Two-dimensional Poiseuille Flow Research Articles

Subcritical transitional flow in two-dimensional plane Poiseuille flow

Recently, subcritical transition to turbulence in the quasi-two-dimensional (quasi-2-D) shear flow with strong linear friction (Camobreco et al., J. Fluid Mech., vol. 963, 2023, R2) has been demonstrated by the 2-D mechanism at $Re = 71\,211$ , and the nonlinear Tollmien–Schlichting (TS) waves related to the edge state were approached independently of initial optimal disturbances. For 2-D plane Poiseuille flow, transition to the fully developed turbulence requires that the Reynolds number is several times larger than the critical Reynolds number $Re_c$ (Markeviciute & Kerswell, J. Fluid Mech., vol. 917, 2021, A57). In this paper, we observed the subcritical transitional flow in 2-D plane Poiseuille flow driven by the nonlinear TS waves by both linear and nonlinear optimal disturbances ( $Re < Re_c$ ) with different quantitative edge states. The nonlinear optimal disturbances could trigger the sustained subcritical transitional flow for $Re \geqslant 2400$ . The initial energy for nonlinear optimal disturbance is more efficient than the linear optimal disturbance in reaching the subcritical transitional flow for $2400 \leqslant Re \leqslant 5000$ . Moreover, the initial energy of linear optimal disturbance is larger than the energy of its edge state. The nonlinear TS waves along the edge state are formed by the nonlinear optimal disturbances to trigger transitional flow, which agrees well with the main conclusions of Camobreco et al. (J. Fluid Mech., vol. 963, 2023, R2), while the required $Re$ of 2-D plane Poiseuille flow is much smaller.

Impact of wall-slip on the soluto-marangoni instability in a two-fluid system in a channel–Creeping flow scenario

The role of wall-slip on the soluto-Marangoni instability that emerges in a two-fluid creeping two-dimensional Poiseuille flow is assessed within the framework of Orr-Sommerfeld analysis and species-transport system. Different wall-slip scenarios, namely S1 (slip at the upper wall), S2 (slip at the lower wall), S1S2 (symmetric slip at the upper and lower walls) are considered. The system instability is characterized by identifying unstable and stable zones in the m-n (viscosity ratio - depth ratio) plane for each of the slip conditions at the wall using asymptotic analysis for long-wave disturbances. Numerical simulations based on Chebyshev spectral collocation method reveals the emergence of two modes (M1 and M2) and corresponding stability features are displayed for typical values of Marangoni numbers, n, m and slip lengths. The occurrence of short-wave modes and the features of exchanging dominance between the short-wave modes in the presence of wall-slip for different slip condition are elucidated and the physical mechanism responsible for triggering the instability in the system are revealed through energy budget analysis. The channel system hosting less viscous fluid adjacent to the upper wall demonstrates interesting characteristics such as (among other things): stabilizing/destabilizing the emerging M1/M2 modes in system with no-slip by imposing S1/S2/S1S2 wall conditions and increasing the slip length; exchange of dominance of unstable modes beyond a threshold slip length for S1 slip type. The study thus presents possible means of stabilizing/destabilizing a creeping channel flow system by designing the walls of the channel with appropriate slip type relevant to the application.

Physical mechanisms of red blood cell splenic filtration.

The splenic interendothelial slits fulfill the essential function of continuously filtering red blood cells (RBCs) from the bloodstream to eliminate abnormal and aged cells. To date, the process by which 8 [Formula: see text]m RBCs pass through 0.3 [Formula: see text]m-wide slits remains enigmatic. Does the slit caliber increase during RBC passage as sometimes suggested? Here, we elucidated the mechanisms that govern the RBC retention or passage dynamics in slits by combining multiscale modeling, live imaging, and microfluidic experiments on an original device with submicron-wide physiologically calibrated slits. We observed that healthy RBCs pass through 0.28 [Formula: see text]m-wide rigid slits at 37 °C. To achieve this feat, they must meet two requirements. Geometrically, their surface area-to-volume ratio must be compatible with a shape in two tether-connected equal spheres. Mechanically, the cells with a low surface area-to-volume ratio (28% of RBCs in a 0.4 [Formula: see text]m-wide slit) must locally unfold their spectrin cytoskeleton inside the slit. In contrast, activation of the mechanosensitive PIEZO1 channel is not required. The RBC transit time through the slits follows a [Formula: see text]1 and [Formula: see text]3 power law with in-slit pressure drop and slip width, respectively. This law is similar to that of a Newtonian fluid in a two-dimensional Poiseuille flow, showing that the dynamics of RBCs is controlled by their cytoplasmic viscosity. Altogether, our results show that filtration through submicron-wide slits is possible without further slit opening. Furthermore, our approach addresses the critical need for in vitro evaluation of splenic clearance of diseased or engineered RBCs for transfusion and drug delivery.

