Creeping Flow Conditions Research Articles

Creeping flow of Herschel-Bulkley fluids in collapsible channels: A numerical study

In this paper, the steady flow of a viscoplastic fluid is modeled in a planar channel equipped with a deformable segment in the middle of an otherwise rigid plate. The fluid is assumed to obey the Herschel-Bulkley model which accounts for both the yield stress and the shear-thinning behavior of physiological fluids such as blood. To accommodate the large deformations of the flexible segment, it is assumed to obey the twoparameter Mooney-Rivlin hyperelastic model. The so-called fluid-structure interaction problem is then solved numerically, under creeping-flow conditions, using the finite element package, COMSOL. It is found that the yield stress leads to a larger wall deformation and a higher pressure drop as compared with Newtonian fluids. This behavior is predicted to intensify if the fluid is shear-thinning. That is, for a given yield stress, the pressure drop and the wall deformation both increase with an increase in the degree of the fluid's shear-thinning behavior.

Heat Transfer Enhancement By Elastic Turbulence in a Micro Curvilinear Channel

With the increasing level of microelectronic techniques, such as MEMS, the effective heat transfer become crucial to their performance. Genernally, the flow in microscale behaves laminar because of the extremely low Reynolds number (Re), which results in very low heat transfer effeciceny. So far, an increasingly efforts have been paid to develop techiniques to improve the heat transfer therein. As is well known in the conventional scale, flow instabilities and turbulence could significantly promote the mixing and heat transfer. Therefore, a great deal of efforts have been paid to to perturb the low Reynolds number flows in order to increase the contact interfaces between fluids with different temperature in microchannel. For example, some efforts are paid to design and utilize specially designed microchannel to excite the unstable flow motion. Another alternative way is to adopt different kinds of working fluid such as nano-fluid, non-Newtonian fluid and so forth. The viscoelastic fluid, a kind of Non-Newtonian fluid, shows nontrivial flow phenomena such as die swell effect (Barus effect), rod climbing effect (Weissenberg effect), Kaye effect, tubeless siphon effect, etc. Moreover, it has been proven that purely elastic flow instabilities or elastic turbulence is possible to be induced under inertialess or creeping flow conditions. This provides another efficient way to realize heat transfer enhancement, which is also the focus of present paper. In this paper, we carried out experimental and numerical investigations on the heat transfer characteristics of viscoelastic fluid and Newtonian fluid in three dimensional (3D) curvilinear microchannel. For the experiments, microelectronics manufacturing technology was taken to fabricate microchips. Platinum films were embedded in a Polydimethyl-siloxane (PDMS) microchannel in order to measure micro heater and channel wall temperature. The polymer solutions (PAAM) were the chosen viscoelastic fluid and sucrose solutions (Newtonian fluid) were used for comparison. For the numerical simulation, the log-conformation algorithm were implemented into the open source software OpenFOAM in order to simulate the viscoelastic flow at high Weissenberg number. Before simulating the viscoelastic fluid flow passing through the target channel, the numerical solver was firstly validated by two benchmark problems i.e., flow around a cylinder and passing a cross-slot by comparing the obtained dimensionless drag coefficient and flow patterns with those in the existing literatures. Combing the experimental and numerical results, the present identified the heat transfer enhancement by the occurrence of elastic instability and turbulence in a curved microchannel.

Microfluidic converging/diverging channels optimised for homogeneous extensional deformation.

In this work, we optimise microfluidic converging/diverging geometries in order to produce constant strain-rates along the centreline of the flow, for performing studies under homogeneous extension. The design is examined for both two-dimensional and three-dimensional flows where the effects of aspect ratio and dimensionless contraction length are investigated. Initially, pressure driven flows of Newtonian fluids under creeping flow conditions are considered, which is a reasonable approximation in microfluidics, and the limits of the applicability of the design in terms of Reynolds numbers are investigated. The optimised geometry is then used for studying the flow of viscoelastic fluids and the practical limitations in terms of Weissenberg number are reported. Furthermore, the optimisation strategy is also applied for electro-osmotic driven flows, where the development of a plug-like velocity profile allows for a wider region of homogeneous extensional deformation in the flow field.

