Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free-free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.     

Non-Linear Two Dimensional Double Diffusive Convection in a Rotating Porous Layer Saturated by a Viscoelastic Fluid

Double diffusive convection in a rotating anisotropic porous layer, saturated by a viscoelastic fluid, heated from below and cooled from above has been studied making linear and non-linear stability analyses. The fluid and solid phases are considered to be in equilibrium. In momentum equation, we have employed the Darcy equation which includes both time derivative and Coriolis terms. The linear theory based on normal mode method is considered to find the criteria for the onset of stationary and oscillatory convection. A weak non-linear analysis based on minimal representation of truncated Fourier series analysis containing only two terms has been used to find the Nusselt number and Sherwood number as functions of time. We have solved the finite amplitude equations using a numerical scheme. The results obtained, during the above analyses, have been presented graphically and the effects of various parameters on heat and mass transfer have been discussed. Finally, we have drawn the steady and unsteady streamlines, isotherms, and isohalines for various parameters.     

The onset of double diffusive convection in a viscoelastic fluid layer

The onset of double diffusive convection in a viscoelastic fluid layer is studied using a linear and a weak nonlinear stability analyses. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. There is a competition between the processes of thermal diffusion, solute diffusion and viscoelasticity that causes the convection to set in through oscillatory mode rather than stationary. The effect of Deborah number, retardation parameter, solutal Rayleigh number, Prandtl number, Lewis number on the stability of the system is investigated. It is shown that the critical frequency increases with Deborah number and solutal Rayleigh number while it decreases with retardation parameter and Lewis number. The nonlinear theory based on the truncated representation of Fourier series method is used to find the heat and mass transfers. The transient behaviour of the Nusselt number and Sherwood number is investigated by solving the finite amplitude equations using Runge-Kutta method. The effect of viscoelastic parameters on heat and mass transfer is brought out.     

Linear and non-linear analyses of convection in a micropolar fluid occupying a porous medium

Linear and weakly non-linear analyses of convection in a micropolar fluid occupying a high-porosity medium are performed. The Brinkman–Eringen momentum equation is considered. The linear and non-linear analyses are, respectively, based on the normal mode technique and truncated representation of Fourier series. The linear theory for a two-phase system reiterates that the preferred mode of convection is stationary as in the case of a single-phase system. An autonomous system of differential equations representing cellular convection arising in the study is considered to analyse the critical points. The Nusselt number is obtained as a function of micropolar and porous medium parameters.     

An analytical study of linear and non-linear convection in Boussinesq-Stokes suspensions

The Rayleigh-Benard situation in Boussinesq-Stokes suspensions is investigated using both linear and non-linear stability analyses. The linear and non-linear analyses are based on a normal mode solution and minimal representation of double Fourier series, respectively. The effect of suspended particles on convection is delineated against the background of the results of the clean fluid. The realm of non-linear convection warrants the quantification of heat transfer and this has been achieved on the Rayleigh-Nusselt plane. Possibility of aperiodic convection is discussed.     

Finite deformation analysis of linearly viscoelastic simple structures2

In order to determine the effect of finite deformations on the stability and non-linear time-deflection behaviour of linearly viscoelastic uniaxially stressed structures, a series of simple rigid-bar-spring dashpot models were analysed ‘exactly’. The material representation was also kept as simple as possible using the standard three-element solid model.Results obtained indicate that the relaxation behaviour of such a structure depends only on its material properties. The creep response is influenced not only by the load level but most significantly by the instantaneous non-linear elastic characteristics of the structure. For structures exhibiting instantaneous elastic local instability a ‘critical time’ may be defined beyond which equilibrium is impossible. The definition for ‘safe-load-limit’ or viscoelastic critical force usually used in linear stability analyses of viscoelastic columns is generalized.     

Linear and non-linear double diffusive convection in a rotating porous layer using a thermal non-equilibrium model

Double diffusive convection in a fluid-saturated rotating porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium, using both linear and non-linear stability analyses. The Darcy model that includes the time derivative and Coriolis terms is employed as momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for stationary, oscillatory and finite amplitude convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of thermal and solute diffusions that causes the convection to set in through either oscillatory or finite amplitude mode rather than stationary. The effect of solute Rayleigh number, porosity modified conductivity ratio, Lewis number, diffusivity ratio, Vadasz number and Taylor number on the stability of the system is investigated. The non-linear theory based on the truncated representation of Fourier series method predicts the occurrence of subcritical instability in the form of finite amplitude motions. The effect of thermal non-equilibrium on heat and mass transfer is also brought out.     

Linear and Non-linear Double Diffusive Convection in a Fluid-Saturated Anisotropic Porous Layer with Cross-Diffusion Effects

The double diffusive convection in a horizontal anisotropic porous layer saturated with a Boussinesq binary fluid, which is heated and salted from below in the presence of Soret and DuFour effects is studied analytically using both linear and non-linear stability analyses. The linear analysis is based on the usual normal mode technique, while the non-linear analysis is based on a minimal representation of double Fourier series. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number, wavenumbers for stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory. The effects of anisotropy parameter, solute Rayleigh number, and Soret and DuFour parameters on the stationary, oscillatory convection, and heat and mass transfer are shown graphically. Some known results are recovered as special cases of the present problem.     

