Linear and nonlinear stability analyses for the onset of time-dependent convection in a horizontal layer of a porous medium saturated by a couple-stress non-Newtonian nanofluid, intercalated between two thermally insulated plates, are presented. Brinkman and Maxwell–Garnett formulations are adopted for nanoscale effects. A modified Darcy formulation that includes the time derivative term is used for the momentum equation. The nanofluid is assumed to be dilute and this enables the porous medium to be treated as a weakly heterogeneous medium with variation of thermal conductivity and viscosity, in the vertical direction. The general transport equations are solved with a Galerkin-type weighted residuals method. A perturbation method is deployed for the linear stability analysis and a Runge–Kutta–Gill quadrature scheme for the nonlinear analysis. The critical Rayleigh number, wave numbers for the stationary and oscillatory modes, and frequency of oscillations are obtained analytically using linear theory, and the nonlinear analysis is executed with minimal representation of the truncated Fourier series involving only two terms. The effect of various parameters on the stationary and oscillatory convection behavior is visualized. The effect of couple-stress parameter on the stationary and oscillatory convections is also shown graphically. It is found that the couple-stress parameter has a stabilizing effect on both the stationary and oscillatory convections. Transient Nusselt number and Sherwood number exhibit an oscillatory nature when time is small. However, at very large values of time, both Nusselt number and Sherwood number values approach their steady-state values. The study is relevant to the dynamics of biopolymers in solution in microfluidic devices and rheological nanoparticle methods in petroleum recovery.

线性和非线性稳定性分析在多孔介质的水平层中随时间变化的对流开始时被耦合应力非牛顿纳米流体饱和，插入两个绝热板，呈现。纳米级效应采用 Brinkman 和 Maxwell-Garnett 公式。包含时间导数项的修改达西公式用于动量方程。假设纳米流体是稀释的，这使得多孔介质能够被视为在垂直方向上具有热导率和粘度变化的弱异质介质。一般输运方程用伽辽金型加权残差法求解。线性稳定性分析采用微扰方法，非线性分析采用 Runge-Kutta-Gill 正交方案。临界瑞利数、稳态和振荡模式的波数以及振荡频率是使用线性理论分析获得的，非线性分析是使用仅涉及两项的截断傅立叶级数的最小表示来执行的。各种参数对固定和振荡对流行为的影响是可视化的。耦合应力参数对固定和振荡对流的影响也以图形方式显示。研究发现，耦合应力参数对固定对流和振荡对流都有稳定作用。瞬态努塞尔数和舍伍德数在时间较短时表现出振荡性质。然而，在非常大的时间值下，努塞尔特数和舍伍德数值都接近它们的稳态值。该研究与微流体装置中溶液中生物聚合物的动力学和石油开采中的流变纳米颗粒方法有关。耦合应力参数对固定和振荡对流的影响也以图形方式显示。研究发现，耦合应力参数对固定对流和振荡对流都有稳定作用。瞬态努塞尔数和舍伍德数在时间较短时表现出振荡性质。然而，在非常大的时间值下，努塞尔特数和舍伍德数值都接近它们的稳态值。该研究与微流体装置中溶液中生物聚合物的动力学和石油开采中的流变纳米颗粒方法有关。耦合应力参数对固定和振荡对流的影响也以图形方式显示。研究发现，耦合应力参数对固定对流和振荡对流都有稳定作用。瞬态努塞尔数和舍伍德数在时间较短时表现出振荡性质。然而，在非常大的时间值下，努塞尔特数和舍伍德数值都接近它们的稳态值。该研究与微流体装置中溶液中生物聚合物的动力学和石油开采中的流变纳米颗粒方法有关。在非常大的时间值下，努塞尔特数和舍伍德数值都接近它们的稳态值。该研究与微流体装置中溶液中生物聚合物的动力学和石油开采中的流变纳米颗粒方法有关。在非常大的时间值下，努塞尔特数和舍伍德数值都接近它们的稳态值。该研究与微流体装置中溶液中生物聚合物的动力学和石油开采中的流变纳米颗粒方法有关。