Axisymmetric Poiseuille Flow Research Articles

Shear-induced migration in concentrated suspensions: Particle mass conservation, contact pressure and jamming

A two-velocity fluid model is presented to describe particle migration in mono-disperse suspensions of neutrally buoyant particles. In contrast to previous migration models, the proposed formulation allows us to impose explicit boundary conditions on particle velocity, and thereby to satisfy strict mass conservation for the particle phase. In addition, the upper bound on particle volume fraction (jamming limit) is strictly enforced through a non-smooth complementarity condition and the introduction of a particle jamming pressure. The model is applied to an axisymmetric Poiseuille flow and solved using a finite-element method. For that purpose, a specific, fully implicit algorithm based on non-smooth optimisation tools is developed and validated. Preliminary comparisons with experimental data from the literature show promising agreement. In particular, the model properly captures the formation of an inner plug region, in which the suspension is saturated and jammed. • New particle migration model for suspension. • Strict mass conservation for each phase. • Two-velocity description. • Jamming via non-smooth complementarity condition.

Деякі особливості втрати стійкості ламінарною течією в трубі

Despite the seeming simplicity of the steady flow in a pipe of constant radius, the question of the cause and process of the transition remains debatable. Especially since the necessary condition for the stability loss of parabolic profile is not satisfied, and the linear theory of hydrodynamic stability for an axisymmetric Poiseuille flow does not give growing axisymmetric eigen solutions for any Reynolds numbers, since the terms characterizing the interaction of disturbances with the initial velocity profile drop out in the linearized equations of momentum conservation. The report presents the results of the study of stages of convective stability loss for the flow at the initial section of the pipe depending on the variable parameters based on the numerical solution of the three-dimensional system of unsteady Navier-Stokes equations and the equation energy transfer. The variable parameters in this study were: Reynolds number, magnitude and gradient sign of the dynamic viscosity coefficient arising in nonisothermal flows. An analogy of the arising secondary axisymmetric large-scale toroidal vortex structures in the near-wall region to Tollmien-Schlichting waves in the region of the transition of the laminar boundary layer on the plate is shown. The subsequent loss of axisymmetry and stability of these torus-like vortex structures is analyzed, which leads to the formation of fairly regular longitudinal vortex structures downstream, the nonlinear interaction of which leads to chaotization of the flow. The lengths of these sections are determined depending on the Reynolds number, the magnitude and sign of the gradient of the dynamic viscosity coefficient.

On the use of the Lambert function in solving non-Newtonian flow problems

We consider unidirectional flows of ideal or regularized Bingham fluids as well as viscoelastic fluids for which analytical solutions can be derived in terms of the Lambert W function. Explicit expressions are derived for the radius of the yielded region in partially yielded circular and axial Couette flows. Analytical solutions are also derived for the velocity and the volumetric flow rate in the plane and axisymmetric Poiseuille flows of a Windhab fluid, which is a combination of the Bingham and Papanastasiou models, and for the shear stress in the plane Couette flow of an exponential Phan–Thien–Tanner fluid. Finally, the Lambert function is used to solve the Poiseuille flow of a power-law fluid and the Newtonian circular Couette flow with wall slip and non-zero slip yield stress by means of a regularized slip equation, which is valid for any value of the wall shear stress.

Application of the Lambert W function to steady shearing Newtonian flows with logarithmic wall slip

We consider various viscometric flows of a Newtonian fluid, i.e., plane, annular, and circular Couette flows and planar and axisymmetric Poiseuille flows, in the presence of wall slip that follows a logarithmic slip law. We derive analytical solutions in terms of the Lambert W function. The effects of logarithmic slip on these flows are discussed, and comparisons of the results with their Navier-slip counterparts are made.

Numerical Simulation of Non-Newtonian Inelastic Flows in Channel based on Artificial Compressibility Method

In this study, inelastic constitutive modelling is considered for the simulation of shear-thinning fluids through a circular channel. Numerical solutions are presented for power-law inelastic model, considering axisymmetric Poiseuille flow through a channel. The numerical simulation of such fluid is performed by using the Galerkin finite element approach based on artificial compression method (AC-method). Usually, the Naiver-Stoke partial differential equations are used to describe fluid activity. These models consist of two partial differential equations; a continuity equation (mass conservation) and time-dependent conservation of momentum, which are maintained in the cylindrical coordinate system (axisymmetric) flow in current study. The effects of many factors such as Reynolds number (Re) and artificial compressibility parameter (sac) are discussed in this study. In particular, this study confirms the effect of these parameters on the convergence level. To meet the method analysis, Poiseuille flow along a circular channel under an isothermal state is used as a simple test problem. This test is conducted by taking a circular section of the pipe. The Findings reveal that, there is a significant effect from the inelastic parameters upon the the velocity temporal convergence-rates of velocity, while for pressue, the change in convergence is modest. In addition, the rate of convergence is increased as the values of artificial compressibility parameter (sac) are decreased.

