Effect of destabilizing fields on hydrodynamic instabilities in nematics

The effect of destabilizing fields on the roll instability (RI) threshold for shear flow and on the homogeneous instability (HI) threshold for plane Poiseuille flow of nematic HBAB (μ _s>0) is studied on the basis of the continuum theory of nematics for flow cells of infinite lateral width. It turns out that the critical shear rate and wave vector at RI threshold decrease with increasing destabilizing field but do not approach zero at the Freedericksz transition. However calculations show that beyond the Freedericksz threshold HI may be favourable over a range of destabilizing field with shear in the stabilizing role. For plane Poiseuille flow a similar analysis points to the existence of a HI threshold in the presence of destabilizing field beyond the Freedericksz threshold again with shear acting as a stabilizing field. These results are compared with theoretical results obtained previously for MBBA.     

高磁雷诺数下剪切流对双撕裂模中二级磁岛的影响

在高磁雷诺数下,当双撕裂模发展进入快速磁场重联阶段时,会发生二级磁岛不稳定性,加剧磁场能量的释放。本文基于扰动形式的守恒磁流体方程组发展高精度的数值模拟程序,在平板位形下研究反对称位形剪切流对双撕裂模中二级磁岛的影响。结果表明：随着剪切流强度和剪切梯度的增加,二级磁岛的数目以及电流片横纵比变小。此外,较强的极向剪切流能够抑制二级磁岛不稳定性的发生。     

Steady low shear rate cholesteric flow normal to the helical axis

Steady cholesteric flow at low shear rate normal to the helical axis is studied analytically for shear flow and plane Poiseuille flow on the basis of Leslie’s continuum theory. For general asymmetric solutions the angle made by the director at the sample centre with the primary flow is found to profoundly affect the oscillations of the apparent viscosity with pitch for pitches of the order of the sample thickness. The velocity and orientation profiles are also found to change drastically. These considerations may be important in flow experiments on long pitch cholesterics.     

Analytical solutions of the lattice Boltzmann BGK model

Analytical solutions of the two-dimensional triangular and square lattice Boltzmann BGK models have been obtained for the plane Poiseuille flow and the plane Couette flow. The analytical solutions are written in terms of the characteristic velocity of the flow, the single relaxation time , and the lattice spacing. The analytic solutions are the exact representation of these two flows without any approximation. Using the analytical solution, it is shown that in Poiseuille flow the bounce-back boundary condition introduces an error of first order in the lattice spacing. The boundary condition used by Kadanoffet al. in lattice gas automata to simulate Poiseuille flow is also considered for the triangular lattice Boltzmann BGK model. An analytical solution is obtained and used to show that the boundary condition introduces an error of second order in the lattice spacing.     

Polymer solutions in co-rotating Taylor-Couette flow without vorticity

We present experimental results of the flow of dilute and semi-dilute polymer solutions in co-rotating Taylor-Couette cylinders. The experimental set-up consists of a modified Mars II rheometer (Thermo Scientific) with two drive units that are mounted opposite each other. The rotational velocities of the inner and outer cylinders are chosen in a way such that the angular velocity has a 1/r profile and the flow is free of vorticity, but the direction of elongation is not constant, but rotates with the flow. Our particle image velocimetry (PIV) measurements show that for polymer solutions without shear thinning the flow is indeed free of vorticity and is equal to a stagnation point flow at a given position and a given instant in time. In contrast, torque measurements reveal that the stresses are identical to the stresses that are present in a plane shear flow. Thus, we find that for polymer solutions a flow with vorticity and a constant direction of elongation is equal to a flow without vorticity in which the direction of elongation is rotating. Finally, we show that for shear thinning solutions the flow velocity becomes non-monotonic through the gap and resembles a pluglike profile which is known from the Poiseuille flow.     

Effect of an axial magnetic field on the Poiseuille flow of a nematic liquid crystal

The effect of an axial magnetic field on the Poiseuille flow of nematicp-azoxyanisole (PAA) has been computed using the Ericksen-Leslie continuum theory. The apparent viscosity decreases appreciably in the presence of the magnetic field. Orientation and velocity profiles for different shear rates and magnetic fields are presented.     

The effect of magnetic fields and boundary conditions on the shear flow of nematics

The flow of a nematic liquid crystal between plane parallel plates, with one plate moving with uniform velocity relative to the other, is discussed. The apparent viscosity, orientation and velocity profiles are computed forp-azoxyanisole as functions of shear rate and magnetic field for symmetric and asymmetric molecular alignment at the plates. For symmetric homeotropic boundary condition, a magnetic field applied along the flow direction exhibits a threshold reminiscent of a Freedericksz transition in the hydrostatic case. In general the apparent viscosity for the asymmetric boundary condition is less than that for the symmetric case.     

