Peter Vadasz

Emerging Topics in Heat and Mass Transfer in Porous Media

Preface Chapter 1: Wang, L.Q., Xu, M., Wei, X. - Dual-Phase-Lagging and Porous-Medium Heat Conduction Processes Chapter 2: Haji-Sheikh, A. & Minkowycz, W.J. - Heat Transfer Analysis under Local Thermal Non-Equilibrium Conditions Chapter 3: Nield, D. A. - Generalized Heterogeneity Effects on the Onset of Convection in a Porous Medium. Chapter 4: Rees, D.A.S., Selim, A., Ennis-King, J.P. - The Instability of Unsteady Boundary Layers in Porous Media. Chapter 5: Vadasz, P. - Analytical Transition to Weak Turbulence and Chaotic Natural Convection in Porous Media. Chapter 6: Govender, S. - Natural Convection in Gravity-Modulated Porous Media Chapter 7: Pedramrazi, Y., Charrier-Mojtabi, M.C., Mojtabi, A. - Thermal Vibrational Convection in a Porous Medium Saturated by a Pure or Binary Fluid. Chapter 8: Kuznetsov, A.V. - New Developments in Bioconvection in Porous Media: Bioconvection Plumes, Bio-Thermal Convection, and Effects of Vertical Vibration. Chapter 9: Kanafer, K., Vafai, K.- Macromolecular Transport in Arterial Walls: Current and Future Directions. Chapter 10: Khanafer, K., AlAmiri, A., Pop, I., Bull, J.L.- Flow and Heat Transfer in Biological Tissues: Application of Porous Media Theory. Chapter 11: Krishnan, S., Murthy, J.Y., Garimella, S.V. - Metal Foams as Passive Thermal Control Systems. Chapter 12: Vadasz, P. - Nano-fluid Suspensions and Bi-Composite Media as Derivatives of Interface Heat Transfer Modelling in Porous Media. Subject Index

Heat Conduction in Nanofluid Suspensions

The heat conduction mechanism in nanofluid suspensions is derived for transient processes attempting to explain experimental results, which reveal an impressive heat transfer enhancement. In particular, the effect of the surface-area-to-volume ratio (specific area) of the suspended nanoparticles on the heat transfer mechanism is explicitly accounted for, and reveals its contribution to the specific solution and results. The present analysis might provide an explanation that settles an apparent conflict between the recent experimental results in nanofluid suspensions and classical theories for estimating the effective thermal conductivity of suspensions that go back more than one century (Maxwell, J.C., 1891, Treatise on Electricity and Magnetism). Nevertheless, other possible explanations have to be accounted for and investigated in more detail prior to reaching a final conclusion on the former explanation.

Coriolis effect on gravity-driven convection in a rotating porous layer heated from below

Linear stability and weak nonlinear theories are used to investigate analytically the Coriolis effect on three-dimensional gravity-driven convection in a rotating porous layer heated from below. Major differences as well as similarities with the corresponding problem in pure fluids (non-porous domains) are particularly highlighted. As such, it is found that, in contrast to the problem in pure fluids, overstable convection in porous media is not limited to a particular domain of Prandtl number values (in pure fluids the necessary condition is Pr <1). Moreover, it is also established that in the porous-media problem the critical wavenumber in the plane containing the streamlines for stationary convection is not identical to the critical wavenumber associated with convection without rotation, and is therefore not independent of rotation, a result which is quite distinct from the corresponding pure-fluids problem. Nevertheless it is evident that in porous media, just as in the case of pure fluids subject to rotation and heated from below, the viscosity at high rotation rates has a destabilizing effect on the onset of stationary convection, i.e. the higher the viscosity the less stable the fluid. Finite-amplitude results obtained by using a weak nonlinear analysis provide differential equations for the amplitude, corresponding to both stationary and overstable convection. These amplitude equations permit one to identify from the post-transient conditions that the fluid is subject to a pitchfork bifurcation in the stationary convection case and to a Hopf bifurcation associated with the overstable convection. Heat transfer results were evaluated from the amplitude solution and are presented in terms of Nusselt number for both stationary and overstable convection. They show that rotation has in general a retarding effect on convective heat transfer, except for a narrow region of small values of the parameter containing the Prandtl number where rotation enhances the heat transfer associated with overstable convection.

Weak Turbulence and Chaos for Low Prandtl Number Gravity Driven Convection in Porous Media

Low Prandtl number convection in porous media is relevant to modern applications of transport phenomena in porous media such as the process of solidification of binary alloys. The transition from steady convection to chaos is analysed by using Adomians decomposition method to obtain an analytical solution in terms of infinite power series. The practical need to evaluate the solution and obtain numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a finite accuracy. The solution shows a transition from steady convection to chaos via a Hopf bifurcation producing a solitary limit cycle’ which may be associated with an homoclinic explosion. This occurs at a slightly subcritical value of Rayleigh number, the critical value being associated with the loss of linear stability of the steady convection solution. Periodic windows within the broad band of parameter regime where the chaotic solution persists are identified and analysed. It is evident that the further transition from chaos to a high Rayleigh number periodic convection occurs via a period halving sequence of bifurcations.

Convergence and accuracy of Adomian’s decomposition method for the solution of Lorenz equations

Abstract The convergence and accuracy of Adomian’s decomposition method of solution is analysed in the context of its application to the solution of Lorenz equations which govern at lower order the convection in a porous layer (or respectively in a pure fluid layer) heated from below. Adomian’s decomposition method provides an analytical solution in terms of an infinite power series and is applicable to a much wider range of heat transfer problems. The practical need to evaluate the solution and obtain numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task, transform the analytical results into a computational solution evaluated up to a finite accuracy. The analysis indicates that the series converges within a sufficiently small time domain, a result that proves to be significant in the derivation of the practical procedure to compute the infinite power series. Comparison of the results obtained by using Adomian’s decomposition method with corresponding results obtained by using a numerical Runge–Kutta–Verner method show that both solutions agree up to 12–13 significant digits at subcritical conditions, and up to 8–9 significant digits at certain supercritical conditions, the critical conditions being associated with the loss of linear stability of the steady convection solution. The difference between the two solutions is presented as projections of trajectories in the state space, producing similar shapes that preserve under scale reduction or magnification, and are presumed to be of a fractal form.