Axisymmetric Poiseuille flow of a Bingham plastic with rheological parameters varying linearly with pressure

We consider the steady axisymmetric Poiseuille flow of a Bingham plastic under the assumption that both the plastic viscosity and the yield stress vary linearly with pressure. An analytical solution is derived for the case where the growth coefficients of both rheological parameters are equal, which is indeed a reasonable assumption for certain oil-drilling fluids allowing the existence of a separable solution with a cylindrical unyielded core. The conditions for the occurrence of flow and the effects of the growth coefficient on the radius of the unyielded plug, the velocity profiles and the pressure distributions are discussed.

One Dimensional Model for Droplet Ejection Process in Inkjet Devices

In recent years, physics-based computer models have been increasingly applied to design the drop-on-demand (DOD) inkjet devices. The initial design stage for these devices often requires a fast turnaround time of computer models, because it usually involves a massive screening of a large number of design parameters. Thus, in the present study, a 1D model is developed to achieve the fast prediction of droplet ejection process from DOD devices, including the droplet breakup and coalescence. A popular 1D slender-jet method (Egger, 1994) is adopted in this study. The fluid dynamics in the nozzle region is described by a 2D axisymmetric unsteady Poiseuille flow model. Droplet formation and nozzle fluid dynamics are coupled, and hence solved together, to simulate the inkjet droplet ejection. The arbitrary Lagrangian–Eulerian method is employed to solve the governing equations. Numerical methods have been proposed to handle the breakup and coalescence of droplets. The proposed methods are implemented in an in-house developed MATLAB code. A series of validation examples have been carried out to evaluate the accuracy and the robustness of the proposed 1D model. Finally, a case study of the inkjet droplet ejection with different Ohnesorge number (Oh) is presented to demonstrate the capability of the proposed 1D model for DOD inkjet process. Our study has shown that 1D model can significantly reduce the computational time (usually less than one minute) yet with acceptable accuracy, which makes it very useful to explore the large parameter space of inkjet devices in a short amount of time.

Development lengths in Newtonian Poiseuille flows with wall slip

The effect of wall slip on the development of planar and axisymmetric Newtonian Poiseuille flows is studied by means of finite element simulations. The Navier slip law is employed. i.e., it is assumed that the slip velocity varies linearly with the wall shear stress. In addition to the standard definition of the development length, L, as the length required for the maximum velocity to attain 99% of its fully-developed value, the wall development length Lw is also relevant in the presence of slip. This is defined as the length required for the slip velocity to decrease to 1.01% of its fully-developed value. The numerical simulations for the planar and the axisymmetric geometries showed that both L and Lw increase with slip passing through a maximum and vanish at a critical value of the slip parameter corresponding to the full slip case. Moreover, the velocity overshoots and the axial pressure minimum near the entrance become less pronounced as slip becomes stronger. The calculations of L for the planar flow are in good agreement with available results in the literature. An interesting result is that in contrast to the axisymmetric flow in which Lw<L, the planar flow develops more slowly at the wall than at the midplane, i.e. Lw>L in the planar case.

Start-up and cessation Newtonian Poiseuille and Couette flows with dynamic wall slip

Analytical solutions are derived for the start-up and cessation Newtonian Poiseuille and Couette flows with wall slip obeying a dynamic slip model. This slip equation allows for a relaxation time in the development of wall slip by means of a time-dependent term which forces the eigenvalue parameter to appear in the boundary conditions. The resulting spatial problem corresponds to a Sturm–Liouville problem different from that obtained using the static Navier slip condition. The orthogonality condition of the associated eigenfunctions is derived and the solutions are provided for the axisymmetric and planar Poiseuille flows and for the circular Couette flow. The effect of dynamic slip on the flow development is then discussed.

Asymptotic solutions of weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity

We consider both the axisymmetric and planar steady-state Poiseuille flows of weakly compressible Newtonian fluids, under the assumption that both the density and the shear viscosity vary linearly with pressure. The primary flow variables, i.e. the two non-zero velocity components and the pressure, as well as the mass density and viscosity of the fluid are represented as double asymptotic expansions in which the isothermal compressibility and the viscosity–pressure-dependence coefficient are taken as small parameters. A standard perturbation analysis is performed and asymptotic, analytical solutions for all the variables are obtained up to second order. These results extend the solutions of the weakly compressible flow with constant viscosity and those of the incompressible flow with pressure-dependent viscosity. The combined effects of compressibility and the pressure dependence of the viscosity are analyzed and discussed.