Steady viscoelastic film flow over 2D Topography: II. The effect of capillarity, inertia and substrate geometry

We examine the two-dimensional, steady flow of a viscoelastic film under the action of gravity over a substrate with periodic topographical features. We account for the rheology of the viscoelastic material using the exponential Phan–Thien and Tanner (PTT) constitutive model. The conservation equations are solved via the mixed finite element method combined with a quasi-elliptic grid generation scheme, while the viscoelastic stresses are discretized using the EVSS-G/SUPG method. Our scheme allows the computation of accurate steady-state solutions up to high values of Deborah, Reynolds and capillary numbers. We perform a thorough parametric analysis to investigate the effect of the elastic, capillary and inertia forces on the flow characteristics. Our results indicate that surface tension and elasticity affect the film closer to the location with abrupt changes of the substrate topography; the sizes of the capillary ridge before a step down and of the depression before a step up are increased and move upstream as fluid elasticity or interfacial tension increase. It is shown that under creeping flow conditions the length scale of the capillary ridge increases with De following a power law of ¼, which can also be predicted by simple scaling arguments. Inertia has a more global effect on the film affecting larger portions of it, while in its presence the length scale of the capillary features is not affected significantly by the material elasticity. Moreover, it is shown that similarly to the case of Newtonian liquids, high inertia causes the formation of a ridge just after the step up. We also explore the effect of the geometrical characteristics of the substrate as well as its inclination angle and it is shown that the interface shape becomes more deformed as the topography appears wider, deeper or it approaches the vertical plane.

Characterizing flow resistance in 3-dimensional disordered fibrous structures based on Forchheimer coefficients for a wide range of Reynolds numbers

The flow resistance in 3-dimensional fibrous structures are investigated in particle Reynolds number representing flow characteristics with strong inertia. The resistance coefficients are established based on steady state simulations of single-phase processes of water numerically. An automatized simulation process in COMSOL is developed with a MATLAB algorithm in which production runs could be carried for various 3-dimensional fibrous structures. Simulation of flow processes ranging from Reynolds numbers at creeping flow levels to high Reynolds number at approximately 1000 are calculated and a numerical data set is established in order to estimate Forchheimer coefficients which are used to correlate a dimensionless friction factor to a modified Reynolds expression for porous media.The friction factor and dimensionless permeability are calculated for fibrous structures with (i) disordered unidirectional fibers (ii) an isotropic fiber orientation in-plane perpendicular to the flow, and (iii) an isotropic fiber structure in a the 3-dimensional space. Empirical correlations of the friction factor and Reynolds number are used to compare our simulation data in order to assess the validity of our models and flow resistance estimations. The dimensionless permeability is moreover compared to other numerical simulations of flow through fibrous structures in order to assess flow resistance at low Reynolds number.It is concluded that flow resistance in the isotropic fiber arrangement in space is lower than the in-plane isotropic orientation and disordered unidirectional fiber arrangements at creeping flow conditions, however, all friction actors converges towards the same value at higher Reynolds numbers indicating that fiber orientation is independent at high inertia flow regimes. Overall, our numerical simulations agree well to classical empirical formulations for a wide range of Reynolds number. However, the comparison differs considerably depending on the porosity level.

Experimental investigation of the impact of elastic turbulence on heat transfer in a serpentine channel

The characteristics of convective heat transfer and fluid flow within a square cross-section serpentine channel are experimentally studied for two groups of polymeric viscoelastic fluids, shear-thinning and constant-viscosity Boger solutions. The elastic turbulence can be created by the non-linear interaction between elastic stresses generated within the flowing high-molecular-weight polymer solutions and the streamline curvature. In order to confirm elastic turbulence in this geometry, pressure drop across the serpentine channel was measured. The findings indicate that the measurements of non-dimensional pressure-drop increase approximately from 1.48 to 4.82 for viscoelastic solutions compared with the Newtonian fluid over a range of Weissenberg number from 4 to 211. The convective heat transfer enhances due to elastic turbulence by up to 200% for low polymer concentration (dilute) solutions and reaches up to 380% for higher polymer concentration (semi-dilute) solutions under creeping-flow conditions in comparison to that achieved by the equivalent Newtonian fluid flow at low Graetz number (up to 14.6). We propose a modified Weissenberg number which is able to approximately collapse the mean Nusselt number data for each solution group.

