Plane Poiseuille Flow Research Articles

Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

Isogeometric analysis of hydrodynamics of vesicles using a monolithic phase-field approach

In this paper, we present an isogeometric analysis for studying the dynamical behavior of inextensible vesicles under an external fluid flow with inertial forces. We consider a phase-field model Aland, et al. (2014) for the coupled fluid–vesicle problem which enforces global area and volume constraints using a Lagrange multiplier method and employs an extra equation for enforcing local inextensibility condition. Full Navier–Stokes equations are considered and their finite element formulation is presented based on a residual-based variational multiscale method while a standard Galerkin finite element framework is employed for the rest of partial differential equations in the model. We solve the system of PDEs using an implicit, monolithic scheme based on the generalized-α time integration method. Compared to the system of equations considered in Aland, et al. (2014), we reduce the number of equations to be solved by leveraging high continuity of NURBS functions. We also extend the algorithm of the phase-field method to three-dimensional problems. A number of two-dimensional numerical examples which model the dynamics of a vesicle in a quiescent fluid, in a shear flow, and in plane Poiseuille flow with and without obstructions are studied. The resistive immersed surface method is employed for dealing with obstructions. We also consider a 3D example where we study the dynamics of a vesicle in a constricted channel which resembles the situation that a vesicle experiences in a stenosed microchannel.

Structured input–output analysis of transitional wall-bounded flows

Input–output analysis of transitional channel flows has proven to be a valuable analytical tool for identifying important flow structures and energetic motions. The traditional approach abstracts the nonlinear terms as forcing that is unstructured, in the sense that this forcing is not directly tied to the underlying nonlinearity in the dynamics. This paper instead employs a structured-singular-value-based approach that preserves certain input–output properties of the nonlinear forcing function in an effort to recover the larger range of key flow features identified through nonlinear analysis, experiments and direct numerical simulation (DNS) of transitional channel flows. Application of this method to transitional plane Couette and plane Poiseuille flows leads to not only the identification of the streamwise coherent structures predicted through traditional input–output approaches, but also the characterization of the oblique flow structures as those requiring the least energy to induce transition, in agreement with DNS studies, and nonlinear optimal perturbation analysis. The proposed approach also captures the recently observed oblique turbulent bands that have been linked to transition in experiments and DNS with very large channel size. The ability to identify the larger amplification of the streamwise varying structures predicted from DNS and nonlinear analysis in both flow regimes suggests that the structured approach allows one to maintain the nonlinear effects associated with weakening of the lift-up mechanism, which is known to dominate the linear operator. Capturing this key nonlinear effect enables the prediction of a wider range of known transitional flow structures within the analytical input–output modelling paradigm.

Continuous Pathway between the Elasto-Inertial and Elastic Turbulent States in Viscoelastic Channel Flow.

Viscoelastic plane Poiseuille flow is shown to become linearly unstable in the absence of inertia, in the limit of high elasticities, for ultradilute polymer solutions. While inertialess elastic instabilities have been predicted for curvilinear shear flows, this is the first ever report of a purely elastic linear instability in a rectilinear shear flow. The novel instability continues up to a Reynolds number (Re) of O(1000), corresponding to the recently identified elasto-inertial turbulent state believed to underlie the maximum-drag-reduced regime. Thus, for highly elastic ultradilute polymer solutions, a single linearly unstable modal branch may underlie transition to elastic turbulence at zero Re and to elasto-inertial turbulence at moderate Re, implying the existence of continuous pathways connecting the turbulent states to each other and to the laminar base state.

Transient linear stability of pulsating Poiseuille flow using optimally time-dependent modes

Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr–Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien–Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.

