Elastic instabilities between two cylinders confined in a channel

Understanding the behavior of viscoelastic (VE) fluids in confined geometries is crucial for applications in biologic systems, heat transfer devices, enhanced oil recovery, and many others. Here, we perform a systematic steady-state simulation of a VE fluid at low Reynolds numbers through a channel with successive smooth contractions and expansions. We analyze the hydrodynamic performance of the fluid with particular attention to vortex patterns that develop downstream of the contractions. We show that elastic vortices form at higher contraction ratios and that there are critical Weissenberg numbers (Wic) unique to each contraction ratio where the flow shifts from non-vortical to vortical. This Wic increases with an increasing contraction length. The coexistence of elongational-, shear-, and rotational-flow is essential for vortex development and evolution. We also analyzed the effect of the Deborah number (De) on the vortex pattern in a multiple contraction system and observed that the vortex area significantly depends on the distance between the contractions. We show that there are three distinctly different regions in De, in which the flow characteristics change in successive contractions. For high De, the flow in the downstream contraction is significantly affected by the upstream contraction. Our results have implications for the use of VE fluids with various VE properties in different types of porous media.

The nonlinear stability and bifurcation of the one-dimensional channel (Poiseuille) flow is examined for a Johnson-Segalman fluid. The velocity and stress are represented by orthonormal functions in the transverse direction to the flow. The flow field is obtained from the conservation and constitutive equations using the Galerkin projection method. Both inertia and normal stress effects are included. The stability picture is dramatically influenced by the viscosity ratio. The range of shear rate or Weissenberg number for which the base flow is unstable increases from zero as the fluid deviates from the Newtonian limit as decreases. Typically, two turning points are observed near the critical Weissenberg numbers. The transient response is heavily influenced by the level of inertia. It is found that the flow responds oscillatorily. When the Reynolds number is small, and monotonically at large Reynolds number when elastic effects are dominated by inertia.

The LS-STAG immersed boundary cut-cell method modification for viscoelastic flow computations is presented. Rate type viscoelastic flow models (linear and quasilinear) are considered. Rate type viscoelastic flow models (linear and quasilinear) are considered. The obtained numerical method is implemented in the LS-STAG software package developed by the author. This software allows to simulate viscous incompressible flows by using the LS-STAG method and it modifications. The LS-STAG-discretization of extra-stress equations for viscoelastic Maxwell, Jeffreys, upper-convected Maxwell, Maxwell-A, Oldroyd-B, Oldroyd-A, Johnson Segalman fluids was developed. Formulae for differential types of convected time derivatives (Oldroyd, Cotter — Rivlin, Jaumann — Zaremba — Noll derivatives) the LS-STAG discretization was obtained. Normal non-newtonian stresses are computed at the centers of base LS-STAG mesh cells and shear non-newtonian stresses are computed at the cell corners. Time-stepping algorithm is based on the first order predictor-corrector scheme. To validate developed numerical method the test problem about viscoelastic Oldroyd-B flow past a circular airfoil was used. Computational experiments were carried out at Weissenberg number in the range from 0 to 4. The computed values of the drag coefficients and the wake length are in good agreement with the experimental data.

