On the stability threshold of Couette flow for 2D Boussinesq equations

Stability threshold of the two-dimensional Couette flow in the whole plane

Purpose The purpose of this study is to the analyze the transient flow formation in a horizontal concentric-porous-annulus due to the combined impact of the sudden application of azimuthal pressure gradient and rotation of horizontal concentric-porous-tubes forming the concentric-porous annulus. Design/methodology/approach The semi-analytical solution of the momentum equation is derived with the aid of a two-step approach. The two-steps process involves solving the governing equations by using the known Laplace technique and Laplace inversion method of the Riemann-sum approximation. Findings During the course of analysis and numerical computations, it is observed that velocity increases from the outer surface of the inner cylinder (R = 1) to the inner surface of the outer cylinder (R = ?) as angular velocity ? increases. It is also found that skin friction decreases as pressure gradient (P) increases in the first half of the annulus and increases with it in the second half of the annulus. Originality/value Owing to the number of researches done on the Taylor–Couette flow, it becomes imperative to investigate the impact of time, suction and injection and angular velocity on the transient Taylor–Couette flow. A semi-analytical solution is presented.

The Green’s Function Method and Thin Film Growth Model with Couette Flow

In this paper, we perform a Floquet-based linear stability analysis of the centrifugal parametric resonance phenomenon in a Taylor–Couette system subjected to a time-quasiperiodic forcing where both the inner and outer cylinders are oscillating with the same amplitude and different angular velocities given respectively by $\varOmega _0 \cos (\omega _1t)$ and $\varOmega _0 \cos (\omega _2t)$ . In this context, the frequencies $\omega _1$ and $\omega _2$ are incommensurate, where the ratio $\omega _2/\omega _1$ is irrational. Taking into account non-axisymmetric disturbances, a new set of partial differential equations is derived and solved using the spectral method along with the Runge–Kutta numerical scheme. The obtained results in this framework show that this forcing triggers new and numerous reversing and non-reversing Taylor vortex flows arising via either synchronous or period-doubling bifurcations. A rich and complex dynamics is found owing to strong mode competition between these modes that alters significantly the topology of the marginal stability curves. The latter exhibit a multitude of small and condensed parabolas, giving rise to several codimension-two bifurcation points, discontinuities and cusp points in the stability diagrams. Furthermore, a proper tuning of the frequency ratio leads to a significant control of both the instability threshold and the axisymmetric nature of the primary bifurcation. Moreover, using a local quasi-steady analysis when the cylinders are slowly oscillating, intermittent instabilities are detected, characterised by spike-like behaviour in the stability diagrams with several successive growths, dampings and periods of quietness. In this limit case, the inner cylinder drive becomes the responsible forcing of the Taylor vortices’ formation where the calculated critical instability parameters correspond to those of the inner oscillating cylinder case with fixed outer cylinder. The potentially unstable regions between the cylinders are determined on the basis of the Rayleigh discriminant, where an excellent agreement with the linear stability analysis results is pointed out.

In this paper, we propose a complete formulation of the Lattice Boltzmann Method adapted for quantum computing. The classical collision, based on linear equilibrium distribution functions and streaming steps, are reformulated as linear algebraic operations. The inherently non-unitary collision operator is decomposed using Singular Value Decomposition and the Linear Combination of Unitaries technique. Bounce-back boundary conditions are incorporated directly into the collision matrix, while the streaming step is realized through conditional unitary shift operations on spatial registers, controlled by lattice velocity indices encoded in the distribution function register. This formulation ensures that the streaming step remains purely unitary. The resulting quantum circuit is implemented using Qiskit and validated against Couette flow and Poiseuille flow benchmarks. The simulation accurately reproduces the expected velocity profile, with relative errors below 1 0 − 4 . This work establishes a foundational framework for quantum fluid solvers and provides a pathway toward quantum computational fluid dynamics.

Stresses and torque due to Couette flow of micropolar fluid in a striated annulus

Linear and nonlinear enhanced dissipation for the 2-D micropolar equations near Couette flow

Bulk nanobubbles (BNBs), unlike macro bubbles, exhibit extraordinary longevity in water. While their super-stability is well-established, the rheological properties of BNB-water systems remain poorly understood. To address this, we employed molecular dynamics simulations to model nitrogen BNBs with a radius of ∼1.9 nm, systematically varying the volume fraction and evaluating the zero-shear viscosity. Our simulations reveal that BNBs exhibit rheological behaviour akin to that of microbubbles, but with a markedly stronger effect. Specifically, the zero-shear viscosity of BNB-water systems exceeds that of pure water and increases significantly with volume fraction-far surpassing the trends observed in microbubble systems. This pronounced increase resembles the behaviour of dilute charged colloidal dispersions, likely due to the high-density gas and surface charges within the nanobubbles. We derived a relative viscosity model grounded in classical theories to describe this behaviour. Under Couette flow, BNBs become unstable at high shear rates, coalescing into larger bubbles. Prior to coalescence, shear viscosity also rises with increasing volume fraction. To validate our findings, we conducted experiments using nanobubbles with a radius of ∼70 nm at a high concentration of 4.0 × 1013 bubbles per mL. The experimental data closely matched our simulations: at low shear rates, BNBs-water viscosity exceeds that of pure water, while at high shear rates, it drops to baseline levels-likely due to bubble coalescence and gas release. Our experimentally validated model predicts zero-shear viscosity within 8.93% of measured values, underscoring the complex, shear- and concentration-dependent rheology of BNB systems and suggesting current technologies may underestimate actual BNB concentrations.