Characteristics of advection–diffusion–Langmuir adsorption processes in two-dimensional plane Poiseuille flows

The advection–diffusion–Langmuir adsorption processes of a liquid solution, colloid, or suspension occur in many biomedical and chemical engineering fields. The dynamics of the system can be described by the so-called advection–diffusion–reaction (ADR) equations and are greatly influenced by five nondimensional numbers. Up to now, cases over a wider range of parameters have not been thoroughly studied, and the quantitative dependence of the system dynamics on the parameters remains unclear. In this study, we systematically solve the ADR equations in two-dimensional plane Poiseuille flows for cases with selected values of parameters by the finite difference method. We identify two different regimes in terms of the distribution of the maximum adsorption flux and discuss the dominant mechanism of mass transfer and the influences of the nondimensional parameters in each regime. We then propose analytical models to describe the influences of specific parameters on the adsorption equilibrium time. The results of this research may provide a convenient method to identify the dominant processes in the advection–diffusion–Langmuir adsorption system in future studies.

On the Stability and Control of Two-Dimensional Channel Flow

Abstract A direct numerical simulation is used to solve the two-dimensional plane Poiseuille flow for three different stability cases; specifically, they are a supercritical case at a Reynolds number of 10,000, a critical-stable case at a Reynolds number of 5772.22, and subcritical case at a Reynolds number of 1000. The perturbations are developed using the eigenfunctions of Orr–Sommerfeld equation's solution. In many applications, the flow was required to be fully laminar. As a result, a model-based controller is developed for stabilizing the flow field using unsteady strong suction and blowing technique, through distributed slots on the two walls of the channel. This technique is introduced to cancel the propagating wave in the boundary-layers near the two walls. Promising results are achieved for the unstable and critical-stable cases. The stable case is tested. Slight improvements in its growth rate are recorded. In turn, a faster response for the flow field is achieved.

Двумерное течение Пуазейля ньютоновской жидкости с вязкостью, зависящей от температуры

The problem of a two-dimensional Poiseuille flow of a Newtonian fluid with a viscosity exponen-tially dependent on temperature under the influence of longitudinal pressure and temperature gra-dients is reduced to a two-parameter boundary value problem for a third-order ordinary differential equation with respect to a dimensionless unknown characterizing the change in temperature across the channel. In the range of negative values of the dimensionless longitudinal temperature gradient, the solution has two branches. By closing, the branches form a boundary on the parameter plane beyond which Poiseuille-type flows do not exist. The solutions belonging to different branches dif-fer in the flow rate and intensity of heat transfer by the liquid through solid boundaries. The latter circumstance entails deceleration of the flow in the near-wall areas. In the region of positive values of the dimensionless longitudinal temperature gradient, the solution is unique. With an increase in the mentioned parameter, the heat flow to the liquid from the walls increases, which entails the formation of a more bulged velocity profile compared to the isothermal case. At sufficiently large values of the longitudinal temperature gradient, there appears a ‘stick’ flow with a uniform velocity distribution everywhere except for narrow layers near solid boundaries. It is shown that the rate is proportional to the ratio of the densities of the transverse and longitudinal heat flows. Explicit ex-pressions for the pressure and shear stress fields are given. The properties of monotonicity of the velocity and temperature profiles as functions of the transverse coordinate have been studied. It is shown that, regardless of the values of the dimensionless parameters of the problem, the velocity has a positive maximum on the central axis of the channel and monotonically decreases to zero at the solid boundary.

A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows

An exact pairwise hydrodynamic theory is developed for the flow-induced spatial distribution of particles in dilute polydisperse suspensions undergoing two-dimensional unidirectional flows, including shear and planar Poiseuille flows. Coupled diffusive fluxes and a drift velocity are extracted from a Boltzmann-like master equation. A boundary layer is predicted in regions where the shear rate vanishes with thickness set by the radii of the upstream collision cross-sections for pair interactions. An analysis of this region yields linearly vanishing drift velocities and non-vanishing diffusivities where the shear rate vanishes, thus circumventing the source of the singular particle distribution predicted by the usual models. Outside of the boundary layer, a power-law particle distribution is predicted with exponent equal to minus half the exponent of the local shear rate. Trajectories for particles with symmetry-breaking contact interactions (e.g. rough particles, permeable particles, emulsion drops) are analytically integrated to yield particle displacements given by quadratures of hard-sphere (or spherical drop) mobility functions. Using this analysis, stationary particle distributions are obtained for suspensions in Poiseuille flow. The scale for the particle distribution in monodisperse suspensions is set by the collision cross-section of the particles but its shape is almost universal. Results for polydisperse suspensions show size segregation in the central boundary layer with enrichment of smaller particles. Particle densities at the centreline scale approximately with the inverse square root of particle size. A superposition approximation reliably predicts the exact results over a broad range of parameters. The predictions agree with experiments in suspensions up to approximately 20 % volume fraction without fitting parameters.

