A ferrofluid saturated porous layer convection problem is studied in the variable gravitational field for a local thermal nonequilibrium (LTNE) model. Internal heating and variation (increasing or decreasing) in gravity with distance through the layer affected the stability of the convective system. The Darcy model is employed for the momentum equation and the LTNE model for the energy equation. The boundaries are considered to be rigid-isothermal and paramagnetic. For the linear stability analysis of the three-dimensional problem, the normal mode has been applied and the eigenvalue problem is solved numerically using Chebyshev pseudospectral method, while weakly nonlinear analysis is carried out with a truncated Fourier series. The effect of different dimensionless parameters on the Rayleigh number has also been studied. We found that the system becomes unstable on the increasing value of nonlinearity index of magnetization ( $$M_3$$ ), porosity-modified conductivity ratio ( $$\beta $$ ), and internal heat parameter ( $$\xi $$ ). It is observed that the system is stabilized by increasing the value of the Langevin parameter ( $$\alpha _\mathrm{{L}}$$ ), variable gravity coefficient ( $$\delta $$ ), and effective heat transfer parameter ( $$H_1$$ ). Runge–Kutta–Gill method has been used for solving the finite-amplitude equations to study the transient behavior of the Nusselt number. Streamlines and isotherms patterns are determined for the steady case and are presented graphically.

中文翻译：

---

具有可变重力和内部热源的 LTNE 模型的铁磁流体层的线性和弱非线性分析

针对局部热非平衡 (LTNE) 模型，研究了可变重力场中的铁磁流体饱和多孔层对流问题。内部加热和重力随穿过层距离的变化（增加或减少）影响对流系统的稳定性。动量方程采用达西模型，能量方程采用 LTNE 模型。边界被认为是刚性等温的和顺磁性的。对于三维问题的线性稳定性分析，应用了正常模式，并使用切比雪夫伪谱方法数值求解了特征值问题，而使用截断傅立叶级数进行了弱非线性分析。还研究了不同无量纲参数对瑞利数的影响。我们发现系统随着磁化非线性指数 ( $$M_3$$ )、孔隙率修正电导率比 ( $$\beta $$ ) 和内热参数 ( $$\xi $$ ) 的增加而变得不稳定. 观察到系统通过增加朗之万参数 ( $$\alpha _\mathrm{{L}}$$ )、变重力系数 ( $$\delta $$ ) 和有效传热参数的值来稳定( $$H_1$$ )。Runge-Kutta-Gill 方法已被用于求解有限振幅方程以研究 Nusselt 数的瞬态行为。流线和等温线模式是为稳定情况确定的，并以图形方式呈现。观察到系统通过增加朗之万参数 ( $$\alpha _\mathrm{{L}}$$ )、变重力系数 ( $$\delta $$ ) 和有效传热参数的值来稳定( $$H_1$$ )。Runge-Kutta-Gill 方法已被用于求解有限振幅方程以研究 Nusselt 数的瞬态行为。流线和等温线模式是为稳定情况确定的，并以图形方式呈现。观察到系统通过增加朗之万参数 ( $$\alpha _\mathrm{{L}}$$ )、变重力系数 ( $$\delta $$ ) 和有效传热参数的值来稳定( $$H_1$$ )。Runge-Kutta-Gill 方法已被用于求解有限振幅方程以研究 Nusselt 数的瞬态行为。流线和等温线模式是为稳定情况确定的，并以图形方式呈现。