Search Results (13)

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 $(R_i\in [-1,1\left]\right),$ representing $\pm 2 °\mathrm{C}$ 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 ($\mid R_i\mid \le 1$) using residual connections, adaptive collocation, and curriculum learning—overcoming standard PINN divergence (errors $>28\%$). The model is validated against analytical ($R_i=0$) and RK4 numerical ($R_i\ne 0$) solutions, achieving $L_2$ errors $\le 0.009\%$ and $L_{\mathrm{\infty }}$ errors $\le 0.023\%$. Results show that stable stratification $(R_i>0)$ 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 $(R_i<0)$ 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 $5–10\times$ 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). Full article

An efficient numerical approach based on weighted-average finite differences is used to solve the Newtonian plane Couette flow with wall slip, obeying a dynamic slip law that generalizes the Navier slip law with the inclusion of a relaxation term. Slip is exhibited only along the fixed lower plate, and the motion is triggered by the motion of the upper plate. Three different cases are considered for the motion of the moving plate, i.e., constant speed, oscillating speed, and a single-period sinusoidal speed. The velocity and the volumetric flow rate are calculated in all cases and comparisons are made with the results of other methods and available results in the literature. The numerical outcomes confirm the damping with time and the lagging effects arising from the Navier and dynamic wall slip conditions and demonstrate the hysteretic behavior of the slip velocity in following the harmonic boundary motion. Full article

We study electrohydrodynamic (EHD) linear (in)stability of microfluidic channel flows, i.e., the stability of interface between two shearing viscous (perfect) dielectrics exposed to an electric field in large aspect ratio microchannels. We then apply our results to particular microfluidic systems known as two-liquid electroosmotic (EO) pumps. Our novel results are detailed analytical expressions for the growth rate of two-dimensional EHD modes in Couette–Poiseuille flows in the limit of small Reynolds number (R); the expansions to both zeroth and first order in R are considered. The growth rates are complicated functions of viscosity-, height-, density-, and dielectric-constant ratio, as well as of wavenumbers and voltages. To make the results useful to experimentalists, e.g., for voltage-control EO pump operations, we also derive equations for the impending voltages of the neutral stability curves that divide stable from unstable regions in voltage–wavenumber stability diagrams. The voltage equations and the stability diagrams are given for all wavenumbers. We finally outline the flow regimes in which our first-order-R voltage corrections could potentially be experimentally measured. Our work gives insight into the coupling mechanism between electric field and shear flow in parallel-planes channel flows, correcting an analogous EHD expansion to small R from the literature. We also revisit the case of pure shear instability, when the first-order-R voltage correction equals zero, and replace the renowned instability mechanism due to viscosity stratification at small R with the mechanism due to discontinuity in the slope of the unperturbed velocity profile. Full article

The electrohydrodynamic plane Couette–Poiseuille flow instability of two superposed conducting and dielectric viscous incompressible fluids confined between two rigid horizontal planes under the action of a normal electric field and pressure gradient through Brinkman porous medium has been analytically investigated. The lower plane is stationary, while the upper one is moving with constant velocity. The details of the base state mathematical model and the linearized model equations for the perturbed state are introduced. Following the usual procedure of linear stability analysis for viscous fluids, we derived two non-dimensional modified Orr–Sommerfeld equations and obtained the associated boundary and interfacial conditions suitable for the problem. The eigenvalue problem has been solved using asymptotic analysis for wave numbers in the long-wavelength limit to obtain a very complicated novel dispersion relation for the wave velocity through lengthy calculations. The obtained dispersion equation has been solved using Mathematica software v12.1 to study graphically the effects of various parameters on the stability of the system. It is obvious from the figures that the system in the absence of a porous medium and/or electric field is more unstable than in their presence. It is found also that the velocity of the upper rigid boundary, medium permeability, and Reynolds number have dual roles on the stability on the system, stabilizing as well as destabilizing depending on the viscosity ratio value. The electric potential, dielectric constant and pressure gradient are found to have destabilizing influences on the system, while the porosity of the porous medium, density ratio and Froude number have stabilizing influences. A depth ratio of less than one has a dual role on the stability of the system, while it has a stabilizing influence for values greater than one. It is observed that the viscosity stratification brings about a stabilizing as well as a destabilizing effect on the flow system. Full article

We present the theoretical description of plane Couette flow based on the previously proposed equations of vortex fluid, which take into account both the longitudinal flow and the vortex tubes rotation. It is shown that the considered equations have several stationary solutions describing different types of laminar flow. We also discuss the simple model of turbulent flow consisting of vortex tubes, which are moving chaotically and simultaneously rotating with different phases. Using the Boussinesq approximation, we obtain an analytical expression for the stationary profile of mean velocity in turbulent Couette flow, which is in good agreement with experimental data and results of direct numerical simulations. Our model demonstrates that near-wall turbulence can be described by a coordinates-independent coefficient of eddy viscosity. In contrast to the viscosity of the fluid itself, this parameter characterizes the turbulent flow and depends on Reynolds number and roughness of the channel walls. Potentially, the proposed model can be considered as a theoretical basis for the experimental measurement of the eddy viscosity coefficient. Full article