Non-linear convection due to compositional and thermal buoyancy in Earth's outer core

The linear stability of convection due to compositional and thermal buoyancy in Earth's outer core has been investigated. We have obtained the values of Takens-Bogdanov bifurcation points by plotting graphs of neutral curves corresponding to stationary and oscillatory convection for different values of physical parameters. We have derived a non-linear two-dimensional Landau-Ginzburg equation with real coefficients near the onset of stationary convection at a supercritical pitchfork bifurcation and two non-linear one-dimensional coupled Landau-Ginzburg type equations with complex coefficients near the onset of oscillatory convection at a supercritical Hopf bifurcation. We have studied Nusselt number contribution from a Landau-Ginzburg equation at the onset of stationary convection. We have discussed the stability regions of standing and travelling waves. We have also discussed the occurrence of secondary instabilities such as Eckhaus, zigzag and Benjamin-Feir instabilities. We have also derived the non-linear amplitude equation near the Takens-Bogdanov bifurcation point.     

Non-linear vibration of a single link viscoelastic Cartesian manipulator

In this present work, the non-linear behavior of a single-link flexible visco-elastic Cartesian manipulator is studied. The temporal equation of motion with complex coefficients of the system is obtained by using D’Alembert's principle and generalized Galarkin method. The temporal equation of motion contains non-linear geometric and inertia terms with forced and non-linear parametric excitations. It may also be found that linear and non-linear damping terms originated from the geometry of the large deformation of the system exist in this equation of motion. Method of multiple scales is used to determine the approximate solution of the complex temporal equation of motion and to study the stability and bifurcation of the system. The response obtained using method of multiple scales are compared with those obtained by numerically solving the temporal equation of motion and are found to be in good agreement. The response curves obtained using viscoelastic beams are compared with those obtained from a linear Kelvin-Voigt model and also with an equivalent elastic beam. The effect of the material loss factor, amplitude of base excitation, and mass ratio on the steady state responses for both simple and subharmonic resonance conditions are investigated.     

A non-linear uniaxial integral constitutive equation incorporating rate effects,creep and relaxation

A previously proposed first order non-linear differential equation for uniaxial viscoplasticity, which is non-linear in stress and strain but linear in stress and strain rates, is transformed into an equivalent integral equation. The proposed equation employs total strain only and is symmetric with respect to the origin and applies for tension and compression. The limiting behavior for large strains and large times for monotonic, creep and relaxation loading is investigated and appropriate limits are obtained. When the equation is specialized to an overstress model it is qualitatively shown to reproduce key features of viscoplastic behavior. These include: initial linear elastic or linear viscoelastic response: immediate elastic slope for a large instantaneous change in strain rate normal strain rate sensitivity and non-linear spacing of the stress-strain curves obtained at various strain rates; and primary and secondary creep and relaxation such that the creep (relaxation) curves do not cross. Isochronous creep curves are also considered. Other specializations yield wavy stress-strain curves and inverse strain rate sensitivity. For cyclic loading the model must be modified to account for history dependence in the sense of plasticity.     

Thermal Instability in a Porous Medium Layer Saturated with a Viscoelastic Nanofluid

The onset of convection in a horizontal layer of a porous medium saturated with a viscoelastic nanofluid was studied in this article. The modified Darcy model was applied to simulate the momentum equation in porous media. An Oldroyd-B type constitutive equation was used to describe the rheological behavior of viscoelastic nanofluids. The model used for the viscoelastic nanofluid incorporates the effects of Brownian motion and thermophoresis. The onset criterion for stationary and oscillatory convection was analytically derived. The effects of the concentration Rayleigh number, Prandtl number, Lewis number, capacity ratio, relaxation, and retardation parameters on the stability of the system were investigated. Oscillatory instability is possible in both bottom- and top-heavy nanoparticle distributions. Results indicated that there is competition among the processes of thermophoresis, Brownian diffusion, and viscoelasticity that causes the convection to set in through oscillatory rather than stationary modes. Regimes of stationary and oscillatory convection for various parameters were derived and are discussed in detail.     

Nonlinear vibration of viscoelastic sandwich plates under narrow-band random excitations

The effect of the narrow-band random excitation on the non-linear response of sandwich plates with an incompressible viscoelastic core is investigated. To model the core, both the transverse shear strains and rotations are assumed to be moderate and the displacement field in the thickness direction is assumed to be linear for the in-plane components and quadratic for the out-of-plane components. In connection to the moderate shear strains considered for the core, a non-linear single-integral viscoelastic model is also used for constitutive modeling of the core. The fifth-order perturbation method is used together with the Galerkin method to transform the nine partial differential equations to a single ordinary integro-differential equation. Converting the lower-order viscoelastic integral term to the differential form, the fifth-order method of multiple scale is applied together with the method of reconstitution to obtain the stochastic phase-amplitude equations. The Fokker–Planck–Kolmogorov equation corresponding to these equations is then solved by the finite difference method, to determine the probability density of the response. The variation of root mean square and marginal probability density of the response amplitude with excitation deterministic frequency and magnitudes are investigated and the bimodal distribution is recognized in narrow ranges of excitation frequency and magnitude.     