Transient dispersion process of active particles

Active particles often swim in confined environments. The transport mechanisms, especially the global one as reflected by the Taylor dispersion model, are of great practical interest to various applications. For the active dispersion process in confined flows, previous analytical studies focused on the long-time asymptotic values of dispersion characteristics. Only several numerical studies preliminarily investigated the temporal evolution. Extending recent studies of Jiang & Chen (J. Fluid Mech., vol. 877, 2019, pp. 1–34; vol. 899, 2020, A18), this work makes a semi-analytical attempt to investigate the transient process. The temporal evolution of the local distribution in the confined-section–orientation space, drift, dispersivity and skewness, is explored based on moments of distributions. We introduce the biorthogonal expansion method for solutions because the classic integral transform method for passive transport problems is not applicable due to the self-propulsion effect. Two types of boundary condition, the reflective condition and the Robin condition for wall accumulation, are imposed respectively. A detailed study on spherical and ellipsoidal swimmers dispersing in a plane Poiseuille flow demonstrates the influences of the swimming, shear flow, initial condition, wall accumulation and particle shape on the transient dispersion process. The swimming-induced diffusion makes the local distribution reach its equilibrium state faster than that of passive particles. Although the wall accumulation significantly affects the evolution of the local distribution and the drift, the time scale to reach the Taylor regime is not obviously changed. The shear-induced alignment of ellipsoidal particles can enlarge the dispersivity but impacts slightly on the drift and the skewness.

Integrating multiple physical properties of microchannel gas flow to extend the Navier–Stokes equations over a wide Knudsen number range

Gas flow in microchannels can be predicted by the Navier–Stokes equations with slip boundary conditions, but only limited to a slightly rarefied flow regime. To improve that, the considerations of the effective mean free path and the volume diffusion phenomena were introduced to the non-kinetic model by previous studies separately. In this study, these two effects, along with the newly proposed wall-to-wall-collision effect, are integrated to extend the Navier–Stokes equations for the planar Poiseuille flows over a wide Knudsen number (Kn) range. The dimensionless mass flow rates calculated by the proposed model can be consistent among different working fluids or flow conditions and mostly agree with the experimental data with Kn≅0.03−57. This analysis facilitates an understanding of the mutual effects on the physical properties of microchannel gas flows and shows a promising prospect for developing a non-kinetic model for highly rarefied flows.

Stability of two-layer flows past slippery surfaces. I. Horizontal channels

We study the role of wall slip on the stability of a two-layered plane Poiseuille flow. The equations of motion for the base flow state are derived, and a linear stability analysis is carried out to arrive at the fourth-order Orr–Sommerfeld equations for the fluid layers. An asymptotic analysis is conducted for wavenumbers in the long wavelength limit. The Orr–Sommerfeld equations are solved numerically using a multidomain Chebyshev Collocation Method to arrive at the spectrum of eigenvalues and their associated eigenfunctions. The effect of wall slip on the stability characteristics of the flow system is examined in greater detail. It is observed that slip brings about a stabilizing, as well as a destabilizing effect on the flow system.

Contribution of Particle–Wall Distance and Rotational Motion of a Single Confined Elliptical Particle to the Effective Viscosity in Pressure-Driven Plane Poiseuille Flows

Rheological properties of the suspension flow, especially effective viscosity, partly depend on spatial arrangement and motion of suspended particles. It is important to consider effective viscosity from the microscopic point of view. For elliptical particles, the equilibrium position of inertial migration in confined state is unclear, and there are few studies on the relationship between dynamics of suspended particles and induced local effective viscosity distribution. Contribution of a single circular or elliptical particle flowing between parallel plates to the effective viscosity was studied, focusing on the particle–wall distance and particle rotational motion using the two-dimensional regularized lattice Boltzmann method and virtual flux method. As a result, confinement effects of the elliptical particle on the equilibrium position of inertial migration were summarized using three definitions of confinement. In addition, the effects of particle shape (aspect ratio and confinement) on the effective viscosity were assessed focusing on the particle–wall distance. The contribution of particle shape to the effective viscosity was found to be enhanced when the particle flowed near the wall. Focusing on the spatial and temporal variation of relative viscosity evaluated from wall shear stress, it was found that the spatial variation of the local relative viscosity was larger than temporal variation regardless of the aspect ratio and particle–wall distance.

