Stationary Shear Flow Research Articles

Flow states of two dimensional active gels driven by external shear.

Using a minimal hydrodynamic model, we theoretically and computationally study the Couette flow of active gels in straight and annular two-dimensional channels subject to an externally imposed shear. The gels are isotropic in the absence of externally- or activity-driven shear, but have nematic order that increases with shear rate. Using the finite element method, we determine the possible flow states for a range of activities and shear rates. Linear stability analysis of an unconfined gel in a straight channel shows that an externally imposed shear flow can stabilize an extensile fluid that would be unstable to spontaneous flow in the absence of the shear flow, and destabilize a contractile fluid that would be stable against spontaneous flow in the absence of shear flow. These results are in rough agreement with the stability boundaries between the base shear flow state and the nonlinear flow states that we find numerically for a confined active gel. For extensile fluids, we find three kinds of nonlinear flow states in the range of parameters we study: unidirectional flows, oscillatory flows, and dancing flows. To highlight the activity-driven spontaneous component of the nonlinear flows, we characterize these states by the average volumetric flow rate and the wall stress. For contractile fluids, we only find the linear shear flow and a nonlinear unidirectional flow in the range of parameters that we studied. For large magnitudes of the activity, the unidirectional contractile flow develops a boundary layer. Our analysis of annular channels shows how curvature of the streamlines in the base flow affects the transitions among flow states.

Structural Causes of the Non-Newtonian Behavior of Fluid Systems

This review considers the well-known results of rheological studies, first of all, the viscosity and flow curves of various dispersed and polymer systems. The existing explanations of the causes of non-Newtonian flow are presented. Various rheological models are described, in which a change in viscosity is associated with a change in the structure of a substance. The new structural rheological model proposed by us is a generalization of the Casson and Cross rheological models. Its scope includes various structured systems: suspensions, emulsions, polymer solutions and melts, micellar solutions, and liquid crystals. Our attention was focused on the features of the stationary shear flow of non-Newtonian systems.

High-Degree Collisional Moments of Inelastic Maxwell Mixtures—Application to the Homogeneous Cooling and Uniform Shear Flow States

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to determine the collisional moments of the second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is demonstrated that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit.

Approach towards local isotropy in statistically stationary turbulent shear flows

We analyse the approach towards local isotropy in statistically stationary turbulent shear flows using the transport equations for the fourth-order moments of the velocity derivative. It is found that terms of these equations representing the large-scale contribution associated with the uniform mean velocity gradient gradually decrease as the Taylor microscale Reynolds number $Re_\lambda$ increases, and finally disappear when $Re_\lambda$ is sufficiently large. This gradual weakening of the large-scale effect is accompanied by a gradual approach towards local isotropy of the small-scale motion. The rate at which local isotropy is approached depends on the weakening of the large-scale forcing, which is controlled by the magnitude of the non-dimensional velocity shear parameter $S^*$ ( $\equiv \overline {u_1^2}({{\partial {{\bar U}_1}}}/{{\partial {x_2}}})/{\bar {\varepsilon }_{iso}}$ , where $\bar {\varepsilon }_{iso}$ is the isotropic mean turbulent energy dissipation rate, $\overline {u_1^2}$ is the streamwise velocity variance, and ${\partial {{\bar U}_1}/\partial {x_2}}$ is the uniform mean velocity gradient in the transverse direction). In particular, we show that the approach towards local isotropy can be recast in the form $C\, Re_\lambda ^{-1}$ , where $C$ is the product of $S^*$ and a ratio of transverse-to-streamwise velocity derivative variances. This is consistent with the behaviour of the normalized third-order moments of transverse velocity derivatives. With the further use of the transport equations for the eighth- and twelfth-order velocity derivative moments, it is found that the even moments of transverse velocity derivatives can significantly affect the rate at which local isotropy is approached, especially for higher orders.

Molecular Dynamics Investigation of Hyaluronan in Biolubrication.

Aqueous solution of strongly hydrophilic biopolymers is known to exhibit excellent lubrication properties in biological systems, such as the synovial fluid in human joints. Several mechanisms have been proposed on the biolubrication of joints, such as the boundary lubrication and the fluid exudation lubrication. In these models, mechanical properties of synovial fluid containing biopolymers are essential. To examine the role of such biopolymers in lubrication, a series of molecular dynamics simulations with an all-atom classical force field model were conducted for aqueous solutions of hyaluronan (hyaluronic acid, HA) under constant shear. After equilibrating the system, the Lees-Edwards boundary condition was imposed, with which a steady state of uniform shear flow was realized. Comparison of HA systems with hydrocarbon (pentadecane, PD) solutions of similar mass concentration indicates that the viscosity of HA solutions is slightly larger in general than that of PDs, due to the strong hydration of HA molecules. Effects of added electrolyte (NaCl) were also discussed in terms of hydration. These findings suggest the role of HA in biolubirication as a load-supporting component, with its flexible character and strong hydration structure.