Investigation of Turbulent Transition in Plane Couette Flows Using Energy Gradient Method

The energy gradient method has been proposed with the aim of better understanding the mechanism of flow transition from laminar flow to turbulent flow. In this method, it is demonstrated that the transition to turbulence depends on the relative magnitudes of the transverse gradient of the total mechanical energy which amplifies the disturbance and the energy loss from viscous friction which damps the disturbance, for given imposed disturbance. For a given flow geometry and fluid properties, when the maximum of the function $K$ (a function standing for the ratio of the gradient of total mechanical energy in the transverse direction to the rate of energy loss due to viscous friction in the streamwise direction) in the flow field is larger than a certain critical value, it is expected that instability would occur for some initial disturbances. In this paper, using the energy gradient analysis, the equation for calculating the energy gradient function $K$ for plane Couette flow is derived. The result indicates that $K$ reaches the maximum at the moving walls. Thus, the fluid layer near the moving wall is the most dangerous position to generate initial oscillation at sufficient high $\operatorname{Re}$ for given same level of normalized perturbation in the domain. The critical value of $K$ at turbulent transition, which is observed from experiments, is about 370 for plane Couette flow when two walls move in opposite directions (anti-symmetry). This value is about the same as that for plane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concluded that the critical value of $K$ at turbulent transition is about 370-389 for wall-bounded parallel shear flows which include both pressure (symmetrical case) and shear driven flows (anti-symmetrical case).     

No Existence and Smoothness of Solution of the Navier-Stokes Equation

The Navier-Stokes equation can be written in a form of Poisson equation. For laminar flow in a channel (plane Poiseuille flow), the Navier-Stokes equation has a non-zero source term (∇²u(x, y, z) = F_x (x, y, z, t) and a non-zero solution within the domain. For transitional flow, the velocity profile is distorted, and an inflection point or kink appears on the velocity profile, at a sufficiently high Reynolds number and large disturbance. In the vicinity of the inflection point or kink on the distorted velocity profile, we can always find a point where ∇²u(x, y, z) = 0. At this point, the Poisson equation is singular, due to the zero source term, and has no solution at this point due to singularity. It is concluded that there exists no smooth orphysically reasonable solutions of the Navier-Stokes equation for transitional flow and turbulence in the global domain due to singularity.     

A study in the developing region of square jet

The developing region of a turbulent square jet is investigated using high-resolution particle image velocimetry (PIV). The mean velocity and turbulence stresses are presented in various horizontal planes, along the jet centerline covering the initial region of the jet as well as the transition to the self-similar region. To study the flow structure away from the central plane, velocity measurements in two additional horizontal planes, one located halfway from the jet central plane toward the edge and the other at the edge of the square jet, are also examined. Analysis of the instantaneous velocity fields reveal the presence of an arrow-like feature in the square jet due to the higher instability generated in the jet shear layer compared with a round jet. To elucidate the imprints of the vortex structures present in the jets, a swirling strength-based vortex identification methodology is applied on a large ensemble of instantaneous velocity fields. Statistical analysis of the number of vortex cores, and their size and rotational strength in the measurement plane is undertaken. Vortex population at the edge was found to be very different compared with that in the central plane.     

Chemical instability induced by a shear flow

We predict a new type of instability induced by shear flow in chemical systems. A homogeneous steady state solution of a reaction-diffusion system loses stability in a Poiseuille flow. The instability appears as the speed of the flow increases beyond a certain threshold. This results in a steady pattern moving with the average fluid velocity. The chemical reaction consists of two species (activator and inhibitor) moving with identical velocities. Contrary to Turing's instability, the pattern arises when the activator has a higher diffusivity than the inhibitor.     

Self-sustaining nonlinear dynamo process in Keplerian shear flows

A three-dimensional nonlinear dynamo process is identified in rotating plane Couette flow in the Keplerian regime. It is analogous to the hydrodynamic self-sustaining process in nonrotating shear flows and relies on the magnetorotational instability of a toroidal magnetic field. Steady nonlinear solutions are computed numerically for a wide range of magnetic Reynolds numbers but are restricted to low Reynolds numbers. This process may be important to explain the sustenance of coherent fields and turbulent motions in Keplerian accretion disks, where all its basic ingredients are present.     

Transport phenomena in a partially lonized Lorentzian magnetoplasma

Some transport coefficients are evaluated for a homogeneous, partially ionized Lorentzian plasma in the presence of a uniform external magnetic field. The electronion collisions are taken into account by means of the modified Fokker-Planck equation and the electron-neutral collision frequency, ν_en is taken as velocity (ω) dependent or velocity independent, depending on the energy range under consideration. The variation of the transport coefficients with magnetic field is determined for ν_en ∝ ω^s (s being a positive or negative integer) and one finds that qualitatively this behavior does not change by changing either the collision frequency or the velocity dependence of the collision frequency; however for weak magnetic fields the magnitudes of these transport coefficients increase with the decrease in ν_en or s, whereas for strong magnetic fields the transverse components of the transport coefficients decrease and the Hall components tend to saturate with the decrease of either the collision frequency or s.     