Simulation of the Particle Dynamics in the Two-Dimensional Poiseuille Flow with Low Reynolds Number

Simulation of the Particle Dynamics in the Two-Dimensional Poiseuille Flow with Low Reynolds Number

Taylor dispersion of elongated rods at small and large rotational Péclet numbers

This work analyzes the Taylor dispersion of elongated rods in a two-dimensional Poiseuille flow between parallel, rigid walls at small and large rotational Peclet numbers. Asymptotic expressions for the dispersivity and mean advection speed in these limits are derived. The approach could be used to analyze dispersion of rods in other types of unidirectional flows.

Symmetric boundary condition for the MPS method with surface tension model

In order to supplement the investigation of boundary conditions for the MPS method, a general computational algorithm of symmetric boundary condition for MPS is presented to simulate the symmetric problems. By the MPS method with the symmetric boundary condition (the SBC-MPS method), only half or quarter of the full computational domain needs to be set up, thus, the computational cost is able to be slashed dramatically. In the SBC-MPS method, mirror particles are introduced, which will be arranged symmetrically according to the positions of real liquid particles near to the symmetric plane, but not involved in the solution of physical properties. The function of these mirror particles is to compensate the values for the calculation of real liquid particles. As numerical tests, various of problems including two-dimensional Poiseuille Flow, three-dimensional dam-break, and some surface tension dominant problems, such as droplet wetting / impinging to a solid surface and binary droplets collision have been systematically studied by the SBC-MPS method. As a result, the algorithm shows good robustness, stability and accuracy, which demonstrates a great potential to improve the computational efficiency for dealing with symmetric problems.

Taylor dispersion of elongated rods

Particles transported in fluid flows, such as cells, polymers, or nanorods, are rarely spherical. In this study, we numerically and theoretically investigate the dispersion of an initially localized patch of passive elongated Brownian particles constrained to one degree of rotational freedom in a two-dimensional Poiseuille flow, demonstrating that elongated particles exhibit an enhanced longitudinal dispersion. In a shear flow, the rods translate due to advection and diffusion and rotate due to rotational diffusion and their classical Jeffery's orbit. The magnitude of the enhanced dispersion depends on the particle's aspect ratio and the relative importance of its shear-induced rotational advection and rotational diffusivity. When rotational diffusion dominates, we recover the classical Taylor dispersion result for the longitudinal spreading rate using an orientationally averaged translational diffusivity for the rods. However, in the high-shear limit, the rods tend to align with the flow and ultimately disperse more due to their anisotropic diffusivities. Results from our Monte Carlo simulations of the particle dispersion are captured remarkably well by a simple theory inspired by Taylor's original work. For long times and large Peclet numbers, an effective one-dimensional transport equation is derived with integral expressions for the particles' longitudinal transport speed and dispersion coefficient. The enhanced dispersion coefficient can be collapsed along a single curve for particles of high aspect ratio, representing a simple correction factor that extends Taylor's original prediction to elongated particles.

Study on the binding focusing state of particles in inertial migration

In this paper, the inertial migration of particles in a periodic straight channel is studied numerically based on the immersed boundary-lattice Boltzmann method in two-dimensional and three-dimensional Poiseuille flow. In the model, on considering the inertial migration of particles, the interaction of a soft particle and a hard particle that are similar in size is studied. We observe an interesting particle interaction phenomenon named the binding focusing state (BFS). That is, when the focused soft particle gets close to the hard particle, both particles will switch their original runways, and then run together on a new equilibrium position. To investigate the characteristics of the BFS, the effects of the soft particle deformability, the Reynolds numbers and the initial particle positions on the BFS are studied in detail. The soft particle deformability and Reynolds numbers can affect the formation of the BFS. Moreover, the streamline and pressure around the particles are analyzed to show the formation laws of the BFS; this is more intuitive and aids in understanding the BFS.

The effect of temperature-dependent viscosity and thermal conductivity on velocity and temperature field: an analytical solution using the perturbation technique

This paper proposes a general form of the perturbation expansion method for the governing equations of viscous flow coupled to the temperature evolution. The effect of the variations of viscosity and thermal conductivity with temperature on the temperature and velocity fields in a steady two-dimensional Couette–Poiseuille flow is examined. The presented analytical solution by the perturbation method is validated against a finite difference solution of the governing equations. The numerical and analytical solutions are in good agreement.