Possible oriented transition of multiple-emulsion globules with asymmetric internal structures in a microfluidic constriction

When a globule with a complete symmetry (such as simple spherical droplets and concentric double emulsions) is transiting in a constriction tube, there is only one pattern of the transition. However, for a multiple-emulsion globule with asymmetric internal structures, there are many possible patterns with different pressure drops Δp due to various initial orientations of the inner droplets. In this paper, a boundary integral method developed recently is employed to investigate numerically the possible oriented transition of a globule with two unequal inner droplets in an axisymmetric microfluidic constriction. The transition is driven by an axisymmetric Poiseuille flow with a fixed volume flow rate, and the rheological behaviors of the globule are observed carefully. When the big inner droplet is initially located in the front of the globule, the maximum pressure drop during the transition is always lower than that when it is initially placed in the rear. Thus, a tropism-whereby a globule more easily gets through the constriction when its bigger inner droplet locates in its front initially-might exist, in which the orientating stimulus is the required pressure drops. The physical explanation of this phenomenon has also been analyzed in this paper.

Symmetry breaking and cross-streamline migration of three-dimensional vesicles in an axial Poiseuille flow.

We analyze numerically the problem of spontaneous symmetry breaking and migration of a three-dimensional vesicle [a model for red blood cells (RBCs)] in axisymmetric Poiseuille flow. We explore the three relevant dimensionless parameters: (i) capillary number, Ca, measuring the ratio between the flow strength over the membrane bending mode, (ii) the ratio of viscosities of internal and external liquids, λ, and (iii) the reduced volume, ν=[V/(4/3)π]/(A/4π)3/2 (A and V are the area and volume of the vesicle). The overall picture turns out to be quite complex. We find that the parachute shape undergoes spontaneous symmetry-breaking bifurcations into a croissant shape and then into slipper shape. Regarding migration, we find complex scenarios depending on parameters: The vesicles either migrate towards the center, or migrate indefinitely away from it, or stop at some intermediate position. We also find coexisting solutions, in which the migration is inwards or outwards depending on the initial position. The revealed complexity can be exploited in the problem of cell sorting out and can help understanding the evolution of RBCs' in vivo circulation.

Cessation of viscoplastic Poiseuille flow with wall slip

We solve numerically the cessation of axisymmetric Poiseuille flow of a Herschel–Bulkley fluid under the assumption that slip occurs along the wall. The Papanastasiou regularization of the constitutive equation is employed. As for the slip equation, a power-law expression is used to relate the wall shear stress to the slip velocity, assuming that slip occurs only above a critical wall shear stress, known as the slip yield stress. It is shown that, when the latter is zero, the fluid slips at all times, the velocity becomes and remains uniform before complete cessation, and the stopping time is finite only when the slip exponent s<1. In the case of Navier slip (s=1), the stopping time is infinite for any non-zero Bingham number and the volumetric flow rate decays exponentially. When s>1, the decay is much slower. Analytical expressions of the decay of the flat velocity for any value of s and of the stopping time for s<1 are also derived. Using a discontinuous slip equation with slip yield stress poses numerical difficulties even in one dimensional time-dependent flows, since the transition times from slip to no-slip and vice versa are not known a priori. This difficulty is overcome by regularizing the slip equation. The numerical results showed that when the slip yield stress is non-zero, slip ceases at a finite critical time, the velocity becomes flat only in complete cessation, and the stopping times are finite, in agreement with theoretical estimates.

Combined effects of compressibility and slip in flows of a Herschel–Bulkley fluid

In this work, the combined effects of compressibility and slip in Poiseuille flows of Herschel–Bulkley fluids are investigated. The density is assumed to obey a linear equation of state, and wall slip is assumed to follow Navier’s slip condition with zero slip yield stress. The flow is considered to be weakly compressible so that the transverse velocity component is zero and the pressure is a function of the axial coordinate. Approximate semi-analytical solutions of the steady, creeping, plane and axisymmetric Poiseuille flows are derived and the effects of compressibility, slip, and the Bingham number are discussed. In the case of incompressible flow, it is shown that the velocity may become plug at a finite critical value of the slip parameter which is inversely proportional to the yield stress. In compressible flow with slip, the velocity tends to become plug upstream, which justifies the use of one-dimensional models for viscoplastic flows in long tubes. The case of pressure-dependent slip is also investigated and discussed.

Laminar axisymmetric flow of a weakly compressible viscoelastic fluid

The combined effects of weak compressibility and viscoelasticity in steady, isothermal, laminar axisymmetric Poiseuille flow are investigated. Viscoelasticity is taken into account by employing the Oldroyd-B constitutive model. The fluid is assumed to be weakly compressible with a density that varies linearly with pressure. The flow problem is solved using a regular perturbation scheme in terms of the dimensionless isothermal compressibility parameter. The sequence of partial differential equations resulting from the perturbation procedure is solved analytically up to second order. The two-dimensional solution reveals the effects of compressibility and the other dimensionless numbers and parameters in the flow. Expressions for the average pressure drop, the volumetric flow rate, the total axial stress, as well as for the skin friction factor are also derived and discussed. The validity of other techniques used to obtain approximate solutions of weakly compressible flows is also discussed in conjunction with the present results.

