Convection Of Viscoelastic Fluid Research Articles

Thermal convection of viscoelastic fluids in concentric rotating cylinders: Elastic turbulence and kinetic energy budget analysis

Thermal convection of viscoelastic fluids in concentric rotating cylinders: Elastic turbulence and kinetic energy budget analysis

Rayleigh–Bénard convection of viscoelastic fluids in three-dimension finite cavity

The Rayleigh–Bénard convection (RBC) with viscoelastic fluids has captured wide attention in the past decades, caused by its ubiquitous uses in the heat transfer process. However, the numerical technique for simulating the viscoelastic fluid flows developed slowly until recent years because of the numerical instability raised by the hyperbolic characteristics of the viscoelastic constitutive equation. In the present work, a novel numerical scheme was developed to simulate the three-dimension (3D) RBC with viscoelastic fluids, where the quasi-linear treatment was applied to the hyperbolic terms of the momentum equation and the viscoelastic constitutive equation. The in-house solver was also optimized in the aspect of time cost. The results show that the critical value of convection onset in 3D is near to that in the two-dimension (2D); however, the flow pattern displays the wave characteristics in the horizontal direction. The viscoelastic kinetic energy budget of oscillating convection in 3D still conforms to the energy transport law of that in 2D [Zheng et al., Phys. Rev. Fluids 8, 023303 (2023)].

Influence of elastic instability and elastic turbulence on mixed convection of viscoelastic fluids in a lid-driven cavity

The addition of a minute amount of solid polymers into a Newtonian solvent like water can originate an elastic instability as the Weissenberg number (ratio of the elastic forces to that of the viscous forces) gradually increases. These instabilities further transit to elastic turbulence (ET) regime at high Weissenberg numbers. In this study, we investigate how these elastic instabilities and elastic turbulence would tend to influence the mixed convective heat transfer phenomena in a lid-driven cavity, which is one of the widely studied problems in the domain of transport phenomena. In doing so, extensive two-dimensional time-resolved numerical simulations have been conducted by using an open-source CFD code OpenFOAM. The Oldroyd-B viscoelastic constitutive model is used in this study to realize the rheological behaviour of a constant shear-viscosity viscoelastic fluid or the so-called Boger fluid. To investigate the effect of the shear-thinning behaviour of a viscoelastic fluid on the elastic instability and elastic turbulence phenomena, we have further conducted limited simulations using the FENE-P (finitely extensible non-linear elastic spring with the Peterlin’s approximation) viscoelastic fluid model. A detailed discussion is presented on how the flow dynamics and heat transfer mechanism inside the cavity can be influenced by the Weissenberg number along with the Reynolds, Prandtl, and Richardson numbers. We show that the heat transfer rate inside the cavity can be increased by more than 100% under appropriate conditions if one uses a constant shear-viscosity Oldroyd-B fluid instead of a simple Newtonian fluid due to the presence of elastic instability and elastic turbulence in the former fluid. Furthermore, we have found that the shear-thinning behaviour of a FENE-P viscoelastic fluid tends to suppress these instabilities and turbulence and hence decreases the heat transfer rate inside the cavity. Therefore, one can use the present approach of using a constant shear-viscosity Oldroyd-B fluid as an efficient way to increase the rate of heat and mass transport processes in various systems. Our approach can be served as an alternative possibility to the existing approaches such as the application of an electric or a magnetic field or the use of a nanofluid. However, one needs to be careful in the selection of the polymer type and its degradation behaviour as well as its appropriate concentration in the solution when choosing this approach.

Unsteady free convective heat transfer in third-grade fluid flow from an isothermal vertical plate: A thermodynamic analysis

The current study investigates theoretically and numerically the entropy generation in time-dependent free-convective third-grade viscoelastic fluid convection flow from a vertical plate. The nondimensional conservation equations for mass, momentum and energy are solved using a Crank–Nicolson finite difference method with suitable boundary conditions. Expressions for known values of flow-variables coefficients are also derived for the wall heat transfer and skin friction and numerically evaluated. The effect of Grashof number, Prandtl number, group parameter (product of dimensionless temperature difference and Brinkman number) and third-grade parameter on entropy heat generation is analyzed and shown graphically. Bejan line distributions are also presented for the influence of several control parameters. The computations reveal that with increasing third-grade parameter, the entropy generation decreases and Bejan number increases. Also, the comparison graph shows that contour lines for third-grade fluid vary considerably from the Newtonian fluid. The study is relevant to non-Newtonian thermal materials processing systems.

Chaotic Convection of Viscoelastic Fluid in Porous Medium Under G-Jitter

The present article aims at investigating the effect of gravity modulation on chaotic convection of a viscoelastic fluid in porous media. For this, the problem is reduced into Lorenz system (non-autonomous) by employing the truncated Galerkin expansion method. The system shows transitions from periodic to chaotic behavior on increasing the scaled Rayleigh number R. The amplitude of modulation advances the chaotic nature in the system while the frequency of modulation has a tendency to delay the chaotic behavior which is in good agreement with the results due to [1]. The behavior of the scaled relaxation and retardation parameter on the system is also studied. The phase portrait and time domain diagrams of the Lorenz system for suitable parameter values have been used to analyze the system.