Symmetry-breaking waves and space-time modulation mechanisms in two-dimensional plane Poiseuille flow

Two distinct scenarios of spatial modulation in two-dimensional plane Poiseuille flow have been studied. The first one is based on the identification of a new family of asymmetric Tollmien-Schlichting waves (TSW) breaking the reflectional symmetry about the channel midplane. The second follows the fate of a branch of time-periodic space-modulated waves that exclusively bridge upper-branch TSW-trains of different number of replicas by means of a codimension-2 bifurcation point. These modulated waves may therefore play a relevant role in the strange saddle governing domain-filling turbulent dynamics at high Reynolds numbers.

Linear stability and energy stability of plane Poiseuille flow with isotropic and anisotropic slip boundary conditions

The linear stability and energy stability of the plane Poiseuille flow with the isotropic and anisotropic slip boundary conditions are theoretically analyzed and numerically calculated in this paper. The asymptotic expansions of the critical parameters for the linear stability and energy stability are derived from the eigenvalue equations characterizing the least stable modes. The critical Reynolds number for the linear stability based on 1.5 times of the bulk mean streamwise velocity is found to be Rl2D≈(1+2741l2)R02D when the non-dimensional isotropic slip length l ≪ 1, where R02D≈5772 is the critical Reynolds number under the no-slip boundary condition. The critical Reynolds numbers for the linear stability are calculated for a wide range of anisotropic slip lengths and are found to be no larger than their counterparts in the isotropic slip cases with the same streamwise slip lengths. In the energy stability analyses of the two-dimensional and three-dimensional plane Poiseuille flows with the isotropic slip boundary condition, the critical Reynolds numbers are found to be RlE2D≈(1+14.95l2)R0E2D and RlE3D≈(1+8.37l2)R0E3D, where R0E2D≈87.6 and R0E3D≈49.6 are their counterparts under the no-slip boundary condition. In the three-dimensional plane Poiseuille flow with the anisotropic slip boundary condition, the critical Reynolds number for the energy stability increases with the increase in streamwise slip length lx and with the decrease in spanwise slip length lz, and its first-order approximation is Rlx,lzE3D≈[1+2.41(lx−lz)]R0E3D.

An immersed boundary-lattice Boltzmann method for gaseous slip flow

An immersed boundary (IB)-lattice Boltzmann (LB) method is proposed for microchannel slip flow encountered in microfluidics applications such as microelectromechanical Systems, filtration applications with nanofibers, polymer processing, and unconventional shale gas and coal seam gas applications. The LB method is theoretically analyzed to have an intrinsic ability to model velocity discontinuities at finite Knudsen numbers (Kn) when a sufficiently fine grid spacing and an external continuous perturbation, e.g., the body force of an IB method, are applied. Based on this analysis, an IB method coupled with a LB framework without ghost grids in nonfluid areas is proposed to study gaseous slip flow at finite Kn. In addition, an improved treatment to the suspending grids in nonfluid areas is proposed to assist the IB-LB method. In the simulations of two-dimensional Poiseuille and Couette flows for 0.01 ≤ Kn ≤ 1, the slip flow predicted by the proposed nonghost-grid IB-LB method achieves good agreement with that predicted by the linearized Boltzmann and/or Direct Simulation Monte Carlo methods up to Kn = 0.2. Since the proposed IB-LB method is free of adjustable parameters and slip velocity models, it provides a simple and promising pathway for modeling microscale flow applications for the validated regime, i.e., Kn < = 0.2.

Inertial migration of an electrophoretic rigid sphere in a two-dimensional Poiseuille flow

There has been a recent interest in integrating external fields with inertial microfluidic devices to tune particle focusing. In this work, we analyse the inertial migration of an electrophoretic particle in a two-dimensional Poiseuille flow with an electric field applied parallel to the walls. For a thin electrical double layer, the particle exhibits a slip-driven electrokinetic motion along the direction of the applied electric field, which causes the particle to lead or lag the flow (depending on its surface charge). The fluid disturbance caused by this slip-driven motion is characterized by a rapidly decaying source-dipole field which alters the inertial lift on the particle. We determine this inertial lift using the reciprocal theorem. Assuming no wall effects, we derive an analytical expression for a ‘phoretic lift’ which captures the modification to the inertial lift due to electrophoresis. We also take wall effects into account, at the leading order, using the method of reflections. We find that for a leading particle, the phoretic lift acts towards the regions of high shear (i.e. walls), while the reverse is true for a lagging particle. Using an order-of-magnitude analysis, we obtain different components of the inertial force and classify them on the basis of the interactions from which they emerge. We show that the dominant contribution to the phoretic lift originates from the interaction of the source-dipole field (generated by the electrokinetic slip at the particle surface) with the stresslet field (generated due to particle’s resistance to strain in the background flow). Furthermore, to contrast the slip-driven phenomenon (electrophoresis) from the force-driven phenomenon (buoyancy) in terms of their influence on the inertial migration, we also study a non-neutrally buoyant particle. We show that the gravitational effects alter the inertial lift primarily through the interaction of the background shear with the buoyancy-induced Stokeslet field.