The conditions for the onset of dissipation thermal instability with temperature-dependent viscosity in the plane Couette flow of a Newtonian fluid are analyzed. The studied system consists of a horizontal fluid layer confined between an adiabatic (fixed) lower wall and an isothermal (moving) upper wall. Both the exponential and the linear fluidity models are considered in order to account for the thermodependency of the fluid’s viscosity. The linear stability analysis of the base solution with respect to arbitrarily oriented normal modes is carried out numerically by employing a shooting method. The most unstable disturbances are proven to be stationary longitudinal rolls, and their stability is governed by three dimensionless parameters: the viscous dissipation Rayleigh number, Prandtl number and a parameter that represents the variability of the viscosity with temperature. It is shown that the effect of the variation of the viscosity is to promote the stability of the base flow. As expected, the two viscosity models’ results diverge as the variability of the viscosity increases, and the exponential model is found to be more stable than the linear fluidity one. By considering the thermophysical properties of real fluids, it is shown that viscous dissipation thermal instability precedes hydrodynamic instability. An energy budget analysis is proposed to better understand both the stabilization effect of the thermal variability of the viscosity and differences with viscous dissipation hydrodynamic instability. Full article

In recent years, scale-by-scale energy transport in wall turbulence has been intensively studied, and the complex spatial and interscale transfer of turbulent energy has been investigated. As the enhancement of heat transfer is one of the most important aspects of turbulence from an engineering perspective, it is also important to study how turbulent heat fluxes are transported in space and in scale by nonlinear multi-scale interactions in wall turbulence as well as turbulent energy. In the present study, the spectral transport budgets of turbulent heat fluxes are investigated based on direct numerical simulation data of a turbulent plane Couette flow with a passive scalar heat transfer. The transport budgets of spanwise spectra of temperature fluctuation and velocity-temperature correlations are investigated in detail in comparison to those of the corresponding Reynolds stress spectra. The similarity and difference between those scale-by-scale transports are discussed, with a particular focus on the roles of interscale transport and spatial turbulent diffusion. As a result, it is found that the spectral transport of the temperature-related statistics is quite similar to those of the Reynolds stresses, and in particular, the inverse interscale transfer is commonly observed throughout the channel in both transport of the Reynolds shear stress and wall-normal turbulent heat flux. Full article

Elasto-inertial turbulence (EIT), a new turbulent state found in polymer solutions with viscoelastic properties, is associated with drag-reduced turbulence. However, the relationship between EIT and drag-reduced turbulence is not currently well-understood, and it is important to elucidate the mechanism of the transition to EIT. The instability of viscoelastic fluids has been studied in a canonical wall-bounded shear flow to investigate the transition process of EIT. In this study, we numerically deduced that an instability occurs in the linearly stable viscoelastic plane Couette flow for lower Reynolds numbers, at which a non-linear unstable solution exists. Under instability, the flow structure is elongated in the spanwise direction and regularly arranged in the streamwise direction, which is a characteristic structure of EIT. The regularity of the flow structure depends on the Weissenberg number, which represents the strength of elasticity; the structure becomes disordered under high Weissenberg numbers. In the energy spectrum of velocity fluctuations, a steep decay law of the structure’s scale towards a small scale is observed, and this can be recognized as a ubiquitous feature of EIT. The existence of instability in viscoelastic plane Couette flow supports the idea that the transitional path toward EIT may be mediated by subcritical instability. Full article

A general unified solution of the plane Couette–Poiseuille–Stokes–Womersley incompressible linear fluid flow in a slit in the presence of oscillatory pressure gradients with periodic synchronous vibrating boundaries is presented. Oscillatory flow remains stable and laminar with no-slip boundary conditions applied. Eigenfunction expansion method is used to obtain the exact analytical solution of the general linear inhomogeneous boundary value problem. Fourier expansion of arbitrary harmonic pressure gradient and non-harmonic wall oscillations was used to calculate arbitrary driving of the fluid. In-house developed optimized computational fluid dynamics marching-in-time finite-volume method was used to test and verify all analytical results. A number of particular transients, steady-state and combined flows were obtained from the general analytical result. Generalized Stokes and Womersley flows were solved using the analytical computations and numerical experiments. The combined effects of periodic non-harmonic wall movements with oscillatory pressure gradients offers rich and interesting flow patterns even for a linear Newtonian fluid and may be particularly interesting for pumping-assist microfluidic devices. The main motivation for developing a unified solution of the unsteady laminar planar Couette–Stokes–Poiseuille–Womersley flow originates in a need for, but is not limited to, in-depth exploration of flow patterns in hemodynamic and microfluidic pumping applications. Full article