Large amplitude oscillatory shear (LAOS) behavior of chocolates of different compositions

The complex rheological response of chocolates of different compositions was analyzed in depth from stationary shear flow curves and large amplitude oscillatory shear tests. The samples have a wide range of cocoa content that controls the rheological behavior characterized by a yielding response. The strain amplitude at which the chocolates changed from elastic-dominated to viscous-dominated behavior ranged from 0.2 to 0.4%. In this region, the stress response to the applied sinusoidally oscillating shear strain showed transient overshoots whose values were similar to yield stress calculated using the Casson equation in continuous flow. The nonlinear response at the steady-state was analyzed. The Q0 nonlinearity parameter, obtained from Fourier Transform analysis, increases as the concentration of the solids conferring bimodality to the particle size distribution increases. In the study of intracycle nonlinearities, the analysis of the third-order Chebyshev elastic and viscous coefficients showed that the sample which contains the highest amount of solids different from cocoa exhibited the lowest strain-stiffening and the highest shear-thinning behaviors. Stress bifurcation analysis identified the beginning of the yielding transition and evidenced a nonlinear solid-like behavior before the solid-liquid transition. Analysis of Lissajous–Bowditch curves allowed the static and dynamic contributions of the yielding transition to be identified, and the full Sequence of Physical Process quantitative analysis captures the time-dependent nonlinear response of chocolates over the history of applied deformations.

Analytical Solution of the Ray Equations of Hamilton for Rossby Waves on Stationary Shear Flows

The asymptotic behavior of Rossby waves in the ocean interacting with a shear stationary flow is considered. It is shown that there is a qualitative difference between the problems for the zonal and non-zonal background flow. Whereas only one critical layer arises for a zonal flow, then several critical layers can exist for a non-zonal flow. It is established that the integrated ray equations of Hamilton are equivalent to the asymptotic behavior of the Cauchy problem solution. Explicit analytical solutions are obtained for the tracks of Rossby waves as a function of time and initial parameters of the wave disturbance, as well as the magnitude of the shear and angle of inclination of the flow to the zonal direction. The ray equations of Hamilton are analytically integrated for Rossby waves on a shear flow. The obtained explicit expressions make it possible to calculate in real-time the Rossby wave tracks for any initial wave direction and any shear current inclination angle. It is shown qualitatively that these tracks for a non-zonal flow are strongly anisotropic.

Hydrodynamic dispersion in Hele-Shaw flows with inhomogeneous wall boundary conditions

Single-phase flow inside a Hele-Shaw cell can exhibit inhomogeneous flow fields, for example when actuated by electroosmosis with varying wall mobilities, leading to internal pressure gradients. We derive a two-dimensional dispersion model for a dissolved species in such a non-uniform flow field, utilizing a multiple-scale perturbation approach. The resulting two-dimensional transport equation is an advection–diffusion equation containing an effective dispersion tensor field and additional advection-correction terms. It can be viewed as a generalization of the well-known Taylor–Aris dispersion model. The dispersion model allows for flow fields with both stationary and oscillatory components. For the special case of non-uniform flow induced by both pressure gradients and electroosmosis, we derive expressions for the flow field in the long-wavelength limit. These include arbitrary, time-dependent functions for both the driving field as well as the wall mobilities. We discuss the general characteristics of the model using a sinusoidally varying wall mobility, and derive analytical expressions for the dispersion tensor. Then, in order to validate the model, we compare three-dimensional Lagrangian particle tracing simulations with the dispersion model for several test cases, including stationary and oscillatory shear flow as well as a recirculating flow field. For each test case, a good agreement between the full three-dimensional simulations and the results of the two-dimensional dispersion model is obtained. The presented model has the potential to significantly simplify computations of mass transport in Hele-Shaw flows.

