Effect of Thermal Modulation on the Onset of Convection in Walters B Viscoelastic Fluid-Saturated Porous Medium

The linear stability of Walters B viscoelastic fluid-saturated horizontal porous layer is examined theoretically when the walls of the porous layer are subjected to time-periodic temperature modulation. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and the corresponding wave number. The shift in critical Rayleigh number is calculated as a function of modulation frequency, viscoelastic parameter, and Prandtl number. The effect of all three types of modulations is found to be destabilizing as compared to the unmodulated system. This result is in contrast to the system with other types of fluids. Besides, the influence of physical parameters on the control of convective instability of the system is discussed

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. © 2013 Elsevier Inc.

The Onset of Stationary and Oscillatory Convection in a Horizontal Porous Layer Saturated with Viscoelastic Liquid Heated and Soluted From Below: Effect of Anisotropy

The onset of double diffusive stationary and oscillatory convection in a viscoelastic Oldroyd type fluid saturated in an anisotropic porous layer heated and soluted from below is studied. The flow is governed by the extended Darcy model for Oldroyd fluid. Stability analysis based on the method of perturbations of infinitesimal amplitude is performed using the normal mode technique. The analysis examines the effect of the Darcy Rayleigh number, the solutal Darcy the Rayleigh number, the relaxation time, the retardation time and the Lewis number. Important conclusions include the destabilizing effect of the relaxation time, the Darcy Rayleigh number and the Lewis number and the stabilizing effect of the solutal Darcy Rayleigh number, the retardation time and anisotropy parameter. Some of the results are generalization of the previous findings for isotropic porous medium

Thermal Convection in Non-Fourier Fluids and Application to Liquid Helium II

This thesis examines different conditions for which non-Fourier effects can be significant in the flow of fluids. Non-Fourier fluids of dual-phase-lagging type (DPL) possess a relaxation time and a retardation time, reflecting the delay in the response of the heat flux and the temperature gradient with respect to one another. For non-Fourier fluids of single-phase-lagging type (SPL) the retardation time is zero. Non-Fourier fluids span a wide range of applications, including liquid helium, nanofluids and rarefied gases. The parallels between non-Fourier fluids and polymeric solutions are established. The instability of steady natural convection of a thin layer of non-Fourier fluid (SPL) between two horizontal (and vertical) surfaces maintained at different temperatures is studied. The SPL model is particularly relevant to liquid helium II, and nanofluids with high nanoparticle concentration. Linear stability analysis is employed to obtain the critical state parameters such as critical Rayleigh (Grashof) numbers. In both cases, as the fluid becomes more non-Fourier, oscillatory convection increasingly becomes the mode of preference, compared to both conduction and stationary convection. Critical Rayleigh (Grashof) number decreases for fluids with higher non-Fourier levels. By invoking the role of the eigenvectors to detect and quantify short-time behavior, transient growth of energy of disturbances in is studied. The energy of the perturbations is introduced in terms of the primary variables as a disturbance measure in order to quantify the size of the disturbance. It is found that nonlinearities are not required for the energy growth, and a significant energy growth can be observed even if the flow is stable. The post-critical convective state for Rayleigh-Benard convection is studied using a nonlinear spectral-amplitude-perturbation approach in a fluid layer heated from below. In the spectral method the flow and temperature fields are expanded periodically along the layer and orthonormal shape functions are used in the transverse direction. A combined amplitude-perturbation approach is developed to solve the nonlinear spectral system in the post critical range, even far from the linear stability threshold. Also, to leading order, the Lorenz model is recovered. Comparison with experimental results is made and a very good qualitative agreement is obtained

Conditional and unconditional nonlinear stability in fluid dynamics

In this thesis we examine some of the interesting aspects of stability for some convection problems. Specifically, the first part of the thesis deals with the Bénard problem for various Non-Newtonian fluids, whereas the second part develops a stability analysis for convection in a porous medium. The work on stability for viscoelastic fluids includes nonlinear stability analyses for the second grade fluid, the generalised second grade fluid, the fluid of dipolar type and the fluid of third grade. It is worth remarking that throughout the work the viscosity is supposed to be any given function of temperature, with the first derivative bounded above by a positive constant. The connection between the two parts of the thesis is made through the method used to approach the nonlinear stability analysis, namely the energy method. It is shown in the introductory chapter how this method works and what are its advantages over the linear analysis. Nonlinear stability results established in both Part I and Part II are the best one can get for the considered physical situations. Different choices of energy have been considered in order to achieve conditional or unconditional nonlinear stability results