Lid-driven cavity flow of viscoelastic liquids

The lid-driven cavity flow is a well-known benchmark problem for the validation of new numerical methods and techniques. In experimental and numerical studies with viscoelastic fluids in such lid-driven flows, purely-elastic instabilities have been shown to appear even at very low Reynolds numbers. A finite-volume viscoelastic code, using the log-conformation formulation, is used in this work to probe the effect of viscoelasticity on the appearance of such instabilities in two-dimensional lid-driven cavities for a wide range of aspect ratios (0.125≤Λ=height/length≤4.0), at different Deborah numbers under creeping-flow conditions and to understand the effects of regularization of the lid velocity. The effect of the viscoelasticity on the steady-state results and on the critical conditions for the onset of the elastic instabilities are described and compared to experimental results.

Slow motion of a porous spherical particle with a rigid core in a spherical fluid cavity

We present an analytical study of viscous flow past a porous spherical particle composed of a rigid core inside. We consider that this particle is located inside a spherical fluid cavity filled with incompressible Newtonian fluid, under the creeping flow conditions. The governing equations inside the fluid region and the porous region are governed by Stokes equation and Brinkman equation respectively, supplemented with the corresponding mass conservation. Hydrodynamic drag and torque exerted by the fluid on particle are obtained which are later used to obtain the translational and rotational mobility of the particle inside the spherical cavity. In general the presence of cavity wall retards the particle movement as a result the mobility parameter becomes smaller than unity. The boundary effect is more pronounced when the separation distance between the particle surface and the cavity wall is less. Various limiting cases are obtained which agree with earlier existing results.

Linear stability of shear-thinning fluids in deformable channels: Effect of inertial terms

In the present work, the effect of inertial terms is numerically investigated on the hydroelastic stability of shear-thinning fluids in pressure-driven flow between two parallel plates lined with a Mooney–Rivlin hyperelastic polymeric gel. Having obtained the basic flow for the fluid and the basic deformation for the solid, the vulnerability of the basic solutions are examined when subjected to infinitesimally-small, normal-mode perturbations. By dropping all nonlinear perturbation terms, an eigenvalue problem is obtained, in the form of two coupled fourth-order ODEs, which are solved using the shooting or spectral methods. Based on the numerical results obtained in this work, it is concluded that the Mooney–Rivlin gels having a negative first-normal-stress-difference are more stable than the neo-Hookean gels as far as hydroelastic instability in plane Poiseuille flow is concerned. While under creeping-flow conditions the effect of shear-thinning is predicted to be stabilizing, in inertial flows its effect is predicted to be stabilizing or destabilizing depending on the Reynolds number.

On the velocity discontinuity at a critical volume of a bubble rising in a viscoelastic fluid

We examine the abrupt increase in the rise velocity of an isolated bubble in a viscoelastic fluid occurring at a critical value of its volume, under creeping flow conditions. This ‘velocity discontinuity’, in most experiments involving shear-thinning fluids, has been somehow associated with the change of the shape of the bubble to an inverted teardrop with a tip at its pole and/or the formation of the ‘negative wake’ structure behind it. The interconnection of these phenomena is not fully understood yet, making the mechanism of the ‘velocity jump’ unclear. By means of steady-state analysis, we study the impact of the increase of bubble volume on its steady rise velocity and, with the aid of pseudo arclength continuation, we are able to predict the stationary solutions, even lying in the discontinuous area in the diagrams of velocity versus bubble volume. The critical area of missing experimental results is attributed to a hysteresis loop. The use of a boundary-fitted finite element mesh and the open-boundary condition are essential for, respectively, the correct prediction of the sharply deformed bubble shapes caused by the large extensional stresses at the rear pole of the bubble and the accurate application of boundary conditions far from the bubble. The change of shape of the rear pole into a tip favours the formation of an intense shear layer, which facilitates the bubble translation. At a critical volume, the shear strain developed at the front region of the bubble sharply decreases the shear viscosity. This change results in a decrease of the resistance to fluid displacement, allowing the developed shear stresses to act more effectively on bubble motion. These coupled effects are the reason for the abrupt increase of the rise velocity. The flow field for stationary solutions after the velocity jump changes drastically and intense recirculation downstream of the bubble is developed. Our predictions are in quantitative agreement with published experimental results by Pilz & Brenn (J. Non-Newtonian Fluid Mech., vol. 145, 2007, pp. 124–138) on the velocity jump in fluids with well-characterized rheology. Additionally, we predict shapes of larger bubbles when both inertia and elasticity are present and obtain qualitative agreement with experiments by Astarita & Apuzzo (AIChE J., vol. 11, 1965, pp. 815–820).

