A unified lattice Boltzmann method is employed to investigate Rayleigh–Bénard convection (RBC) subjected to sidewall heating and unipolar charge injection from the bottom wall. The study focuses on how the complex and nonlinear coupling between the buoyancy and Coulomb effects modify the heat transfer, flow structure and the transition between buoyancy- and Coulomb-dominated regimes. The results show that a side-heated wall, in the absence of charge injection, enhances the heat transfer rate and changes the scaling law between the Nusselt number $\textit{Nu}$ and Rayleigh number $Ra$ from $\textit{Nu} = 0.22Ra^{0.29}$ (classical RBC) to $\textit{Nu} = 0.56 Ra^{0.22}$ due to an additional buoyancy effect from the sidewall. When the electric charge is injected from the bottom wall, it is shown that the thermal boundary layer thickness decreases, leading to a further enhancement of heat transfer. Furthermore, systematic simulations over a broad range of $Ra$ and electric Rayleigh numbers $T$ reveal that, at given $T$, $\textit{Nu}$ remains constant when $Ra$ is low, indicating a Coulomb-dominated regime. Beyond a critical value of $Ra$, a power-law relationship between $\textit{Nu}$ and $Ra$ emerges, signifying a transition to the buoyancy-dominated regime. This transition can be well predicted by a dimensionless parameter, which is developed considering buoyancy to Coulomb forces. In addition, by analysing the flow structure using the Fourier mode decomposition, a phase diagram describing the dominant flow modes is proposed. The results demonstrate that the proposed dimensionless parameter not only delineates the transition between the two heat transfer regimes but also accurately captures the flow mode shift. Our findings offer new insights into the complex interaction between buoyancy and Coulomb effects and their influence on heat transfer and flow structure, with potential implications for the design of heat exchangers aimed at actively and efficiently controlling heat transfer.

Rate-dependent viscosity in power-law fluids significantly affects contact line stress singularities and moving contact line behaviour. Contact line forces show more severe divergence for shear-thickening fluids ($n\gt 1$) or remain finite for shear-thinning fluids ($n\lt 1$). Complementing earlier self-similar derivations of spreading laws by Starov et al. (J. Colloid Interface Sci. vol. 257, 2003, pp. 284–290) for shear-thinning drops, we extend the classical Cox-Voinov theory to power-law fluids and obtain explicit dynamic contact angle relationships – results that are more fundamental than previously reported spreading laws. This development provides a unified yet fundamentally distinct description of advancing contact line behaviour across the full range of shear-thinning and shear-thickening rheologies. We show that the apparent dynamic contact angle $\theta _{d}$ depends critically on the characteristic dissipation length $h^{*}\propto U^{n/(n-1)}$, fundamentally altering its dependence on contact line speed $U$. For shear-thinning fluids (n < 1) with less diverging contact line stresses, this length scale yields $\theta _{d}\sim C{a_{\textit{local}}}^{1/3}$ in the familiar Cox–Voinov form in terms of the local capillary number $ \textit{Ca}_{\textit{local}} = (h/h^{*})^{1-n}$, with the contact line motion dissipated within $h^{*}$ extending beyond local wedge height $h$, thereby eliminating the need for a microscopic cutoff. This feature also renders $\theta _{d}$ size dependent and varying with the spreading radius $R$, recovering $R\propto t^{n/(3n+7)}$and $\theta _{d}\propto U^{3n/(2n+7)}$ as previously derived by Starov et al. (2003). For shear-thickening fluids (n > 1) that exhibit more strongly diverging contact line stresses, by contrast, the contact line motion is dissipated within a much narrow region $h^{*}$ that is much smaller than the required microscopic cutoff hm. A complete precursor theory is also developed, showing $ h_{m} \propto U^{-n/(4-n)}$. This leads to $\theta_{d} \propto U^{n/(4-n)}$, making the global spreading behaviour highly sensitive to the contact line microstructure. Importantly, regardless of the microscopic mechanisms, the apparent dynamic contact angle relationship can always be expressed in the analogous Cox–Voinov form $\theta _{d}\sim {\textit{Ca}_{\textit{eff}}}^{1/3}$ in terms of the effective capillary number $\textit{Ca}_{\textit{eff}}=\eta_{\kern-1.5pt f}U/\gamma=(h^*/h_m)^{n-1}$ (with the surface tension γ) based on the microscopic viscosity $\eta_{\kern-1.5pt f}\propto (U/h_{m})^{n-1}$ associated with the local shear rate $ U/h_{m}$ across the cutoff $ h_{m}$. The present Cox-Voinov generalisation can be applied to more realistic rheological laws such as the Carreau model, where self-similar solutions may no longer exist, thereby enabling a direct mapping of non-Newtonian spreading dynamics onto equivalent Newtonian behaviour and offering a more robust framework for the design and control of droplet dynamics in practical applications.

