids or stationary for weakly elastic fluids. The dynamics associated with the nonlinear interaction between the two kinds of instabilities is first analyzed in the framework of a weakly nonlinear theory. For sufficiently elastic fluids, analytical expressions of the nonlinear threshold above which a second hysteretic bifurcation from oscillatory to stationary convective pattern are derived and found to agree with two-dimensional numerical simulations of the full equations. Computations performed with high Rayleigh number indicated that the system exhibits a third transition from steady single-cell convection to oscillatory multi-cellular flows. Moreover, we found that an intermittent oscillation regime may exist with steady state before the emergence of the secondary Hopf bifurcation. For weakly elastic fluids, we determined a second critical value ℜ2Osc(λ1,λ2) above which a Hopf bifurcation from steady convective pattern to oscillatory convection occurs. The well known limit of ℜ2Osc(λ1=0,λ2=0)=390 for Newtonian fluids is recovered, while the fluid elasticity is found to delay the onset of the Hopf bifurcation. The major new findings were presented in the form of bifurcation diagrams as functions of viscoelastic parameters for ℜ up to 420.

Numerical simulation of thermal convection of viscoelastic fluids in an open-top porous medium with constant heat flux

This paper makes a numerical study of the buoyancy-driven convection of a viscoelastic fluid saturated in an open-top porous square box under the constant heat flux boundary condition. The effects of the relaxation and retardation times on the onset of the oscillatory convection, the convection heat transfer rate and the flow pattern transition are investigated. It is shown that a large relaxation time can destabilize the fluid flow leading to an early onset of the thermal convection and a high heat transfer rate, while a large retardation time tends to stabilize the flow and suppress the convection onset and the heat transfer. After the convection sets in, the flow bifurcation appears earlier with the increase of the relaxation time and the decrease of the retardation time, resulting in more complicated flow patterns in the porous medium. Furthermore, with the increase of the ratio of the relaxation time to the retardation time, the fluid may be blocked from flowing through the open-top boundary, which may be caused by the viscoelastic effect. Finally, the comparison of our results with those under isothermal heating boundary conditions reveals that the heat transfer rate corresponding to a constant heat flux boundary is always higher.

An autonomous system for thermal convection of viscoelastic fluids in a porous layer using a thermal nonequilibrium model

Thermal convection of viscoelastic fluids saturating a horizontal porous layer heated from below is analyzed using a thermal nonequilibrium model to take account of the interphase heat transfer between the fluid and the solid. The viscoelastic character of the flow is considered by a modified Darcy’s law. An autonomous system with five differential equations is deduced by applying the truncated Galerkin expansion to the momentum and heat transfer equations. The effects of interphase heat transfer H on the thermal convection of viscoelastic fluids in a porous medium are analyzed and discussed. The results show that the weak interphase heat transfer tends to stabilize the steady convection.

Generalized Lorenz Equation Derived from Thermal Convection of Viscoelastic Fluids in a Loop

A new generalized Lorenz system is presented based on the thermal convection of Oldroyd-B fluids in a circular loop. Two non-dimensional parameters De1 (a measure of the fluid relaxation) and De2 (a measure of the fluid retardation) appear in the equation. Then we study this generalized Lorenz equation numerically and find that the values of De1 and De2 can greatly influence the behavior of the solution. The fluid relaxation (De1) is found to precipitate the onset of periodic solution (limit cycle) in the system and impedes the onset of chaos while the fluid retardation (De2) tends to delay the onset of the periodic solution and precipitate the onset of chaos in the system.