Inertial migration of a neutrally buoyant circular particle in a planar Poiseuille flow with thermal fluids

This work presents an extensive numerical study on the inertial migration of a neutrally buoyant circular particle in a planar Poiseuille flow with thermal convection using the lattice Boltzmann method. The numerical analysis confirms that, once the channel Grashof number Gr reaches a critical value Grc, the particle lateral equilibrium position transits from above the centerline to below. Moreover, the critical Grashof number Grc depends on the channel Reynolds number and the particle-to-channel size ratio. A force scaling analysis is performed, from which a unified Grashof number Gr* is introduced to consider the synergetic effects of thermal convection and particle size. In addition, it is also found that the occurrence of equilibrium position transition is not sensitive to the initial conditions, but the exact transition point depends on the initial conditions, such as the initial position of the particle and the fluid flow state.

Theoretical Study on Poiseuille Flow of Herschel-Bulkley Fluid in Porous Media

This theoretical study analyses the effects of geometrical and fluid parameters on the flow metrices in the Hagen-Poiseuille and plane-Poiseuille flows of Herschel-Bulkley fluid through porous medium which is considered as (i) single pipe/single channel and (ii) multi–pipes/multi-channels when the distribution of pores size in the flow medium are represented by each one of the four probability density functions: (i) Uniform distribution, (ii) Linear distribution of Type-I, (iii) Linear distribution of Type-II and (iv) Quadratic distribution. It is found that in Hagen-Poiseuille and plane-Poiseuille flows, Buckingham-Reiner function increases linearly when the pressure gradient increases in the range 1 - 2.5 and then it ascends slowly with the raise of pressure gradient in the range 2.5 - 5. In all of the four kinds of pores size distribution, the fluid’s mean velocity, flow medium’s porosity and permeability are substantially higher in Hagen-Poiseuille fluid rheology than in plane-Poiseuille fluid rheology and, these flow quantities ascend considerably with the raise of pipe radius/channel width and a reverse characteristic is noted for these rheological measures when the power law index parameter increases. The flow medium’s porosity decreases rapidly when the period of the pipes/channels distribution rises from 1 to 2 and it drops very slowly when the period of the pipes/channels rises from 2 to 11.

Benchmark solution for the stability of plane Couette flow with net throughflow

This paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow. A linear stability analysis has been performed by employing the method of normal modes and the resulting stability equation is solved numerically using the Chebyshev collocation method. Contrary to the stability of plane Couette flow (PCF) to small disturbances for all values of the Reynolds number in the absence of vertical throughflow, it is found that PCF becomes unstable owing to the change in the sign of growth rate depending on the magnitude of throughflow. The critical Reynolds number triggering the instability is computed for different values of throughflow dependent Reynolds number and it is shown that throughflow instills both stabilizing and destabilizing effect on the base flow. It is seen that the direction of throughflow has no influence on the stability of fluid flow. A comparative study between plane Poiseuille flow and PCF has also been carried out and the similarities and differences are highlighted.

Vertical distribution and longitudinal dispersion of gyrotactic microorganisms in a horizontal plane Poiseuille flow

A more concise and accurate generalized Taylor dispersion theory is applied to dispersion of active gyrotactic microorganisms in a plane Poiseuille flow. The joint effect of boundary conditions, cell shape anisotropy, swimming speed, and flow speed leads to the nonmonotonic variations of the phenomenological dispersion coefficients.