Traveling waves in one-dimensional non-linear models of strain-limiting viscoelasticity

In this paper we investigate traveling wave solutions of a non-linear differential equation describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models introduced by Rajagopal. To describe the response of viscoelastic solids we assume a non-linear relationship among the linearized strain, the strain rate and the Cauchy stress. We then concentrate on traveling wave solutions that correspond to the heteroclinic connections between the two constant states. We establish conditions for the existence of such solutions, and find those solutions, explicitly, implicitly or numerically, for various forms of the non-linear constitutive relation.     

Effect of Thermal Modulation on the Onset of Convection in a Viscoelastic Fluid Saturated Porous Layer

The effect of thermal modulation on the onset of convection in a horizontal, anisotropic porous layer saturated by a viscoelastic fluid is investigated by a linear stability analysis. Darcy’s law with viscoelastic correction is used to describe the fluid motion. The perturbation method is used to find the critical Rayleigh number and the corresponding wavenumber for small amplitude thermal modulation. The stability of the system characterized by a correction Rayleigh number is calculated as a function of the thermal and mechanical anisotropy parameters, the viscoelastic parameters and the frequency of modulation. It is found that the onset of convection can be delayed or advanced by the factors represented by these parameters. The results of the problem have possible implications in mantle convection.     

An outline of the non-linear viscoelastic behaviour of wheat flour dough in shear

Creep and creep recovery, stress relaxation and small- and large-amplitude oscillatory shear experiments have been used to study the steady-state flow behaviour and the transient viscoelastic response of wheat flour dough in shear over large ranges of time, stress and strain. The results are discussed with reference to the limited body of reliable literature data. Dough does display a linear viscoelastic domain. The complex character of its non-linear viscoelastic properties is essentially due to the extremely low shear rate limit of the initial Newtonian plateau and to the onset of time-dependent flow behaviour above a certain strain threshold, which explain qualitatively the discrepancies observed in certain cases on a part of the range of the rheological variables explored, despite global self-consistency of the results. Comparison of gluten and dough linear viscoelastic properties shows that dough cannot be viewed simply as a concentrated suspension of starch granules in the hydrated viscoelastic gluten matrix.Paper presented at the second Annual European Rheology Conference (AERC 2005) held in Grenoble, France on April 21–23, 2005.     

A detailed comparison of various FENE dumbbell models

The rheological behaviour of dilute solutions of finitely extensible non-linear elastic (FENE) dumbbells in both steady state and transient shear and simple elongational flow is investigated. Three dumbbell models are compared: the original FENE model with the Warner spring force, which is treated by brownian dynamics simulations, and the FENE-P model based on the Peterlin approximation and the FENE-CR model as suggested by Chilcott and Rallison, which are treated by standard numerical techniques. It is shown that in the linear viscoelastic limit and in steady state flows the behaviour is similar, except for the FENE-CR dumbbell in shear flow, modelling a Boger fluid. In transient flows larger differences appear.     

Stability of conducting liquid flowing down an inclined plane at moderate Reynolds number in the presence of constant electromagnetic field

The stability of a conducting viscous film flowing down an inclined plane at moderate Reynolds number in the presence of electromagnetic field is investigated under induction-free approximation. Using momentum integral method a non-linear evolution equation for the development of the free surface is derived. The linear stability analysis of the evolution equation shows that the magnetic field stabilizes the flow whereas the electric field stabilizes or destabilizes the flow depending on its orientation with the flow. The weakly non-linear study reveals that both the supercritical stability and subcritical instability are possible for this type of thin film flow. The influence of magnetic field on the different zones is very significant, while the impact of electric field is very feeble in comparison.     

Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous medium

Stability analysis of double-diffusive convection for viscoelastic fluid with Soret effect in a porous medium is investigated using a modified-Maxwell-Darcy model. We use the linear stability analysis to investigate how the Soret parameter and the relaxation time of viscoelastic fluid effect the onset of convection and the selection of an unstable wavenumber. It is found that the Soret effect is to destabilize the system for oscillatory convection. The relaxation time also enhances the instability of the system. The effects of Soret coefficient and relaxation time on the heat transfer rate in a porous medium are studied using the nonlinear stability analysis, the variation of the Nusselt number with respect to the Rayleigh number is derived for stationary and oscillatory convection modes. Some previous results can be reduced as the special cases of the present paper.     

Non-linear heat transfer of solids in orthogonal coordinate systems

Solutions to the non-linear partial differential equation of heat conduction, (Poisson type), are obtained in which the conductivity is temperature dependent, by solving a linear partial differential equation and transforming it to the non-linear form using the Kirchhoff transformation. The method applies to any orthogonal coordinate system.

Transformations for handling boundary conditions of the Dirichlet, Neumann, convection and non-zero type are developed. The method is extended to solve a special class of non-linear unsteady-state conduction problems.

Two non-linear examples are solved to illustrate the method.