Pattern formation in reaction-diffusion system in crossed electric and magnetic fields

We consider a reaction-diffusion system in crossed electric and magnetic fields lying on the reaction plane. It is shown that a charge separation along the direction normal to the reaction plane resulting in a diffusional flux may cause a differential flow induced chemical instability and stationary pattern formation on a homogeneous steady state. This pattern is generically different from a Turing pattern modified by the crossed fields. The special role of magnetic field is emphasized. Our theoretical analysis is corroborated by numerical simulation on a reaction-diffusion system in three dimensions.     

Energy inconsistency of Galerkin approximationsfor plane Couette flow

For classical solutions of the incompressible Navier-Stokes equations (NSE) the energybalance between kinetic energy, work done by external forces, and viscous dissipation holds rigorously true. It is shown in this paper that standard Galerkin approximations violate energy balance in the case of plane Couette flow, whereas Poiseuille flow turns out to be energy consistent at any cutoff. The main reason for this discrepancy is seen in the different boundary conditions between the stationary linear shear flow and its disturbances. In our analysis, essentially, we introduce an auxiliary external force field which enforces the finite dimensional Galerkin approximation to fulfil the NSE. It is exemplarily demonstrated how the energy discrepancy decreases when the number of disturbed modes is increased which couple to the stationary shear flow.     

Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe

Generally speaking, rheological properties of materials are specified by their so-called constitutive equations. The simplest constitutive equation for a fluid is a Newtonian one, on which the classical Navier-Stokes theory is based. The mechanical behavior of many fluids is well described by this theory. However, there are many rheologically compli- cated fluids such as polymer solutions, blood and heavy oils which are inadequately de- scribed by a Newtonian constitutive equation that does …     

Monodisperse drop formation in square microchannels

The electrohydrodynamic instability of the interface between two liquids with different physical and electrical properties in plane Poiseuille flow is used to form monodisperse droplets in a square channel. The drop size and formation rate are controlled by simply controlling the flow rates and the amplitude of the electric field applied across the channel.     

Three-dimensional lattice Boltzmann method for simulating blood flow in aortic arch

The three-dimensional （3D） lattice Boltzmann models, 3DQ15, 3DQ19 and 3DQ27, under different wall boundary conditions and lattice resolutions have been investigated by simulating Poiseuille flow in a circular cylinder for a wide range of Reynolds numbers. The 3DQ19 model with improved Fillippova and Hanel （FH） curved boundary condition represents a good compromise between computational efficiency and reliability. Blood flow in an aortic arch is then simulated as a typical haemodynamic application. Axial and secondary fluid velocity and effective wall shear stress profiles in a 180° bend are obtained, and the results also demonstrate that the lattice Boltzmann method is suitable for simulating the flow in 3D large-curved vessels.     

矩形微槽道气体流动的速度分布

矩形微槽道的各个流向截面可以局部近似为平面Poiseuille流动,应用信息保存(IP)方法和直接模拟Monte Carlo(DSMC)方法计算了从连续介质区到自由分子流区的平面Poiseuille流动,利用其结果对Beskok-Karniadadis公式和质量流率动理论因子进行修正和重新拟合,给出在整个稀薄气体流动领域都适用的微槽道气体流动速度分布.     

Understanding the sub-critical transition to turbulence in wall flows

In contrast with free shear flows presenting velocity profiles with inflection points which cascade to turbulence in a relatively mild way, wall bounded flows are deprived of (inertial) instability modes at low Reynolds numbers and become turbulent in a much wilder way, most often marked by the coexistence of laminar and turbulent domains at intermediate Reynolds numbers, well below the range where (viscous) instabilities can show up. There can even be no unstable mode at all, as for plane Couette flow (pCf) or for Poiseuille pipe flow (Ppf) that are currently the subject of intense research. Though the mechanisms involved in the transition to turbulence in wall flows are now better understood, statistical properties of the transition itself are yet unsatisfactorily assessed. A widely accepted interpretation rests on non-trivial solutions of the Navier-Stokes equations in the form of unstable travelling waves and on transient chaotic states associated to chaotic repellors. Whether these concepts typical of the theory of temporal chaos are really appropriate is yet unclear owing to the fact that, strictly speaking, they apply when confinement in physical space is effective while the physical systems considered are rather extended in at least one space direction, so that spatiotemporal behaviour cannot be ruled out in the transitional regime. The case of pCf will be examined in this perspective through numerical simulations of a model with reduced cross-stream (y) dependence, focusing on the in-plane (x, z) space dependence of a few velocity amplitudes. In the large aspect-ratio limit, the transition to turbulence takes place via spatiotemporal intermittency and we shall attempt to make a connection with the theory of first-order (thermodynamic) phase transitions, as suggested long ago by Pomeau.