Wave–sea-ice interactions shape the transition zone between open ocean and pack ice in the polar regions. Most theoretical paradigms, implemented in coupled wave–sea-ice models, predict exponential decay of the wave energy but some recent observations deviate from this behaviour. Expanding on a framework based on wave energy dissipation due to ice–water drag, we account for drifting sea ice to derive an improved model for wave energy attenuation. Analytical solutions replicate the observed non-exponential wave energy decay and the spatial evolution of the effective attenuation rate in Antarctic sea ice.

This work presents an efficient statistical model to simulate expected scalar transport in fractured porous media below the representative elementary volume scale. We focus on embedded, highly conductive, isolated fractures. The statistical integro-differential fracture model (Sid-FM) solves for ensemble-averaged solutions directly, avoiding computationally expensive Monte Carlo simulation. The expected fluid exchange between isolated fractures and the porous matrix is modelled via a non-local kernel function, leading to a set of integro-differential equations. The model is validated against reference data from Monte Carlo simulations for statistically one-dimensional test cases and shows good agreement.

The collisions between elongated particles in turbulence play an important role in various natural and industrial processes. In this study, we establish a theoretical model to estimate the collision kernels of monodisperse elongated passive particles in homogeneous isotropic turbulence. The model is composed of two terms: the collision kernel with fixed relative angles and the angular-volumetric number density of neighbouring pairs, where we first derive under the Gaussian hypothesis of the fluid velocity gradients and then incorporate a non-Gaussianity factor into the formulations. The collision kernels obtained by the present model are in a good agreement with that of direct numerical simulations. Moreover, our model provides insights into the mechanism of the relative alignment between nearby particle pairs and the chain formation in turbulence, from the perspective of particle collisions.

The study investigates the influence of temperature-dependent viscosity on the stability of buoyancy-driven flow in a vertical porous slab bounded by impermeable walls and subject to Robin thermal boundary conditions. Three viscosity–temperature relationships are considered – linear, quadratic and exponential. A normal-mode linear stability analysis is carried out for two porous flow models: Model I, representing Darcy flow with the transient velocity term neglected, and Model II, its counterpart that retains it. The resulting stability eigenvalue problem is solved numerically to obtain neutral stability curves and the critical parameters, with emphasis on the roles of the Prandtl–Darcy number, Biot number and viscosity parameters. The similarities and differences between the two models, as well as those among the viscosity–temperature laws, are examined in detail. For linear and quadratic viscosity variations, Model I predicts unconditional stability irrespective of boundary heat exchange, whereas Model II admits instability when the boundaries are thermally imperfect. In contrast, exponential viscosity variation promotes instability in both models. In Model I, instability is confined to a restricted range of Biot numbers, which depends sensitively on the exponential viscosity parameter, while the flow remains stable for all thermal boundary conditions when this parameter is less than 8.2070. Model II, however, displays qualitatively distinct behaviour characterised by mode transitions and the emergence of two instability regimes of Biot number separated by a stability window.