Local and global transitions to chaos and hysteresis in a porous layer heated from below

The routes to chaos in a fluid saturated porous layer heated from below are investigated by using the weak nonlinear theory as well as Adomians decomposition method to solve a system of ordinary differential equations which result from a truncated Galerkin representation of the governing equations. This representation is equivalent to the familiar Lorenz equations with different coefficients which correspond to the porous media convection. While the weak nonlinear method of solution provides significant insight to the problem, to its solution and corresponding bifurcations and other transitions, it is limited because of its local domain of validity, which in the present case is in the neighbourhood of any one of the two steady state convective solutions. On the other hand, the Adomians decomposition method provides an analytical solution to the problem in terms of infinite power series. The practical need to evaluate numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task transform the otherwise analytical results into a computational solution achieved up to a finite accuracy. The transition from the steady solution to chaos is analysed by using both methods and their results are compared, showing a very good agreement in the neighbourhood of the convective steady solutions. The analysis explains previously obtained computational results for low Prandtl number convection in porous media suggesting a transition from steady convection to chaos via a Hopf bifurcation, represented by a solitary limit cycle at a sub-critical value of Rayleigh number. A simple explanation of the well known experimental phenomenon of Hysteresis in the transition from steady convection to chaos and backwards from chaos to steady state is provided in terms of the present analysis results.

Transitions and chaos for free convection in a rotating porous layer

Abstract The non-linearity which is inherently present in centrifugally driven free convection in porous media raises the problem of multiple solutions existent in this particular type of system. The solution to the non-linear problem is obtained by using a truncated Galerkin method to obtain a set of ordinary differential equation for the time evolution of the Galerkin amplitudes. It is demonstrated that Darcys model when extended to include the time derivative term yields, subject to appropriate scaling, the familiar Lorenz equations although with different coefficients, at a similar level of Galerkin truncation. The system of ordinary differential equations was solved by using Adomians decomposition method. Below a certain critical value of the centrifugally related Rayleigh number the obvious unique motionless conduction solution is obtained. At slightly super-critical values of the centrifugal Rayleigh number a pitchfork bifurcation occurs, leading to two different steady solutions. For highly supercritical Rayleigh numbers transition to chaotic solutions occurs via a Hopf bifurcation. The effect of the time derivative term in Darcys equation is shown to be crucial in this truncated model as the value of Rayleigh number when transition to the non-periodic regime occurs goes to infinity at the same rate as the time derivative term goes to zero. Examples of different convection solutions and the resulting rate of heat transfer are provided.

Convection and stability in a rotating porous layer with alternating direction of the centrifugal body force

An alternating direction of the centrifugal body force results when the axis of rotation is placed within the boundaries of a rotating fluid saturated porous layer. The onset of thermal convection and stability in a fluid saturated porous layer prevailing such conditions is investigated analytically. The marginal stability criterion was evaluated in terms of a critical centrifugal Rayleigh number and a corresponding critical wave number. The effect of the offset distance of the layers cold wall from the axis of rotation on the convection is analyzed, showing that the critical centrifugal Rayleigh and wave numbers increase significantly as the layers cold wall moves away from the rotation axis. This leads eventually to unconditional stability when the layers hot wall coincides with the rotation axis. This unconditional stability prevails when the axis of rotation moves away from the porous domain, so that the imposed temperature gradient opposes the direction of the centrifugal acceleration. Significant effects on the convection pattern are identified as a result of the rotation axis location.

Subcritical transitions to chaos and hysteresis in a fluid layer heated from below

Abstract The route to chaos in a fluid layer heated from below is investigated by using the weak non-linear theory as well as Adomian’s decomposition method to solve a system of ordinary differential equations which result from a truncated Galerkin representation of the governing equations. This representation yields the familiar Lorenz equations. While the weak non-linear method of solution provides significant insight to the problem, to its solution and corresponding bifurcations and other transitions, it is limited because of its local domain of validity, which in the present case is in the neighbourhood of any one (but only one) of the two steady state convective solutions. This method is expected to loose accuracy and gradually breakdown as one moves away from this neighbourhood. On the other hand, Adomian’s decomposition method provides an analytical solution to the problem in terms of an infinite power series. The practical need to evaluate numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task transform the otherwise analytical results into a computational solution achieved up to a finite accuracy. The transition from the steady solution to chaos is analysed by using both methods and their results are compared, showing a very good agreement in the neighbourhood of the convective steady solutions. The analysis explains the computational results, which indicate a transition from steady convection to chaos via a solitary limit cycle followed by a homoclinic explosion at a subcritical value of a Rayleigh number. A transient analysis of the amplitude equation obtained from the weak non-linear solution reveals the mechanism by which the Hopf bifurcation becomes subcritical. A simple explanation of the well-known experimental phenomenon of hysteresis in the transition from steady convection to chaos and backwards from chaos to steady state is provided in terms of the present analysis results.

Stability of free convection in a narrow porous layer subject to rotation

Abstract The stability and onset of convection in a narrow, fluid saturated porous layer subject to a centrifugal body force due to rotation is investigated analytically. The marginal stability criterion is established in terms of a critical centrifugal Rayleigh number and a critical wave number. As a result, the corresponding eigenfunctions are evaluated at the convection threshold.