A linear stability analysis of plane Poiseuille flow of an upper-convected Maxwell (UCM) fluid, bounded between rigid plates separated by a distance $2L$, has been carried out to investigate the interplay of elasticity and inertia on flow stability. The stability is governed by the following dimensionless groups: the Reynolds number $Re=\unicode[STIX]{x1D70C}U_{max}L/\unicode[STIX]{x1D702}$ and the elasticity number $E\equiv W/Re=\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}/(\unicode[STIX]{x1D70C}L^{2})$, where $W=\unicode[STIX]{x1D706}U_{max}/L$ is the Weissenberg number. Here, $\unicode[STIX]{x1D70C}$ is the fluid density, $\unicode[STIX]{x1D702}$ is the fluid viscosity, $\unicode[STIX]{x1D706}$ is the micro-structural relaxation time and $U_{max}$ is the maximum base-flow velocity. The stability is analysed for two-dimensional perturbations using both pseudo-spectral and shooting methods. We also analyse the linear stability of plane Couette flow which, along with the results for plane Poiseuille flow, yields insight into the structure of the complete elasto-inertial eigenspectrum. While the general features of the spectrum for both flows remain similar, plane Couette flow is found to be stable over the range of parameters examined ($Re\leqslant 10^{4},E\leqslant 0.01$). On the other hand, plane Poiseuille flow appears to be susceptible to an infinite hierarchy of elasto-inertial instabilities. Over the range of parameters examined, there are up to seven distinct neutral stability curves in the $Re$–$k$ plane (here $k$ is the perturbation wavenumber in the flow direction). Based on the symmetry of the eigenfunctions for the streamwise velocity about the centreline, four of these instabilities are antisymmetric, while the other three are symmetric. The neutral stability curve corresponding to the first antisymmetric mode is shown to be a continuation (to finite $E$) of the Tollmien–Schlichting (TS) instability already present for Newtonian channel flow. As $E$ is increased beyond $0.0016$, a new elastic mode appears at $Re\sim 10^{4}$, which coalesces with the continuation of the TS mode for a range of $Re$, thereby yielding a single unstable mode in this range. This trend persists until $E\sim 0.0021$, beyond which this neutral curve splits into two separate ones in the $Re$–$k$ plane. The new elastic mode which arises out of this splitting has been found to be the most unstable, with the lowest critical Reynolds number $Re_{c}\approx 1210.9$ for $E=0.0066$. The neutral curves for both the continuation of the original TS mode, and the new elastic antisymmetric mode, form closed loops upon further increase in $E$, which eventually vanish at sufficiently high $E$. For $E\ll 1$, the critical Reynolds number and wavenumber scale as $Re_{c}\sim E^{-1}$ and $k_{c}\sim E^{-1/2}$ for the first two of the symmetric modal families, and as $Re_{c}\sim E^{-5/4}$ for first two of the antisymmetric modal families; $k_{c}\sim E^{-1/4}$ for the third antisymmetric family. The critical wave speed for all of these unstable eigenmodes scales as $c_{r,c}\sim E^{1/2}$ for $E\ll 1$, implying that the modes belong to a class of ‘wall modes’ in viscoelastic flows with disturbances being confined in a thin region near the wall. The present study shows that, surprisingly, even in plane shear flows, elasticity acting along with inertia can drive novel instabilities absent in the Newtonian limit.

A recently developed microfluidic device, the optimized shape cross-slot extensional rheometer or OSCER [S. J. Haward, M. S. N. Oliveira, M. A. Alves, and G. H. McKinley, “Optimized cross-slot flow geometry for microfluidic extensional rheometry,” Phys. Rev. Lett. 109, 128301 (2012)10.1103/PhysRevLett.109.128301], is used to investigate the stability of viscoelastic polymer solutions in an idealized planar stagnation point flow. Aqueous polymer solutions, consisting of poly(ethylene oxide) and of hyaluronic acid with various molecular weights and concentrations, are formulated in order to provide fluids with a wide range of rheological properties. Semi-dilute solutions of high molecular weight polymers provide highly viscoelastic fluids with long relaxation times, which achieve a high Weissenberg number (Wi) at flow rates for which the Reynolds number (Re) remains low; hence the elasticity number El = Wi/Re is high. Lower concentration solutions of moderate molecular weight polymers provide only weakly viscoelastic fluids in which inertia remains important and El is relatively low. Flow birefringence observations are used to visualize the nature of flow instabilities in the fluids as the volumetric flow rate through the OSCER device is steadily incremented. At low Wi and Re, all of the fluids display a steady, symmetric, and uniform “birefringent strand” of highly oriented polymer molecules aligned along the outflowing symmetry axis of the test geometry, indicating the stability of the flow field under such conditions. In fluids of El > 1, we observe steady elastic flow asymmetries beyond a critical Weissenberg number,Wicrit, that are similar in character to those already reported in standard cross-slot geometries [e.g., P. E. Arratia, C. C. Thomas, J. Diorio, and J. P. Gollub, “Elastic instabilities of polymer solutions in cross-channel flow,” Phys. Rev. Lett. 96, 144502 (2006)10.1103/PhysRevLett.96.144502]. However, in fluids with El < 1 we observe a sequence of time-dependent inertio-elastic instabilities beyond a critical Reynolds number, Re crit, characterized by high frequency spatiotemporal oscillations of the birefringent strand. By plotting the critical limits of stability for the various fluids in the Wi-Re operating space, we are able to construct a stability diagram delineating the distinct steady symmetric, steady asymmetric and inertio-elastic flow regimes in this idealized planar elongational flow device.