Numerical modelling of two-dimensional melt fracture instability in viscoelastic flow

Computationally modelling the two-dimensional (2-D) Poiseuille flow along and outside a straight channel with a differential viscoelastic constitutive equation, we demonstrate unstable dynamics involving bifurcations from steady flow to periodic melt fracture (sharkskin instability) and its further transition regime to a chaotic state. The numerical simulation first exposes transition from steady flow to a weak instability of periodic fluctuation, and in the middle of this periodic limit cycle (in the course of increasing flow intensity) a unique bifurcation into the second steady state is manifested. Then, a subcritical (Hopf) transition restoring this stable flow to stronger periodic instability follows, which results from the high stress along the streamlines of finite curvature with small vortices near the die lip. Its succeeding chaotic transition at higher levels of flow elasticity that induces gross melt fracture, seems to take a period doubling as well as quasiperiodic route. By simple geometrical modification of the die exit, we, as well, illustrate reduction or complete removal of sharkskin and melt fractures. The result as a matter of fact suggests convincing evidence of the possible cause of the sharkskin instability and it is thought that this fluid dynamic transition has to be taken into account for the complete description of melt fracture. The competition between nonlinear dynamic transition and other possible origins such as wall slip will ultimately determine the onset of the sharkskin and melt fractures. Therefore, the current study conceivably provides a robust methodology to portray every possible type of melt fracture if combined with an appropriate mechanism that also results in flow instability.

Optimal mixing in two-dimensional stratified plane Poiseuille flow at finite Péclet and Richardson numbers

We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity $\unicode[STIX]{x1D708}$ and density diffusivity $\unicode[STIX]{x1D705}$. The initial diffusive error function density distribution varies continuously so that $\unicode[STIX]{x1D70C}\in [\bar{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D70C}_{0}/2,\bar{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D70C}_{0}/2]$. A constant pressure gradient is imposed in a plane two-dimensional channel of depth $2h$. We consider flows with a finite Péclet number $Pe=U_{m}h/\unicode[STIX]{x1D705}=500$ and Prandtl number $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$, and a range of bulk Richardson numbers $Ri_{b}=g\unicode[STIX]{x1D70C}_{0}h/(\bar{\unicode[STIX]{x1D70C}}U^{2})\in [0,1]$ where $U_{m}$ is the maximum flow speed of the laminar parallel flow, and $g$ is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimisation problems, extending the optimal mixing results of Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximise the time-averaged perturbation kinetic energy developed over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be $T=10$) of a ‘mix-norm’ as first introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46), further discussed by Thiffeault (Nonlinearity, vol. 25, 2012, pp. 1–44) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged kinetic-energy maximising perturbations are significantly suboptimal at mixing compared to the mix-norm minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm minimising optimal perturbations can still trigger substantial mixing for $Ri_{b}\lesssim 0.3$. By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into ‘available potential energy’, which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic-energy reservoirs.

Finite element lattice Boltzmann method for fluid flow through complex fractured media with permeable matrix

Abstract This study proposed an improved finite element based generalized lattice Boltzmann model (FE-GLBM) to simulate the fluid flow in extremely heterogeneous porous media, in which the generalized lattice Boltzmann equations proposed by Guo and Zhao (2002) are solved using the characteristic Galerkin finite element method (GFEM), and the FEM is applied to improve the flexibility of GLBM and to optimize the flow simulation in complex geometries. An image-based automatic meshing method is adopted to generate advanced unstructured mesh of the structure. The proposed numerical model is validated with analytical results, finite-difference results, and previously published data in three cases, including the generalized two-dimensional Poiseuille flow, the lid-driven cavity flow, and the circular cylinder flow, respectively. The sensitivity study of the fluid flow in three-dimensional fractured porous media with a disordered fracture network and permeable matrix is carried out as a practical application example, and the results indicate that the matrix permeability plays an important role in controlling the flow dynamics in fractured porous media.