Numerical simulation of thermal convection of viscoelastic fluids using the grid-by-grid inversion method

Rayleigh–Bénard convection of viscoelastic fluids in a cavity is investigated using a newly developed grid-by-grid inversion method. In the grid-by-grid inversion method the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source. This renders the stress tensor a local function of the velocity gradient tensor as in the case of the Newtonian fluids and makes the algorithms for Newtonian fluids applicable to viscoelastic fluids. To corroborate the accuracy of the grid-by-grid inversion method, a linear stability analysis is performed to find the critical Rayleigh number and the domains of Hopf bifurcation and exchange of stabilities in the parameter space. The numerical results from the grid-by-grid inversion method are found to coincide with those of linear stability analysis exactly. Also considered is the standard benchmark problem of viscoelastic flow past a cylinder placed at the center between two plates to confirm the accuracy of the grid-by-grid inversion method.

Numerical simulation of thermal convection of a viscoelastic fluid in a porous square box heated from below

Thermal convection phenomena of viscoelastic fluids in a closed-top porous square box heated from below are studied numerically based on a modified Darcy’s law, where two characteristic time constants λ and ε are used to describe the viscoelastic properties of the fluid. Results for three set values of λ and ε are obtained for a Darcy-Rayleigh number up to 400, and show very different heat transport characteristics from those of Newtonian fluids. Two types of thermal convection are shown to be possible. Oscillatory convection due to the viscoelastic properties of the fluid precedes the onset of steady convection due to thermal boundary instability. After steady convection sets in, it tends to suppress oscillatory convection. The interaction between the two types of thermal convection causes the flow patterns in the porous layer to be more complicated than those of Newtonian fluids. Furthermore, thermal convection of viscoelastic fluids in porous layers undergoes earlier bifurcations with increase of Ra than those of Newtonian fluids and the occurrence of bifurcation is earlier for larger values of λ∕ε in our study. Finally, our numerical results show that scaling laws exist for thermal convection of viscoelastic fluids in porous layers, and specific scaling laws were obtained by the least-squares fitting technique to describe the Nu versus Ra relationship in different regimes of Ra and for the three cases studied in this work.

Three-dimensional thermal convection of viscoelastic fluids

The influence of inertia and elasticity on the onset and stability of three-dimensional thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The onset and the stability of steady convective patterns, namely rolls, hexagons and squares, are studied in the post-critical range of the Rayleigh number by using an amplitude equation approach. The square pattern is found to be unstable. In contrast to Newtonian fluids, the hexagonal pattern can be stable for a certain range of elasticity.

Spatially modulated thermal convection of viscoelastic fluids.

The thermal convection of modulated viscoelastic flow is examined in this study. The modulation is assumed to be weak enough for a regular perturbation solution to be implemented. In addition to being more accurate, the second-order perturbation results reveal new physical phenomena that could not be predicted by the first-order analysis. Inertia was found to enhance globally the discrepancies between the first- and the second-order perturbation solution. A comparison between the Newtonian and the non-Newtonian solution is carried out and the influences of inertia, modulation amplitude, and wave number are emphasized. The present results show that elasticity has a marked effect on fluid patterns, especially regarding the roll structure and symmetry. The influence of elasticity is greater for larger Rayleigh number and aspect ratio.

Characterization of chaotic thermal convection of viscoelastic fluids

A four-dimensional non-linear dynamical system resulting from a truncated Fourier representation of the conservation and constitutive equations, for an Oldroyd-B fluid is used to examine the viscoelastic flow in the context of the Rayleigh–Benard thermal convection set-up. Numerical simulations are performed and the results are studied by means of time signature, phase portraits, power spectrum, Poincaré map, Lyapunov exponents and bifurcation diagrams. Depending on the flow parameters and initial conditions the asymptotic behavior can be stationary, periodic, quasiperiodic or chaotic. The aim of this paper is to identify the route(s) to chaos, and illustrate the dynamical response of the flow with the change of the control parameters. Fluid elasticity and fluid retardation alter the flow behavior in comparison to inertia-dominated Newtonian flow. Although both flows, Newtonian and viscoelastic, follow similar routes to and from chaos as a function of inertia, elasticity tends to precipitate the onset of chaos. Narrow regions or windows with ordered periodic orbits appear in some regions where the chaotic behavior seems to be dominant. Although the numerical solution can be chaotic, there is a great deal of order, or structure, to deterministic chaos. For example, viscoelastic chaotic attractor is characterized by a higher fractal dimension than the Newtonian chaotic attractor.