A recently developed microfluidic device, the optimized shape cross-slot extensional rheometer or OSCER [S. J. Haward, M. S. N. Oliveira, M. A. Alves, and G. H. McKinley, “Optimized cross-slot flow geometry for microfluidic extensional rheometry,” Phys. Rev. Lett. 109, 128301 (2012)10.1103/PhysRevLett.109.128301], is used to investigate the stability of viscoelastic polymer solutions in an idealized planar stagnation point flow. Aqueous polymer solutions, consisting of poly(ethylene oxide) and of hyaluronic acid with various molecular weights and concentrations, are formulated in order to provide fluids with a wide range of rheological properties. Semi-dilute solutions of high molecular weight polymers provide highly viscoelastic fluids with long relaxation times, which achieve a high Weissenberg number (Wi) at flow rates for which the Reynolds number (Re) remains low; hence the elasticity number El = Wi/Re is high. Lower concentration solutions of moderate molecular weight polymers provide only weakly viscoelastic fluids in which inertia remains important and El is relatively low. Flow birefringence observations are used to visualize the nature of flow instabilities in the fluids as the volumetric flow rate through the OSCER device is steadily incremented. At low Wi and Re, all of the fluids display a steady, symmetric, and uniform “birefringent strand” of highly oriented polymer molecules aligned along the outflowing symmetry axis of the test geometry, indicating the stability of the flow field under such conditions. In fluids of El &amp;gt; 1, we observe steady elastic flow asymmetries beyond a critical Weissenberg number,Wicrit, that are similar in character to those already reported in standard cross-slot geometries [e.g., P. E. Arratia, C. C. Thomas, J. Diorio, and J. P. Gollub, “Elastic instabilities of polymer solutions in cross-channel flow,” Phys. Rev. Lett. 96, 144502 (2006)10.1103/PhysRevLett.96.144502]. However, in fluids with El &amp;lt; 1 we observe a sequence of time-dependent inertio-elastic instabilities beyond a critical Reynolds number, ${\mathop{\rm Re}\nolimits} _{crit} $ Re crit, characterized by high frequency spatiotemporal oscillations of the birefringent strand. By plotting the critical limits of stability for the various fluids in the Wi-Re operating space, we are able to construct a stability diagram delineating the distinct steady symmetric, steady asymmetric and inertio-elastic flow regimes in this idealized planar elongational flow device.