Linear inviscid damping and enhanced dissipation for two dimensional incompressible MHD equations with damping term around Couette flow

When simulating three-dimensional flows interacting with deformable and elastic obstacles, current methods often encounter complexities in the governing equations and challenges in numerical implementation. In this work, we introduce a novel numerical formulation for simulating incompressible viscous flows at low Reynolds numbers in the presence of deformable interfaces. Our method employs a vorticity-stream vector formulation that significantly simplifies the fluid solver, transforming it into a set of coupled Poisson problems. The body–fluid interface is modeled using a phase field, allowing for the incorporation of various free-energy models to account for membrane bending and surface tension. In contrast to existing three-dimensional approaches, such as lattice Boltzmann methods or boundary-integral techniques, our formulation is lightweight and grounded in classical fluid mechanics principles, making it implementable with standard finite-difference techniques. We demonstrate the capabilities of our method by simulating the evolution of a single vesicle or droplet in Newtonian Poiseuille and Couette flows under different free-energy models, successfully recovering canonical axisymmetric shapes and stress profiles. Although this work primarily focuses on single-body dynamics in Newtonian suspending fluids, the framework can be extended to include body forces, inertial effects, and viscoelastic media.

Ideal magnetohydrodynamics around couette flow: Long time stability and vorticity–current instability

Transition threshold of Couette flow for 2D Boussinesq equations

Abstract In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel with non‐slip boundary condition. Compared to the case , we establish the stability in the context of long wave associated with the frequency range by developing the resolvent estimate argument. The new ingredient is to discover the key division point at in the frequency interval (0,1) by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval (0,1), and then we establish the space–time estimates on the low‐frequency and the intermediate frequency , respectively. As an application of the space–time estimates, we obtain the non‐linear transition threshold to be . Meanwhile, we also show that when the frequencies , the enhanced dissipation effect occurs for the linearized Navier–Stokes equations.

Stability of hydromagnetic Couette flow in an anisotropic porous medium with oblique principal axes and constant wall transpiration

Bounds on turbulent averages in shear flows can be derived from the Navier–Stokes equations by a mathematical approach called the background method. Bounds that are optimal within this method can be computed at each Reynolds number $ \textit{Re}$ by numerically optimising subject to a spectral constraint, which requires a quadratic integral to be non-negative for all possible velocity fields. Past authors have eased computations by enforcing the spectral constraint only for streamwise-invariant (2.5-D) velocity fields, assuming this gives the same result as enforcing it for three-dimensional (3-D) fields. Here, we compute optimal bounds over 2.5-D fields and then verify, without doing computations over 3-D fields, that the bounds indeed apply to 3-D flows. One way is to directly check that an optimiser computed using 2.5-D fields satisfies the spectral constraint for all 3-D fields. A second way uses a criterion we derive that is based on a theorem of Busse (1972 Arch. Ration. Mech. Anal., vol. 47, pp. 28–35) for energy stability analysis of models with certain symmetry. The advantage of checking this criterion, as opposed to directly checking the 3-D constraint, is lower computational cost and natural extrapolation of the criterion to large $ \textit{Re}$ . We compute optimal upper bounds on friction coefficients for the wall-bounded Kolmogorov flow known as Waleffe flow and for plane Couette flow. This requires lower bounds on dissipation in the first model and upper bounds in the second. For Waleffe flow, all bounds computed using 2.5-D fields satisfy our criterion, so they hold for 3-D flows. For Couette flow, where bounds have been previously computed using 2.5-D fields by Plasting & Kerswell (2003 J. Fluid Mech., vol. 477, pp. 363–379), our criterion holds only up to moderate $ \textit{Re}$ , so at larger $ \textit{Re}$ we directly verify the 3-D spectral constraint. Over the $ \textit{Re}$ range of our computations, this confirms the assumption by Plasting & Kerswell that their bounds hold for 3-D flows.