Yielding the yield-stress analysis: a study focused on the effects of elasticity on the settling of a single spherical particle in simple yield-stress fluids.

The sedimentation of a single particle in materials that exhibit simultaneously elastic, viscous and plastic behavior is examined in an effort to explain phenomena that contradict the nature of purely yield-stress materials. Such phenomena include the loss of the fore-and-aft symmetry with respect to an isolated settling particle under creeping flow conditions and the appearance of the "negative wake" behind it. Despite the fact that similar observations have been reported in studies involving viscoelastic fluids, researchers conjectured that thixotropy is responsible for these phenomena, as the aging of yield-stress materials is another common feature. By means of transient calculations, we study the effect of elasticity on both the fluidized and the solid phase. The latter is considered to behave as an ideal Hookean solid. The material properties of the model are determined under the isotropic kinematic hardening framework via Large Amplitude Oscillatory Shear (LAOS) measurements. In this way, we are able to predict accurately the unusual phenomena observed in experiments with simple yield-stress materials, irrespective of the appearance of slip on the particle surface. Viscoelasticity favors the formation of intense shear and extensional stresses downstream of the particle, significantly changing the entrapment mechanism in comparison to that observed in viscoplastic fluids. Therefore, the critical conditions under which the entrapment of the particle occurs deviate from the well-known criterion established theoretically by Beris et al. (1985) and verified experimentally by Tabuteau et al. (2007) for similar materials under conditions that elastic effects are negligible. Our predictions are in quantitative agreement with published experimental results by Holenberg et al. (2012) on the loss of the fore-aft symmetry and the formation of the negative wake in Carbopol with well-characterized rheology. Additionally, we propose simple expressions for the Stokes drag coefficient, as a function of the gravity number, Yg (related to the Bingham number), for different levels of elasticity and for its critical value, under which entrapment of particles occurs. These criteria are in agreement with the results found in the recent work by Ahonguio et al. (2014). Finally, we propose a method to quantify experimentally the elastic effects in viscoplastic particulate systems.

Stokes resistance of a porous spherical particle in a spherical cavity

The boundary effect on the asymmetrical motion of a porous spherical particle in an eccentric spherical cavity is investigated in the quasi-steady limit under creeping flow conditions. The porous particle translates and rotates in the viscous fluid, located within the spherical cavity, normal to the line connecting their centers. The fluid inside the porous particle is governed by the Brinkman equation. A tangential stress jump condition at the interface between the fluid and the porous particle is applied. A semi-analytical approach based on a collocation technique is used. Due to the linearity of the present problem, the flow variables for the clear fluid region are constructed by superposing basic solutions of two problems: the first one is the regular solution inside the cavity region in the absence of the porous particle where a first system of coordinates has its origin at the center of the cavity, while the second problem is the regular solution in the infinite region outside the spherical porous particle where a second coordinate system with its origin at the center of the porous particle is used. Numerical results displaying the resistance coefficients acting on the particle are obtained with good convergence for various values of the physical parameters of the problem. The results are tabulated and represented graphically. The findings demonstrate that the collocation results of the resistance coefficients are in good agreement with the corresponding results for the impermeable solid particle.