A recently developed microfluidic device, the optimized shape cross-slot extensional rheometer or OSCER [S. J. Haward, M. S. N. Oliveira, M. A. Alves, and G. H. McKinley, “Optimized cross-slot flow geometry for microfluidic extensional rheometry,” Phys. Rev. Lett. 109, 128301 (2012)10.1103/PhysRevLett.109.128301], is used to investigate the stability of viscoelastic polymer solutions in an idealized planar stagnation point flow. Aqueous polymer solutions, consisting of poly(ethylene oxide) and of hyaluronic acid with various molecular weights and concentrations, are formulated in order to provide fluids with a wide range of rheological properties. Semi-dilute solutions of high molecular weight polymers provide highly viscoelastic fluids with long relaxation times, which achieve a high Weissenberg number (Wi) at flow rates for which the Reynolds number (Re) remains low; hence the elasticity number El = Wi/Re is high. Lower concentration solutions of moderate molecular weight polymers provide only weakly viscoelastic fluids in which inertia remains important and El is relatively low. Flow birefringence observations are used to visualize the nature of flow instabilities in the fluids as the volumetric flow rate through the OSCER device is steadily incremented. At low Wi and Re, all of the fluids display a steady, symmetric, and uniform “birefringent strand” of highly oriented polymer molecules aligned along the outflowing symmetry axis of the test geometry, indicating the stability of the flow field under such conditions. In fluids of El > 1, we observe steady elastic flow asymmetries beyond a critical Weissenberg number,Wicrit, that are similar in character to those already reported in standard cross-slot geometries [e.g., P. E. Arratia, C. C. Thomas, J. Diorio, and J. P. Gollub, “Elastic instabilities of polymer solutions in cross-channel flow,” Phys. Rev. Lett. 96, 144502 (2006)10.1103/PhysRevLett.96.144502]. However, in fluids with El < 1 we observe a sequence of time-dependent inertio-elastic instabilities beyond a critical Reynolds number, Re crit, characterized by high frequency spatiotemporal oscillations of the birefringent strand. By plotting the critical limits of stability for the various fluids in the Wi-Re operating space, we are able to construct a stability diagram delineating the distinct steady symmetric, steady asymmetric and inertio-elastic flow regimes in this idealized planar elongational flow device.

SPH simulations of a viscoelastic flow around a periodic array of cylinders confined in a channel

A modified Darcy’s law for viscoelastic flows of highly dilute polymer solutions through porous media

In this study, numerical simulation is conducted to understand the two-dimensional viscoelastic flows past two side-by-side circular cylinders at a Reynolds number of 100. The Peterlin approximation of the finitely extensible nonlinear elastic model is adopted to describe the non-linear modulus of elasticity and the finite extendibility of polymer macromolecules. The flow behavior and time-averaged forces that act on the two cylinders are investigated over a wide range of parameter space, i.e., the Weissenberg number (We), from 0 to 8, and the spacing between the two cylinders (LD), from 0.1D to 3.0D (D denotes the diameter of each cylinder). Similar to the corresponding Newtonian flow, the viscoelastic flow gradually undergoes six transitions as LD increases. However, these transitions are delayed in the viscoelastic flow, particularly at a high We. As a result, three distinct flow modes remain within the above-mentioned LD range at a high We. With increasing We, the total drag acting on the two cylinders increases for all LD values, and the repulsive force between the two cylinders gradually decreases for a lower LD value but increases for a higher LD value. Both the intensity and frequency of force fluctuation decrease as We increases. The findings of the present study may provide new insight into the multi-body wake dynamics in the viscoelastic flow.

Improving the understanding of mechanisms involved in a low miscible displacement efficiency is significant for a wide spectrum of applications in subsurface from environment such as groundwater remediation and sequestration to energy extraction such as enhanced oil recovery and geothermal recovery. Two key limiting factors to the efficiency are viscous fingering (VF) instability and dead‐end pores in porous media. Previous research on VF simply assumes all pores are well connected and fluids can be mobilized by convection. However, fluids trapped in dead‐end pores, such as nonaqueous phase liquids (NAPLs) in groundwater remediation, are inaccessible to convection, resulting in even less efficient displacements. Instead of the classic convection‐diffusion/dispersion equation, in this work, we use a fundamentally different capacitance model to incorporate the mass transfer between two pore types in miscible displacements. The hybrid pseudospectral and high‐order finite difference methods are employed to solve the governing equations in a fixed reference frame for simulating the flow dynamics. We found that the viscous fingering instabilities in well‐connected pore network led to an unstable, nonuniform distribution of trapped NAPLs in dead‐end pores network. Different with a traditional view that NAPLs are easily cleaned up, our study indicates the existence of dead‐end pores results in an inefficient cleanup of NAPLs in swept area because of the slow mass transfer of trapped NAPLs between two pore types. A simple model is developed to accurately predict the NAPL concentration behind finger trailing front, which has not been previously examined. Six flow regimes, four of which are new, are then identified.