Effect of Thermal Conductivity and Thickness of the Walls on the Natural Convection in a Horizontal Viscoelastic Jeffreys Fluid Layer

It is a common practice to use ideal thermal boundary conditions to investigate natural convection. These correspond to very good conducting walls and to very bad conducting walls. In particular, this has been the case in natural convection of viscoelastic fluids. In this paper, these conditions are generalized by taking into account the finite thermal conductivities and thicknesses of the walls in the natural convection of a viscoelastic Jeffreys fluid heated from below. The goal is to present more realistic results related to experimental conditions. The critical Rayleigh number Rc, the frequency of oscillation ωc, and the wavenumber kc have been plotted varying the properties of the walls from the case of very good thermal conductivity to very poor thermal conductivity. In order to understand the convective phenomena, two parameters are fixed and the other one varied among the nondimensional relaxation time F, the relative retardation time E, and the Prandtl number Pr of the viscoelastic fluid. The role of the relative retardation time E on the thermal instability is discussed in detail.

Peculiarity in the Rayleigh-Bénard convection of viscoelastic fluids

The Rayleigh-Bénard convection of viscoelastic fluids in a cavity is simulated using the grid-by-grid inversion method. The grid-by-grid inversion method allows the algorithms for Newtonian fluids to be applicable to viscoelastic fluids using a technique of deferred correction. Until now, the detailed characteristics of thermal convection in a cavity have not been investigated sufficiently. Employing the grid-by-grid inversion method, we have found a peculiar phenomenon specific to the viscoelastic fluids, which cannot be predicted by the linear and nonlinear stability analysis. Contrary to the cases of Newtonian fluids and viscoelastic fluids in the parameter range of exchange of stabilities, the convection cell of viscoelastic fluids in the parameter range of Hopf bifurcation changes the circulation direction at each temporal period of oscillatory convection.

Numerical Solution for Ohmic Soret-Dufour Heat and Mass Mixed Convection of Viscoelastic Fluid over a Stretching Sheet with Multimedia Physical Features

AbstractIn this study, an analysis was performed for numerical Soret-Dufour heat and mass transfer of a steady laminar boundary-layer flow with Ohmic dissipation, electromagnetic hydrodynamic (EMHD...

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.

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.

Natural Convection of Viscoelastic Fluid from a Cone Embedded in a Porous Medium with Viscous Dissipation

We study natural convection from a downward pointing cone in a viscoelastic fluid embedded in a porous medium. The fluid properties are numerically computed for different viscoelastic, porosity, Prandtl and Eckert numbers. The governing partial differential equations are converted to a system of fourth order ordinary differential equations using the similarity transformations and then solved together by using the successive linearization method (SLM). Many studies have been carried out on natural convection from a cone but they did not consider a cone embedded in a porous medium with linear surface temperature. The results in this work are validated by the comparison with other authors.

Numerical Investigation of Transient Natural Convection of Viscoelastic Fluid Due to a Nonlinearly Stretching Surfaces with a Power-Law Surface Temperature

Transient boundary layer flow and heat transfer of a viscoelastic fluid due to non-linearly stretching surface subject to a power-law surface temperature are investigated numerically. The dimensionless unsteady, coupled and nonlinear partial governing differential equations are solved by an implicit finite difference scheme of Crank-Nicolson type. The effects of various parameters entering into the problem on the flow and heat transfer transient characteristics are studied. It was found that, the velocity inside the boundary layer increases as the viscoelastic parameter increases, and accordingly the skin friction coefficient increases. Meanwhile, an opposite behavior is observed for the temperature profiles. The power-law index has significant influences on the temperature profiles and the Nusselt. The computed numerically results are compared with previously reported work and were found to be in excellent agreement. This paper is the first attempt in introducing the heat transfer of a viscoelastic fluid due to non-linearly stretching surface subject to a power-law surface temperature and this model is essential in many patents based on drawing devices for viscoelastic fluid over a non-linearly stretching surface subject to a power-law surface temperature.

English

The thermal instability of a Rivlin-Ericksen viscoelastic fluid, acted upon by uniform vertical rotation and heated from below, is investigated. Following linearized stability theory and normal mode analysis, the mathematical analysis of the governing equations of Rivlin-Ericksen viscoelastic fluid convection with a uniform vertical rotation is performed. It is shown that for the cases of rigid boundaries the complex growth rate  of oscillatory perturbations, neutral or unstable for all wave numbers, must lie inside a semi-circle, in the right-hand half of a complex  -plane with the center at the origin. This prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a rotatory Rivlin-Ericksen viscoelastic fluid heated from below. Furthermore, the conditions necessary for the existence of oscillatory motions of growing amplitude in the present configuration and the sufficient condition for the validity of the Principle of Exchange of Stabilities are established.

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.

The convective instability of Maxwell fluid-saturated porous layer using a thermal non-equilibrium model

A linear stability analysis is carried out to study viscoelastic fluid convection in a horizontal porous layer heated from below and cooled from above when the solid and fluid phases are not in a local thermal equilibrium. The modified Darcy–Brinkman–Maxwell model is used for the momentum equation and two-field model is used for the energy equation each representing the solid and fluid phases separately. The conditions for the onset of stationary and oscillatory convection are obtained analytically. Linear stability analysis suggests that, there is a competition between the processes of viscoelasticity and thermal diffusion that causes the first convective instability to be oscillatory rather than stationary. Elasticity is found to destabilize the system. Besides, the effects of Darcy number, thermal non-equilibrium and the Darcy–Prandtl number on the stability of the system are analyzed in detail.

MHD Mixed Convection of Viscoelastic Fluid Over a Stretching Sheet with Ohmic Dissipation

AbstractA magnetic hydrodynamic (MHD) mixed convection of an incompressible viscoelastic fluid over a stretching sheet with ohmic dissipation is studied. The buoyant effect and the electric number E1 couple with magnetic parameter M to represent the dominance of the ohmic effect are presented in governing equations which is the main contribution by this study. The similarity transformation, the finitedifference method have been used to analyze the present problem. The numerical solutions of the flow velocity distributions, temperature profiles and the important wall unknown values of f″(0) and θ′(0) are carried out.

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.