Mass transfer around slender drops in an extensional flow: Inertial effects at large peclet numbers

AbstractThe exchange of mass between a slender drop and a viscous liquid in an axisymmetric extensional flow and at large Peclet numbers is examined. Four cases are discussed: an inviscid drop and a viscous drop, under both creeping flow conditions and when the external flow has a weak amount of inertia. Solutions are presented as a function of the capillary number (Ca ≫ 1), the viscosity ratio (κ ≪ 1), the external Reynolds number (Re ≪ 1), and the Peclet number (Pe ≫ 1). The results suggest that the steady mass transfer rate, to or from the slender drop, is proportional to the elongation rate to the (relative) high power of 3/2. In general, as inertia increases, the drop becomes thinner and longer, and the average tangential surface velocity and the surface area increase, resulting in larger mass transfer rates, especially near the breakup point.

On the origin of extrusion instabilities: Linear stability analysis of the viscoelastic die swell

It is well-known that by increasing the flow rate in polymer extrusion, the flow becomes unstable and the smooth extrudate surface becomes wavy and disordered to an increasing degree. In order to investigate the mechanisms responsible for these instabilities we perform a linear stability analysis of the steady extrusion of a viscoelastic fluid flowing through a planar die under creeping flow conditions. We consider the Phan–Thien–Tanner (PTT) model to account for the viscoelasticity of the material. We employ the mixed finite element method combined with an elliptic grid generator to account for the deformable shape of the interface. The generalized eigenvalue problem is solved using Arnoldi's algorithm. We perform a thorough parametric study in order to determine the effects of all material properties and rheological parameters. We investigate in detail the effect of the interfacial tension and the presence of a deformable interface. It is found that the presence of a finite surface tension destabilizes the flow as compared to the case of the stick–slip flow. We recognize two modes, which become unstable beyond a critical value of the Weissenberg number and perform an energy analysis to examine the mechanisms responsible for the destabilization of the flow and compare against the mechanisms that have been suggested in the literature.

Interfacial instability in two-layer Couette–Poiseuille flow of viscoelastic fluids

The temporal linear stability analysis of shear flow of two-layer viscoelastic fluids, both described using the upper convected Maxwell (UCM) model, is carried out. The plane Couette flow under creeping flow conditions both in the absence and presence of pressure gradient is examined to study the role of unmatched elasticities in the interfacial instability. As the most critical disturbance is known to be of finite wavelength of the order of channel width, the numerical analysis examines the entire range of disturbance wavenumbers and the stability map is constructed in the parametric space of fluid Weissenberg numbers to indicate the region of stable interface. With a focus on optical fiber coating process, the analysis investigates the role of pressure gradient imposed on the plate-driven flow. Both the adverse and favorable pressure gradients are analyzed to identify the region for stable coating flow. In the presence of pressure gradient, the region of stable interface broadens when the high elasticity fluid occupies the region of low shear rate, which is the bottom (top) layer for the adverse (favorable) pressure gradient. For the two fluids of unmatched rheology, in addition to elastic instability, the role of viscosity stratification leading to a jump in the shear rate at interface is also examined. The adverse pressure gradient has a stabilizing effect when more viscous fluid is more elastic, whereas the favorable pressure gradient tends to stabilize the interface when the less viscous fluid has higher elasticity than the more viscous fluid.

A DLM/FD/IB method for simulating compound vesicle motion under creeping flow condition

In this article we present first a new distributed Lagrange multiplier/fictitious domain (DLM/FD) method for simulating fluid–particle interaction in Stokes flow. A conjugate gradient method driven by both pressure and distributed Lagrange multiplier, called one-shot method, has been developed to solve the discrete Stokes problem while enforcing the rigid body motion within the region occupied by the particle. The methodology is validated by comparing the numerical results of a neutrally buoyant particle of either a circular or elliptic shape with the associated Jeffery's solutions. We have successively combined the above methodology with an immersed boundary (IB) method and an elastic membrane modeled by a spring network to simulate the dynamics of a compound vesicle. In simple shear flow under creeping flow condition, the results are consistent with those obtained in literature. In Poiseuille flow, the compound vesicle motion is dominated by the motion of the vesicle membrane as expected and stays in the central region of the channel.