The influence of aspect ratio on flow states in the buoyancy-driven turbulence with free slip boundaries

Buoyancy driven turbulence with free-slip top and bottom plates in a confined box is studied via direct numerical simulations (DNS). The Rayleigh number is fixed to Ra=108 and the Prandtl number is Pr=10. The length/height aspect ratio Γy is fixed to 2, while the depth/height aspect ratio Γx varies from 0.125 to 0.3. With the variation of Γx, the hysteresis-like behavior of the heat and momentum transfer is found due to the emergence of two stable states of flow organization. For small Γx, the sheared flow occurs with disordered plumes occupying the entire bulk region, while it is replaced by convection rolls with increasing Γx. The striking feature is that the heat and momentum transport is greatly suppressed in the shear flow (SF) state, compared with that in the convection roll (CR) state. The turbulent kinetic energy budget is performed to delineate the relative contributions of physical processes that govern energy transport. It is found that the kinetic energy in the SF state is mainly produced by the buoyancy force, while it is dominated by the shear production near the plates in the CR state. During energy transport, it seems that the buoyancy force plays a more important role in the SF state than in the CR state, particularly in the bulk. Furthermore, through the analysis of dynamic mode decomposition of temperature fields, we illustrate that the flow in the SF state is dominated by the cloud-like spatially coherent structures with low frequencies, while the low-frequency roll structures play a leading role in the CR state. The flow state transition from SF to CR corresponds to the process that the cloud-like coherent structures form roll structures.

Stationary Shear Flows of Nematic Liquid Crystals: A Comprehensive Study via Ericksen–Leslie Model

We apply the Ericksen–Leslie dynamic model to investigate stationary solutions of planar shear flows for nematic liquid crystals. Nematic material is extremely rich and involves several parameters to characterize the basic local properties. The Ericksen–Leslie model of the nematic material includes Frank’s parameters for Oseen–Frank energy and Leslie’s dynamic parameters. Even in this simple setting of shear flows, the dynamics supported by the material exhibit quick rich behavior. With the aid of a Hamiltonian formulation and phase plane portraits, we are able to provide a rather complete picture about the possible solutions. In particular, we establish the existence of multiple solutions for the boundary value problem in many situations. We also try to explain the physical reasons for some of the results.

Течение Куэтта горячего вязкого газа

Найдено новое точное решение уравнений движения вязкого газа для плоского стационарного сдвигового течения горячего (800-1500 K) газа между движущимися с разными скоростями параллельными пластинами (аналог несжимаемого течения Куэтта). Одна из пластин считалась теплоизолированной. Для зависимости коэффициента вязкости от температуры принята формула Сазерленда. В отличие от других известных точных решений, вместо аналогии Рейнольдса (предположение о линейной связи между коэффициентами вязкости и теплопроводности) для вычисления коэффициента теплопроводности использована более точная формула, имеющая в рассматриваемом диапазоне температур ту же точность, что и формула Сазерленда (2 %). С использованием полученного точного решения исследовано качественное влияние сжимаемости на напряжение трения и на профили температуры и скорости. Показано, что (если одна из пластин теплоизолирована) сжимаемость газа приводит к увеличению напряжения трения. Проведено сравнение нового точного решения с известным точным решением (V. N. Golubkin, G. B. Sizykh, 2018), полученным с использованием формулы Сазерленда для коэффициента вязкости и аналогии Рейнольдса для коэффициента теплопроводности. Обнаружено, что оба решения приводят к одинаковым выводам о качественном влиянии сжимаемости на напряжение трения и на профили температуры и скорости. Однако прирост напряжения трения, вызванный сжимаемостью, при использовании аналогии Рейнольдса оказался недооцененным в два раза. Это показывает, что предположение о линейной связи между коэффициентами вязкости и теплопроводности может приводить к заметным количественным ошибкам.

Effective forcing for direct numerical simulations of the shear layer of turbulent free shear flows

A numerically efficient configuration to simulate turbulent flows is to use triply periodic domains, with numerical forcing techniques to sustain turbulence. Previous homogeneous shear turbulence simulations considered only idealized homogeneous shear flows and not the statistically stationary shear turbulence observed in practical free shear flows. In contrast, the current study mathematically derives the complete forcing technique from the large scales of the turbulent free shear flows. Different statistically stationary free shear flows are considered in this study, namely, a nearly homogeneous shear turbulent flow, turbulent mixing layer, a turbulent planar jet, and a turbulent round jet. The simulations are performed on triply periodic, statistically homogeneous cubic domains in the vicinity of the shear layer in the self-similar region. An a priori analysis is performed to calculate the effects of the different forcing terms and to predict the expected turbulence quantities. The forcing technique is then used to perform direct numerical simulations at different Reynolds numbers. Numerical results for the different cases are discussed and compared with results from experiments and other simulations of free shear turbulent flows. Anisotropy is observed both in the components of velocity and vorticity, with stronger Reynolds number dependence in the anisotropy of vorticity. Energy spectra obtained from the present homogeneous shear turbulence agree well with the spectra from temporally evolving shear layers. The results also highlight the effects of the additional forcing terms that were neglected in previous studies and the role of shear convection and the associated splitting errors in the unbounded evolution of previous numerical simulations.