The current study investigates the Couette flow of viscous liquid in parallel walls through a deformable permeable medium. The connected occurrence of liquid flow and the deformable, permeable solid medium is estimated. The result of applicable parameters on the viscous liquid and solid displacement is examined in particular. The influences acquired for the current flow characteristic show numerous exciting behaviours that warrant additional investigation on the deformable permeable medium. Also, the effectiveness of drag, planar wall, slip parameter, and volume fraction on the liquid flow is examined with the help of graphs. Major Findings: Comparing the present article results with classical Poiseuille flow, it is noticed that velocity and skin friction are enhanced because of the movement of the boundary. Most of the available research focuses on undeformable porous media. However, under healthy conditions, most of the biological systems contain deformable tissue layers. For example, Synovial fluid flow in newly replaced joints may be studied by examining Couette flow past a deformable porous layer.

The current study investigates the MHD Couette flow of Casson fluid between two similar plates filled through a deformable permeable medium. The correlated phenomenon of the liquid flow and solid deformable permeable medium is considered. The effect of suitable parameters on the liquid velocity and solid displacement is examined in detail. The influences found for the existing flow characteristic indicated numerous exciting behaviours that warrant additional analysis of the deformable permeable media. The importance of drag, magnetic parameter, Casson parameter, upper plate velocity, slip parameter and skin friction on the fluid flow is discussed with the support of graphs. Furthermore, we noticed that skin friction increases with enhancement in the fluid fraction in the porous layer. Alexion and Kapellos1 repeat a similar analysis in the case of plane Couette-Poiseuille flows further into a homogeneous poroelastic layer. Major Findings: Magnetic fields can influence blood vessels in living organisms, potentially reducing inflammation in affected areas, thereby aiding in pain relief and tissue repair. Motivated by these effects, we have conducted a study on fluid flow through deformable porous layers. Furthermore, magnetic fields can impact cellular permeability, enhancing the absorption of nutrients and facilitating the removal of waste products.

We investigate flow instability produced by viscosity and density discontinuities at the interface separating two Newtonian fluids in generalised Couette–Poiseuille (GCP) flow. The base flow, driven by counter-moving plates and an inclined pressure gradient at angle $0^\circ \leqslant \phi \leqslant 90^\circ$ , exhibits a twisted, two-component velocity profile across the layers, characterised by the Couette–Poiseuille magnitude parameter $0^\circ \leqslant \theta \leqslant 90^\circ$ . Plane Couette–Poiseuille (PCP) flow at $ \phi = 0^\circ$ is considered as a special case. Flow/geometry parameters are $(\phi ,\theta )$ , a Reynolds number $Re$ and the viscosity, depth and density ratios $(m,n,r)$ , respectively. A mapping from the GCP to PCP extended Orr–Sommerfeld equations is found that simplifies the numerical study of interfacial-mode instabilities, including determination of shear-mode critical parameters. For interfacial modes, unstable regions in $(m,n,r)$ space are delineated by three distinct surfaces found via long-wave analysis, with the exception of strict Couette flow where the $(m,n)$ surface asymptotically vanishes with $\theta \rightarrow 0^\circ$ . In interfacial stable regions but with unstable shear modes, one-layer PCP stability can be identified with a cut-off $\theta$ that conforms to canonical PCP stability. Competition between the interfacial-mode reversal phenomenon and the shear-mode cut-off behaviour is discussed. Extending to the full GCP configuration with the mapping algorithms applied, we systematically chart how pressure-gradient inclination and perturbation wavefront angle shift the balance between interfacial and shear instabilities in a specific case.

We use deep Physics-Informed Neural Networks (PINNs) to simulate stratified forced convection in plane Couette flow. This process is critical for atmospheric boundary layers (ABLs) and oceanic thermoclines under global warming. The buoyancy-augmented energy equation is solved under two boundary conditions: Isolated-Flux (single-wall heating) and Flux–Flux (symmetric dual-wall heating). Stratification is parameterized by the Richardson number (Ri∈ [−1,1]), representing ±2 °C thermal perturbations. We employ a decoupled model (linear velocity profile) valid for low-Re, shear-dominated flow. Consequently, this approach does not capture the full coupled dynamics where buoyancy modifies the velocity field, limiting the results to the laminar regime. Novel contribution: This is the first deep PINN to robustly converge in stiff, buoyancy-coupled flows (∣Ri∣≤1) using residual connections, adaptive collocation, and curriculum learning—overcoming standard PINN divergence (errors >28%). The model is validated against analytical (Ri=0) and RK4 numerical (Ri≠0) solutions, achieving L2 errors ≤0.009% and L∞ errors ≤0.023%. Results show that stable stratification (Ri>0) suppresses convective transport, significantly reduces local Nusselt number (Nu) by up to 100% (driving Nu towards zero at both boundaries), and induces sign reversals and gradient inversions in thermally developing regions. Conversely, destabilizing buoyancy (Ri<0) enhances vertical mixing, resulting in an asymmetric response: Nu increases markedly (by up to 140%) at the lower wall but decreases at the upper wall compared to neutral forced convection. At 5–10× lower computational cost than DNS or RK4, this mesh-free PINN framework offers a scalable and energy-efficient tool for subgrid-scale parameterization in general circulation models (GCMs), supporting SDG 13 (Climate Action).