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.

Inertia-less viscoelastic channel flow displays a supercritical nonnormal mode elastic instability due to finite-size perturbations despite its linear stability. The nonnormal mode instability is determined mainly by a direct transition from laminar to chaotic flow, in contrast to normal mode bifurcation leading to a single fastest-growing mode. At higher velocities, transitions to elastic turbulence and further drag reduction flow regimes occur accompanied by elastic waves in three flow regimes. Here, we demonstrate experimentally that the elastic waves play a key role in amplifying wall-normal vorticity fluctuations by pumping energy, withdrawn from the mean flow, into wall-normal fluctuating vortices. Indeed, the flow resistance and rotational part of the wall-normal vorticity fluctuations depend linearly on the elastic wave energy in three chaotic flow regimes. The higher (lower) the elastic wave intensity, the larger (smaller) the flow resistance and rotational vorticity fluctuations. This mechanism was suggested earlier to explain elastically driven Kelvin-Helmholtz-like instability in viscoelastic channel flow. The suggested physical mechanism of vorticity amplification by the elastic waves above the elastic instability onset recalls the Landau damping in magnetized relativistic plasma. The latter occurs due to the resonant interaction of electromagnetic waves with fast electrons in the relativistic plasma when the electron velocity approaches light speed. Moreover, the suggested mechanism could be generally relevant to flows exhibiting both transverse waves and vortices, such as Alfven waves interacting with vortices in turbulent magnetized plasma, and Tollmien-Schlichting waves amplifying vorticity in both Newtonian and elasto-inertial fluids in shear flows.

In the field of computational fluid dynamics, finding the analytical solutions of partial differential equations governing complex problems presents a challenge to the researchers. Our basic aim is to find the symmetries of Lei- group theory of viscoelastic fluid flows to obtain the results to the PDE’s system including the continuity equation, momentum equations and constitutive equations related with viscoelastic stresses, under suitable initial and boundary conditions.This paper presents the passing hydrodynamics actions of the viscoelastic fluid flow in pipes through porous media related with Darcy-Brinkman model. The research is lectured by means of analytical results and also numerical results of the problems of given differential equations associated by means of initial and boundary conditions, happening in the knowledge of viscoelastic flow in pipes through porous media. The solutions of problem of this paper are obtained adopting the Lie group of transformation theoretic approach and analytical solution is obtained under the symmetries for the equations through Lie group technique.

Detailed measurements of the flow instability of dilute shear-thinning viscoelastic aqueous solutions, with relatively low zero-shear viscosities, in an obstructed microchannel flow are reported. We examine the flow behaviour resulting from a 100 μm post placed in the channel centreline over a range of Reynolds numbers ( $$5<Re<300$$ ) and Weissenberg numbers ( $$20<Wi<10^3$$ ). Micro-particle image velocimetry measurements show the onset of an upstream instability within a Reynolds number range and at a critical elasticity number corresponding to polymer concentrations above 25 ppm of long-chain polyacrylamide. The instability results in significant local fluctuations in the flow field approaching 30 % of the mean velocity. The magnitude of the local viscosity ratio in the region upstream of the post is proposed as a driving mechanism for the instability which resembles a buckling flow. Additionally, the classical instability owing to separation and vortex formation downstream of the post in Newtonian flow is suppressed and a very long stable wake is observed extending over 10 post-diameters downstream.