Topological study of steady state, three dimensional flow over a backward facing step

The topology and evolution of flow over a backward facing step in three dimensional channel flow is examined from creeping flow conditions up to Re=950 with aspect ratio 1:40 and expansion ratio 1:2. Direct numerical simulations were performed via the parallel finite element code “FEM-3D”. Analysis of the results is performed using the three dimensional theory of separation. Results reveal the existence of a primary spanwise vortex that emanates from the lateral wall and is present for all Reynolds numbers immediately downstream from the step. Frictional (shear) stresses along the lateral wall create multiple tornado like vortices. Another spanwise vortex along the top wall appears at Re=400. As the Reynolds number increases, the top vortex increases both in spanwise and streamwise length and stunts the growth of the primary vortex. The downstream motion of the top vortex releases frictional stress thus destroying some critical points upstream while creating new ones downstream. Finally, at Re=900 and 950, the primary and top vortices are twisting severely.

Multi-scale permeability of deformable fibrous porous media

The contribution of fiber dynamics and clustering to the effective permeability in hierarchical fibrous media is poorly understood, due to the complex fluid–structure interactions taking place across fiber, yarn and textile scales. In this work, a two-dimensional model for fiber deformation subject to out-of-plane movement restrictions is derived for creeping flow conditions by analogy with non-Brownian suspensions of particles with confining potentials. This leads to a homogeneous Fokker–Planck equation in a phase space of fiber configurations, for the probability density function of the fiber displacements. A fiber clustering criterion is then defined using autoconvolution functions of the local probability densities, which yields the local change in fiber-scale permeability according to a topological description of the porous media instead of the typical geometric description. The resulting multi-scale hydrodynamic system is numerically solved by a coupled method, where the Stokes flow at yarn-scale is solved with a finite volume method and the mesoscopic model that recovers information from the fiber-scale is solved by a lattice Boltzmann method. The fiber-scale permeability is characterized in terms of porosity, dimensionless shear rate and dimensionless out-of-plane forces. When assessed in terms of a reduced viscosity related to Brinkman׳s closure for porous media, the mesoscopic model shows that deformable fibrous porous media qualitatively behave like dense particle suspensions. For low volume fractions a non-Newtonian reduced viscosity exhibiting shear-thinning and low- and high-shear plateaux is obtained. For high volume fractions and high shear rates the out-of-plane forces lead to shear thickening. The results on steady fiber-scale permeability are presented in the form of phase diagrams which show that in the typical range of parameters for textiles, the effective permeability of the deformable case can be up 60% lower than that of the rigid case due to the formation of fiber clusters.

Volume-averaged macroscopic equation for fluid flow in moving porous media

Darcy’s law and the Brinkman equation are two main models used for creeping fluid flows inside moving permeable particles. For these two models, the time derivative and the nonlinear convective terms of fluid velocity are neglected in the momentum equation. In this paper, a new momentum equation including these two terms are rigorously derived from the pore-scale microscopic equations by the volume-averaging method. It is shown that Darcy’s law and the Brinkman equation can be reduced from the derived equation under creeping flow conditions. Using the lattice Boltzmann equation (LBE) method, the macroscopic equations are solved for the problem of a porous circular cylinder moving along the centerline of a channel. Galilean invariance of the equations are investigated both with the intrinsic phase averaged velocity and the phase averaged velocity. The results demonstrate that the commonly used phase averaged velocity cannot be considered, while the intrinsic phase averaged velocity should be chosen for porous particulate systems. In addition, the Poiseuille flow in a porous channel is simulated using the LBE method with the improved equations, and good agreements are obtained when compared with the finite-difference solutions.

Collision rates of small ellipsoids settling in turbulence

AbstractWe propose that the collision rates of non-spherical particles settling in a turbulent environment are significantly higher than those of spherical particles of the same mass and volume. The theoretical argument is based on the dependence of the particle drag force on the particle orientation, thus varying gravitational settling velocities, which can remain different until contact due to the particle inertia. Therefore, non-spherical particles can collide with large relative velocities. Direct numerical simulations (DNS) of streamwise decaying isotropic turbulence seeded with small, heavy, rotationally symmetric ellipsoids of five different aspect ratios are performed to confirm these arguments. The motion of 21 million ellipsoids is tracked by a Lagrangian particle solver assuming creeping flow conditions and neglecting the influence of the particles on the flow. We find that ellipsoids collide considerably more often than spherical particles of the same volume and mass due to a drastically increased mean relative velocity at contact.