Dynamics of a pair of paramagnetic particles in shear flows under a uniform magnetic field

We numerically investigated the particle–particle interaction and relative motion of a pair of equal-sized magnetic particles in simple shear and plane Poiseuille flows. Two-dimensional numerical models were created by using direct numerical simulations, which are based on the finite element method and arbitrary Lagrangian–Eulerian approach with full consideration of particle–particle, particle–magnetic field, particle–flow field interactions. The effects of direction and strength of magnetic field on the dynamics of the particles in simple shear and plane Poiseuille flows were investigated, respectively. In the simple shear flow, the presence of magnetic field can show stabilizing or destabilizing effect on the particle trajectories, depending on the direction of the magnetic field. Specifically, the particles initially located at closed trajectories moved closer and closer when the magnetic field is applied at 0°and 135°, while they moved further apart and separated when the field is applied at 90°. In the plane Poiseuille flow, the magnetic field changed the relative motion of two particles: it induced two particles to form a chain when a strong magnetic field is applied at 0°and 135°; it separated the two particles when a strong magnetic field is applied at 45°and 90°. This work offers insights toward understanding the mechanisms of particle–particle interaction in magnetorheological fluids in simple shear or plane Poiseuille flows under a uniform magnetic field.

Discrete unified gas kinetic scheme simulation of microflows with complex geometries in Cartesian grid

Microscale gas flow attracts significant research interest in recent years since it is concerned with a wide range of engineering applications. It is noted that the Navier–Stokes equations-based scheme and the standard lattice Boltzmann method both encounter a great challenge in the simulation of such flows. The newly developed discrete unified gas kinetic scheme (DUGKS) has been demonstrated to be capable of modeling microflows, but presently it is mainly limited to the problems with straight boundaries. In this study, the ghost-cell (GC) immersed boundary method is introduced to the DUGKS for handling curved boundaries. The most attractive feature of the GC method is to set a ghost point inside the solid domain, at which the information is unknown and will be extrapolated linearly from the corresponding wall and image nodes. As for the two latter points, the distribution functions are first evaluated by the inverse distance weighted (IDW) method and then should be corrected according to the impenetrability condition and Maxwellian diffuse-scattering rule. Three typical test cases, including the plane Poiseuille flow, cylindrical Couette flow and flow through porous media are simulated to validate the present IDW-GC-DUGKS. The results demonstrate the accuracy and feasibility of the method for the gaseous microflows.

The centre-mode instability of viscoelastic plane Poiseuille flow

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Early evolution of optimal perturbations in a viscosity-stratified channel

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Wave focusing and related multiple dispersion transitions in plane Poiseuille flows

Motivated by the recent discovery of a dispersive-to-nondispersive transition for linear waves in shear flows, we accurately explored the wavenumber-Reynolds number parameter map of the plane Poiseuille flow in the limit of least-damped waves. We have discovered the existence of regions of the map where the dispersion and propagation features vary significantly from their surroundings. These regions are nested in the dispersive, low-wavenumber part of the map. This complex dispersion scenario demonstrates the existence of linear dispersive focusing in wave envelopes evolving out of an initial, spatially localized, three-dimensional perturbation. An asymptotic wave packet's representation, based on the saddle-point method, allows to enlighten the nature of the packet's morphology, in particular, the arrow-shaped structure and spatial spreading rates. A correlation is also highlighted between the regions of largest dispersive focusing and the regions which are most subject to strong nonlinear coupling in observations.

Self-sustaining and propagating mechanism of localized wave packet in plane-Poiseuille flow

In this Letter, it is revealed that the convection velocity of a localized wave packet in two-dimensional plane-Poiseuille flow is determined by the solitary wave at the centerline of a vortex dipole, which is evinced by subtracting the base flow from the mean flow. The fluctuation component propagates obeying the local dispersion relation of the mean flow and oscillates with a global frequency selected by the upstream marginal absolute instability and, hence, is a traveling wave mode. The vortex dipole provides an unstable region for the fluctuation waves to grow up, and the Reynolds stress of the fluctuation waves leads to streaming components enhancing the vortex dipole. By applying localized initial disturbances, a nonzero wave-packet density is achieved at the threshold state, suggesting a first-order transition.

Internal hydraulic jump in plane Poiseuille two-layer flow: theoretical, numerical and experimental study

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