Floquet and Nonlinear Energy Stability of Periodically Heated Thermosolutal Convection in a Shear-Thinning Micropolar Porous Layer

Abstract

The stability of a horizontal porous layer saturated by a binary, shear-thinning micropolar fluid is examined when the imposed temperature difference is sinusoidally modulated. A Darcy-Brinkman-micropolar formulation is coupled to thermal and solutal transport, while the apparent viscosity follows a regularized Carreau law. The conductive state is reduced by a normal-mode Galerkin projection to a periodic linear system. Its monodromy matrix supplies the Floquet multipliers and the neutral thermal Rayleigh number. A weighted energy functional is then used without linearizing the advective terms; the resulting endpoint matrix inequalities provide a sufficient nonlinear decay threshold. For the reference parameter set, the unmodulated oscillatory threshold is 71.97. At modulation frequency 8, amplitudes 0.2 and 0.4 reduce the threshold to 65.39 and 59.36, respectively, through a subharmonic Floquet tongue. Micropolar coupling raises both the Floquet and energy thresholds, whereas stronger shear thinning contracts the finite-strain energy-stable region. Because the regularized viscosity equals its zero-shear value at the conductive state, the power-law index does not enter infinitesimal onset; it enters the nonlinear margin through amplitude-dependent dissipation. The separation between the Floquet and energy boundaries identifies a parameter interval in which linear decay is not yet accompanied by a global finite-amplitude guarantee.

Introduction

The text develops a mathematical framework for studying buoyancy-driven convection in a periodically heated, double-diffusive porous layer containing a micropolar, shear-thinning Carreau fluid.

• Background: Porous-medium convection is governed by the competition between buoyancy, drag, and diffusion. When temperature and concentration gradients coexist, unequal diffusivities can produce both stationary and oscillatory double-diffusive instabilities.

• Micropolar effects: The fluid has an additional microrotation field and couple stresses. Spin–vorticity coupling, spin diffusion, and spin damping modify the effective mechanical resistance and therefore the convection threshold.

• Non-Newtonian behavior: A Carreau viscosity law is used to represent shear thinning. Importantly, the base conductive state has zero shear, so the linear critical Rayleigh number is identical to the corresponding Newtonian value. However, shear thinning strongly affects finite-amplitude behavior and nonlinear heat transport.

• Periodic heating: The imposed temperature difference varies as 1+?cos?(Ωt), making the stability problem time-periodic. Consequently, Floquet theory, rather than ordinary instantaneous eigenvalue analysis, is required. Harmonic, subharmonic, and oscillatory instability can occur depending on the forcing.

• Mathematical model: The formulation combines incompressibility, Brinkman drag, micropolar momentum and spin equations, temperature and concentration transport equations, and a regularized Carreau viscosity.

• Boundary conditions: Impermeable, isothermal, isosolutal boundaries with strong microrotation anchoring are imposed. These conditions also ensure that the nonlinear advection terms do not contribute to the perturbation-energy balance.

• Modal reduction: A first Galerkin approximation reduces the governing equations to a two-variable, periodically forced system for temperature and concentration amplitudes. Under rapid spin relaxation, microrotation can be eliminated, producing an effective modal resistance M(a,μ).

• Key linear result: Because μr(0)=1, the Carreau parameters n and Cu disappear from the infinitesimal stability equations. Thus, shear thinning does not directly change the linear onset threshold, despite being important for nonlinear stability.

• Floquet stability: The fundamental matrix over one forcing period gives Floquet multipliers. Neutral stability occurs when the largest Floquet growth exponent reaches zero. The critical thermal Rayleigh number is obtained by varying RT and minimizing over the horizontal wave number.

• Nonlinear energy stability: The text then motivates a separate energy-stability analysis because linear neutrality only concerns infinitesimal disturbances. A finite gap can exist between the linear Floquet threshold and the sufficient nonlinear/global stability threshold.

Conclusion

A unified Floquet-energy analysis has been developed for periodically heated thermosolutal convection in a shear-thinning micropolar porous layer. The linearized Galerkin equations form a genuinely periodic system, and their monodromy spectrum reveals a subharmonic instability tongue. For the baseline case, modulation with ?=0.4 and ?=8 lowers the critical thermal Rayleigh number from 71.97 to 59.36. Micropolar coupling is stabilizing because it increases the modal resistance after spin relaxation, raising both R_F and R_E. The nonlinear energy threshold remains lower than the Floquet threshold and is frequency independent under the endpoint estimate, thereby separating a certified global-decay region from an interval requiring sharper analysis or direct nonlinear simulation. Regularized shear thinning leaves infinitesimal onset independent of n because the conductive state has zero shear. Its influence appears at finite amplitude: decreasing n reduces the conditional energy threshold through the strain-dependent viscosity. This separation of roles provides a consistent mathematical interpretation of why linear calculations can miss the principal rheological sensitivity of non-Newtonian porous convection.

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Copyright

Copyright © 2026 Subhash Kumar, Dr. H. K. Pandey. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.