Stationary inviscid limit to shear flows

In this article we establish a density result for certain stationary shear flows, μ(y), that vanish at the vertical boundaries of a box. We construct stationary solutions to 2D Navier-Stokes that are ε-close in L∞ to the given shear flow. Our construction is based on a coercivity estimate for the Rayleigh operator, R[v], which is based on a decomposition made possible by the vanishing of μ at the boundaries. As a corollary, we obtain, for each ε<<1, a one parameter family of solutions to the NS(ε) system which approximate the Couette flow.

Крупномасштабная слоистая стационарная конвекция вязкой несжимаемой жидкости под действием касательных напряжений на верхней границе. Исследование полей температуры и давления

Изучается новое точное решение переопределенной системы уравнений Обербека-Буссинеска, которое описывает стационарное сдвиговое течение вязкой несжимаемой жидкости в бесконечном слое. Приведенное точное решение является обобщением класса Остроумова-Бириха для слоистого однонаправленного потока. В предложенном решении горизонтальные скорости зависят только от поперечной координаты $z$. Поле температуры и поле давление являются трехмерными. В отличие от решения Остроумова-Бириха, в представленном в статье решении горизонтальные градиенты температуры являются линейными функциями от координаты $z$. Такая структура точного решения позволяет найти нетривиальное решение уравнений Обербека-Буссинеска посредством тождественного равенства нулю уравнения несжимаемости. Данное точное решение пригодно для исследования крупномасштабных течений вязкой несжимаемой жидкости квазидвумерными уравнениями. Конвективное движение жидкости обусловлено заданием касательных напряжений на свободной границе слоя. Неоднородные тепловые источники заданы на обеих границах. Давление в жидкости на верхней границе совпадает с атмосферным давлением. В статье основное внимание уделяется исследованию полей температуры и давления, которые описываются многочленами трех переменных. Детально обсуждаются особенностей распределения профилей температуры и давления, которые являются многочленами седьмой и восьмой степени соответственно. Для анализа свойств температуры и давления используются алгебраические методы для исследования числа корней на отрезке. Показано, что фоновая температура и фоновое давление являются немонотонными функциями. Температурное поле расслаивается на зоны, которые формируют термоклин и тепловой пограничный слой около границ слоя жидкости. Исследование свойств поля давления показали, что оно расслаивается на одну, две или три зоны относительно отсчетного значения (атмосферного давления).

Spectral instability of characteristic boundary layer flows

In this paper, we construct growing modes of the linearized Navier-Stokes equations about generic stationary shear flows of the boundary layer type in a regime of sufficiently large Reynolds number: $R \to \infty$. Notably, the shear profiles are allowed to be linearly stable at the infinite Reynolds number limit, and so the instability presented is purely due to the presence of viscosity. The formal construction of approximate modes is well-documented in physics literature, going back to the work of Heisenberg, C.C. Lin, Tollmien, Drazin and Reid, but a rigorous construction requires delicate mathematical details, involving for instance a treatment of primitive Airy functions and singular solutions. Our analysis gives exact unstable eigenvalues and eigenfunctions, showing that the solution could grow slowly at the rate of $e^{t/\sqrt {R}}$. A new, operator-based approach is introduced, avoiding to deal with matching inner and outer asymptotic expansions, but instead involving a careful study of singularity in the critical layers by deriving pointwise bounds on the Green function of the corresponding Rayleigh and Airy operators.

Packing, alignment and flow of shape-anisotropic grains in a 3D silo experiment

Granular material flowing through bottlenecks, like the openings of silos, tend to clog and thus inhibit further flow. We study this phenomenon in a three-dimensional hopper for spherical and shape-anisotropic particles by means of x-ray tomography. The x-ray tomograms provide information on the bulk of the granular filling, and allows us to determine the particle positions and orientations inside the silo. In addition, it allows us to calculate local packing densities in different parts of the container. We find that in the flowing zone of the silo particles show a preferred orientation and thereby a higher order. Similarly to simple shear flows, the average orientation of the particles is not parallel to the streamlines but encloses a certain angle with it. In most parts of the hopper, the angular distribution of the particles did not reach the one corresponding to stationary shear flow, thus the average orientation angle in the hopper deviates more from the streamlines than in stationary shear flows. In the flowing parts of the silo, shear induced dilation is observed, which is more pronounced for elongated grains than for nearly spherical particles. The clogged state is characterized by a dome, i.e. the geometry of the layer of grains blocking the outflow. The shape of the dome depends on the particle shape.