Studies In Small Scale Thermal Convection

The effect of non-Fourier heat transfer and partial-slip boundary conditions in Rayleigh-Bénard are analyzed theoretically. Non-Fourier fluids possess a relaxation time that reflects the delay in the response of the heat flux to a change in the temperature gradient while the partial slip boundary condition assumes that the fluid velocity and temperature are not equal to that of the wall. Both non-Fourier and partial-slip effects become important when small-scale heat transfer applications are investigated such as convection around micro- and nano-devices as suggested by the extended heat transport equations from kinetic theory. Other applications are also known to exhibit one or both of these effects such as low-temperature liquids, nanofluids, granular flows, rarefied gases and polymer flows. Small scale effects are measured by the Knudsen number. From this, non-Fourier effects can be estimated, measured non-dimensionally by the Cattaneo number and modelled using the frame indifferent Cattaneo-Vernotte equation which is derived from Oldroyd’s upper-convected derivative. Linear stability of non-Fourier fluids shows that the neutral stability curve possesses a stationary Fourier branch and an oscillatory branch intersecting at some wave number, where for small relaxation time, no change in stability is expected from that of a Fourier fluid. As the relaxation time increases to a critical value, both stationary and oscillatory convection become equally probable. Past this value, oscillatory instability is expected to occur at a smaller Rayleigh number and larger wave number than for a Fourier fluid. Non-linear analysis of weakly non-Fourier fluids shows that near the onset of convection, the convective roll intensity is stronger than for a Fourier fluid. The bifurcation to convection changes from subcritical to supercritical as the Cattaneo number increases and the instability of the convection state for a non-Fourier fluid is shown to occur via a Hopf bifurcation at lower Rayleigh number and higher Nusselt number than for a Fourier fluid. When hydrodynamic slip and temperature jump boundary conditions are considered, a significant reduction in the minimum critical Rayleigh number and corresponding wave number are found. Depending on the sign used for second-order coefficients, critical conditions can be greater than or less than that for first-order boundary conditions

Numerical representations of fluid mixing

The work contained within this thesis is concerned with a theoretical investigatiop of both laminar and thermally driven types of cavity flow, together with an analysis of their associated mixing processes which find applications to Industrial mixing and also to the environment. The mixing efficiency has been viewed from two perspectives namely the tracking of a selection of fluid particles, and also the simulation of the dispersive mixing of a coloured fluid element as carried along by the flow. This thesis also incorporates features of both Newtonian and a wide range of non-Newtonian fluids

Effect of time-periodic Boundary temperatures/body Force on Rayleigh-Benard Convection in a Ferromagnetic Fluid

We discuss the thermal instability in a layer of a ferromagnetic fluid when the boundaries of the layer are subjected to synchronous/asynchronous imposed time-periodic boundary temperatures (ITBT)/ time-periodic body force (TBF). Only infinitesimal disturbances are considered. The Venezian approach is adopted in arriving at the critical Rayleigh and wave numbers for small amplitudes of ITBT. A pertur- bation solution in powers of the amplitude of the applied temperature field is obtained. When the ITBT at the two walls are synchronized then, for moderate frequency values, the role of magnetization in inducing sub-critical instabilities is delineated. A similar role is shown to be played by the Prandtl number. The magnetization parameters and Prandtl number have the opposite effect at large frequencies. The system is most stable when the ITBT is asynchronous. The effect of TBF on the onset of convection is found to be qualitatively similar to the effect of an asynchronous ITBT. Low Prandtl number fluids are shown to be more easily vulnerable to destabilization by TBF compared to very large Prandtl number fluids. The problem has relevance in many ferromagnetic fluid applications wherein regulation of thermal convection is called for