The modified Stokes second problem for incompressible upper-convected Maxwell (UCM) fluids with linear dependence of viscosity on the pressure is analytically and numerically investigated. The fluid motion, between infinite horizontal parallel plates, is generated by the lower wall, which oscillates in its plane. The movement region of the fluid is symmetric with respect to the median plane, but its motion is asymmetric due to the boundary conditions. Closed-form expressions are found for the steady-state components of start-up solutions for non-dimensional velocity and the corresponding non-trivial shear and normal stresses. Similar solutions for the simple Couette flow are obtained as limiting cases of the solutions corresponding to the motion due to cosine oscillations of the wall. For validation, it is graphically proved that the start-up solutions (numerical solutions) converge to their steady-state components. Solutions for motions of ordinary incompressible UCM fluids performing the same motions are obtained as special cases of present results using asymptotic approximations of standard Bessel functions. The time needed to reach the permanent or steady state is also determined. This time is higher for motions of ordinary fluids, compared with motions of liquids with pressure-dependent viscosity. The impact of physical parameters on the fluid motion and the spatial–temporal distribution of start-up solutions are graphically investigated and discussed. Ordinary fluids move slower than fluids with pressure-dependent viscosity. Full article

In this paper, a Multi Relaxation Time Lattice Boltzmann scheme is used to describe the evolution of a non-Newtonian fluid. Such method is coupled with an Immersed-Boundary technique for the transport of arbitrarily shaped objects navigating the flow. The no-slip boundary conditions on immersed bodies are imposed through a convenient forcing term accounting for the hydrodynamic force generated by the presence of immersed geometries added to momentum equation. Moreover, such forcing term accounts also for the force induced by the shear-dependent viscosity model characterizing the non-Newtonian behavior of the considered fluid. Firstly, the present model is validated against well-known benchmarks, namely the parabolic velocity profile obtained for the flow within two infinite laminae for five values of the viscosity model exponent, n = 0.25, 0.50, 0.75, 1.0, and 1.5. Then, the flow within a squared lid-driven cavity for Re = 1000 and 5000 (being Re the Reynolds number) is computed as a function of n for a shear-thinning (n < 1) fluid. Indeed, the local decrements in the viscosity field achieved in high-shear zones implies the increment in the local Reynolds number, thus moving the position of near-walls minima towards lateral walls. Moreover, the revolution under shear of neutrally buoyant plain elliptical capsules with different Aspect Ratio (AR = 2 and 3) is analyzed for shear-thinning (n < 1), Newtonian (n = 1), and shear-thickening (n > 1) surrounding fluids. Interestingly, the power law by Huang et al. describing the revolution period of such capsules as a function of the Reynolds number and the existence of a critical value, Re${}_c$, after which the tumbling is inhibited in confirmed also for non-Newtonian fluids. Analogously, the equilibrium lateral position $y_{eq}$ of such neutrally buoyant capsules when transported in a plane-Couette flow is studied detailing the variation of $y_{eq}$ as a function of the Reynolds number as well as of the exponent n. Full article

The onset of turbulence in subcritical shear flows is one of the most puzzling manifestations of critical phenomena in fluid dynamics. The present study focuses on the Couette flow inside an infinitely long annular geometry where the inner rod moves with constant velocity and entrains fluid, by means of direct numerical simulation. Although for a radius ratio close to unity the system is similar to plane Couette flow, a qualitatively novel regime is identified for small radius ratio, featuring no oblique bands. An analysis of finite-size effects is carried out based on an artificial increase of the perimeter. Statistics of the turbulent fraction and of the laminar gap distributions are shown both with and without such confinement effects. For the wider domains, they display a cross-over from exponential to algebraic scaling. The data suggest that the onset of the original regime is consistent with the dynamics of one-dimensional directed percolation at onset, yet with additional frustration due to azimuthal confinement effects. Full article

Wall-bounded flows experience a transition to turbulence characterized by the coexistence of laminar and turbulent domains in some range of Reynolds number R, the natural control parameter. This transitional regime takes place between an upper threshold $R_{\mathrm{t}}$ above which turbulence is uniform (featureless) and a lower threshold $R_{\mathrm{g}}$ below which any form of turbulence decays, possibly at the end of overlong chaotic transients. The most emblematic cases of flow along flat plates transiting to/from turbulence according to this scenario are reviewed. The coexistence is generally in the form of bands, alternatively laminar and turbulent, and oriented obliquely with respect to the general flow direction. The final decay of the bands at $R_{\mathrm{g}}$ points to the relevance of directed percolation and criticality in the sense of statistical-physics phase transitions. The nature of the transition at $R_{\mathrm{t}}$ where bands form is still somewhat mysterious and does not easily fit the scheme holding for pattern-forming instabilities at increasing control parameter on a laminar background. In contrast, the bands arise at $R_{\mathrm{t}}$ out of a uniform turbulent background at a decreasing control parameter. Ingredients of a possible theory of laminar-turbulent patterning are discussed. Full article