Perturbation solutions of Poiseuille flows of weakly compressible Newtonian liquids

Both the planar and axisymmetric isothermal Poiseuille flows of weakly compressible Newtonian liquids with constant shear and bulk viscosities are solved up to the second-order. A linear equation of state is assumed and a perturbation analysis of the primary flow variables is performed using compressibility as the perturbation parameter. The effects of compressibility, the bulk viscosity, the aspect ratio, and the Reynolds number on the velocity and pressure fields are studied and comparisons are made with available analytical results.

Weakly compressible Poiseuille flows of a Herschel–Bulkley fluid

In this work, we derive approximate semi-analytical solutions of the steady, creeping, weakly compressible plane and axisymmetric Poiseuille flows of a Herschel–Bulkley fluid. Since the flow is weakly compressible, the radial velocity component is assumed to be zero and the derivatives of the axial velocity with respect to the axial direction are assumed to be much smaller than those with respect to the radial direction. The axial velocity is then given by an expression similar to that holding for the incompressible flow, the only difference being that the pressure-gradient is a function of the axial coordinate and satisfies a non-linear equation involving the density of the fluid. In the present work, a linear as well as an exponential equation of state, relating the density of the fluid to the pressure, are considered. The pressure distribution along the flow direction is calculated by means of numerical integration and the two-dimensional axial velocity can then be constructed. The effects of compressibility, the equation of state, the Bingham number and the power-law exponent on the solutions are investigated.

Some simple unsteady unidirectional flows of a binary mixture of incompressible Newtonian fluids

The problems dealing with some simple unsteady unidirectional flows of a mixture of two incompressible Newtonian fluids are investigated. By using the constitutive equations appeared in the literature for binary mixtures of chemically inert incompressible Newtonian fluids, the equations governing the motion of the binary mixture are reduced to a system of coupled partial differential equations. By means of integral transforms, the exact solutions of these equations are obtained for the following three problems: (i) unsteady Couette flow, (ii) unsteady plane Poiseuille flow, (iii) unsteady axisymmetric Poiseuille flow.

Resonance behavior of viscoelastic fluids in Poiseuille flow and application to flow enhancement

Resonance-like phenomena in axisymmetric Poiseuille flows of viscoelastic fluids are studied. It is shown that instantaneous flow velocities of the upper convected Maxwell fluid with a spectrum of relaxation times drastically increase at certain frequencies of the oscillating pressure gradient. This effect becomes more pronounced as the tube radius gets smaller. Non-linear constitutive structures are also considered to show that the most effective mean flow rate enhancement for pulsating flows can be reached at the resonance frequencies.

Hydrodynamic interactions of spherical particles in quadratic Stokes flows

The quasi-steady hydrodynamic Stokes drag force and torque exerted on each of N non-identical particles immersed in a general quadratic undisturbed flow field at infinity is analytically investigated. Explicit results are given for the case of two spherical particles. In particular, expressions are obtained for these forces and torques in terms of: (1) the linear and angular velocities of each of the particles together with the local shear rate of the quadratic flow, all evaluated at the center of volume of the two-sphere system; and (2) the position-independent triadic shear-rate gradient characterizing the undisturbed quadratic flow field. These tensorial (matrix) formulas, obtained by the method of reflections for the intrinsic Stokes resistance coefficients for each of the spheres, are expressed nondimensionally in terms of the respective sphere radii, the center-to-center distance 2 H between spheres, a unit vector e lying along the line-of-centers, and a characteristic length-scale R 0 of the undisturbed quadratic flow field — typically the radius of a tube bounding an axisymmetric Poiseuille flow in which the two spheres are suspended. The reflection scheme is accurate within an error of O r/H α r/R 0 β , wherein α+ β≤6 and in which r is a characteristic sphere radius. The respective translational velocities of the two spheres are derived for the particular case where each is freely suspended in an unbounded Poiseuille flow (i.e., wall effects are neglected). It is shown in this case that a net radial motion of the pair of neutrally buoyant spheres ensues across the undisturbed streamlines as a consequence of the quadratic nature of the flow field (coupled with the interparticle hydrodynamic interaction). The direction of net migration calculated for the two spheres is from low to high shear rates. This contrasts with shear-induced migration phenomena observed in concentrated suspensions subjected to inhomogeneous shearing flows, where the direction of particle migration is from high to low shear rates. The two opposing effects acting in concert may help explain how a steady-state radial particle concentration distribution can be achieved for suspensions of non-Brownian particles.