On the flux Richardson number in stably stratified turbulence

The flux Richardson number $R_{f}$ (often referred to as the mixing efficiency) is a widely used parameter in stably stratified turbulence which is intended to provide a measure of the amount of turbulent kinetic energy $k$ that is irreversibly converted to background potential energy (which is by definition the minimum potential energy that a stratified fluid can attain that is not available for conversion back to kinetic energy) due to turbulent mixing. The flux Richardson number is traditionally defined as the ratio of the buoyancy flux $B$ to the production rate of turbulent kinetic energy $P$. An alternative generalized definition for $R_{f}$ was proposed by Ivey &amp; Imberger (J. Phys. Oceanogr., vol. 21, 1991, pp. 650–658), where the non-local transport terms as well as unsteady contributions were included as additional sources to the production rate of $k$. While this definition precludes the need to assume that turbulence is statistically stationary, it does not properly account for countergradient fluxes that are typically present in more strongly stratified flows. Hence, a third definition that more rigorously accounts for only the irreversible conversions of energy has been defined, where only the irreversible fluxes of buoyancy and production (i.e. the dissipation rates of $k$ and turbulent potential energy ($E_{PE}^{\prime }$)) are used. For stationary homogeneous shear flows, all of the three definitions produce identical results. However, because stationary and/or homogeneous flows are typically not found in realistic geophysical situations, clarification of the differences/similarities between these various definitions of $R_{f}$ is imperative. This is especially true given the critical role $R_{f}$ plays in inferring turbulent momentum and heat fluxes using indirect methods, as is commonly done in oceanography, and for turbulence closure parameterizations. To this end, a careful analysis of two existing direct numerical simulation (DNS) datasets of stably stratified homogeneous shear and channel flows was undertaken in the present study to compare and contrast these various definitions. We find that all three definitions are approximately equivalent when the gradient Richardson number $Ri_{g}\leqslant 1/4$. Here, $Ri_{g}=N^{2}/S^{2}$, where $N$ is the buoyancy frequency and $S$ is the mean shear rate, provides a measure of the stability of the flow. However, when $Ri_{g}&gt;1/4$, significant differences are noticeable between the various definitions. In addition, the irreversible formulation of $R_{f}$ based on the dissipation rates of $k$ and $E_{PE}^{\prime }$ is the only definition that is free from oscillations at higher gradient Richardson numbers. Both the traditional definition and the generalized definition of $R_{f}$ exhibit significant oscillations due to the persistence of linear internal wave motions and countergradient fluxes that result in reversible exchanges between $k$ and $E_{PE}^{\prime }$. Finally, we present a simple parameterization for the irreversible flux Richardson number $R_{f}^{\ast }$ based on $Ri_{g}$ that produces excellent agreement with the DNS results for $R_{f}^{\ast }$.

On the spectral instability of parallel shear flows

This short note is to announce our recent results [2,3] which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number R→∞. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows.

Extensional behavior of rod suspension in dilute polymer solution

Extensional viscosity of rod particle suspensions in polymer solutions is studied experimentally. Rod particle suspensions were prepared by dispersing FeOOH rods in polyacrylamide (molecular weight of 5-6 M) solutions in a glycerin-water mixture. The diameter of rod particles was 100 nm and the aspect ratio was 4.3, 8.7 and 15.6. Particle volume fraction was 0.005-0.02. The extensional viscosity was measured by the capillary thinning method using the commercially available CaBER. Under the experimental condition both the polymer solution and particle suspensions are dilute. Particle-particle interaction is neglected in the solutions which are stretched by the extensional flow at the bulk. The result shows that extensional viscosity of the rod suspension in polymer solution decreases with the increase in particle volume fraction. The decrease is ascribed to the change in polymer conformation from the stretched state in the bulk flow to the less stretched state in shear flow developed near the particle to match the no-slip condition at the particle surface.

Fluctuations in the uniform shear flow state of a granular gas

Using kinetic theory we analyse the fluctuations of the total internal energy of a granular gas under stationary uniform shear flow, and find that they are coupled to fluctuations in the different components of the total pressure tensor. Explicit expressions for all the possible one- and two-time cross and auto correlations of the fluctuations are obtained for the two dimensional case. For the range of inelasticity considered, good agreement is found between the theoretical predictions and Molecular Dynamics simulations.