Hopf bifurcation in thermal convection of viscoelastic fluids within finite domains

Hopf bifurcation in thermal convection is an interesting phenomenon peculiar to the viscoelastic fluids. Employing a general constitutive relation which encompasses Maxwell model, Jeffreys model, Oldroyd-B model and Phan-Thien-Tanner model, the range of viscoelastic parameters where the Hopf bifurcation arises is determined when the viscoelastic fluid is confined in finite domains, which are compatible with the experimental situations. Employing the power series method, Landau equation is derived which determines the temporal evolution of the convection intensity in finite domains with nonslip sidewalls when the Hopf bifurcation occurs. The comparison of these results with the experimental data may be used to guide the selection of constitutive equations and to estimate viscoelastic parameter values.

Thermal convection of viscoelastic fluid with Biot boundary conduction

Two-dimensional unsteady natural convection of a non-linear fluid represented by Criminale–Erickson–Filbey (CEF) fluid model in a square cavity is studied in the fluid for Rayleigh–Benard convection case. The governing vorticity and energy transport equations are solved numerically either simple explicit and ADI methods, respectively. The two-dimensional convective motion is generated by buoyancy forces on the fluid in a square cavity, when the vertical walls are either perfectly insulated or conducted with Biot boundary conduction condition. The contributions of the elastic and shear dependent characteristics of the liquid to the non-Newtonian behaviour are investigated on the temperature distribution and heat transfer. The effect of the Weissenberg (which is a measure of the elasticity of the fluid), Rayleigh and Biot numbers on the temperature and streamline profiles are delineated and this has been documented first time for the viscoelastic fluid.

Non-linear overstability in the thermal convection of viscoelastic fluids

The existence and stability of elastic overstability in the presence of non-negligible inertia for an Oldroyd-B fluid are examined for the Rayleigh-Bernard thermal convection problem. The study is based on the four-dimensional non-linear dynamical system presented by Khayat (J. Non-Newtonian Fluid Mech., 53 (1994) 227) which constitutes a generalization of the classical Lorenz system for a Newtonian fluid. It is shown that elastic overstability can only set in once the Deborah number exceeds a critical value which depends on the Prandtl number and fluid retardation. Fluid elasticity is found to precipitate the onset of overstability while retardation tends to delay it. The conditions of existence of the corresponding Hopf bifurcation are examined as functions of fluid elasticity, retardation and thermal conductivity. The stability of the periodic orbit (in phase space) is investigated using center manifold theory. It is found that the orbit is asymptotically stable to perturbations about the conductive state, with the initial period of oscillation decreasing with Deborah number, reaching a minimum, and increasing asymptotically toward a constant value.

Chaos and overstability in the thermal convection of viscoelastic fluids

The onset of aperiodic or chaotic behaviour in viscoelastic fluids is examined in the context of the Rayleigh-Benard thermal convection setup. A truncated Fourier representation of the conservation and constitutive equations, for an Oldroyd-B fluid, leads to a four-dimensional system that constitutes a generalization of the classical Lorenz system for a Newtonian fluid. It is found that, to the order of the present truncation and below a critical Deborah number De c, the critical Rayleigh number Ra c, for the onset of steady thermal convection does not depend on fluid elasticity or retardation. For De > De c, it is shown that steady convection does not exist, with the fluid becoming overstable instead. Fluid overstability, namely when the convective cell structure is time periodic, and which is attributed to fluid elasticity, is found to set in at a Rayleigh number that depends on the Deborah number and fluid retardation, and may be much smaller than Ra c. It is also found that fluid elasticity tends to destabilize the convective cell structure, precipitating the onset of chaotic motion, at a Rayleigh number that may be well below that corresponding to Newtonian fluids.