Dissipative soft modes in viscous nematodynamics

This paper presents ordered rate constitutive theories in Lagrangian description for compressible as well as incompressible homogeneous, isotropic thermoviscoelastic solid matter with memory in which the material derivative of order m of the deviatoric stress tensor and heat vector are functions of temperature, temperature gradient, time derivatives of the conjugate strain tensor up to any desired order n, and the material derivatives of up to order m−1 of the stress tensor. The thermoviscoelastic solids described by these theories are called ordered thermoviscoelastic solids with memory due to the fact that the constitutive theories are dependent on orders m and n of the material derivatives of the conjugate stress and strain tensors. The highest orders of the material derivative of the conjugate stress and strain tensors define the order of the thermoviscoelastic solid. The constitutive theories derived here show that the material for which these theories are applicable have fading memory. As is well known, the second law of thermodynamics must form the basis for deriving constitutive theories for all deforming matter (to ensure thermodynamic equilibrium during evolution), since the other conservation and balance laws are independent of the constitution of the matter. The entropy inequality expressed in terms of Helmholtz free energy density \({\Phi}\) does not provide a mechanism to derive a constitutive theory for the stress tensor when its argument tensors are stress and strain rates in addition to others. With the decomposition of the stress tensor into equilibrium and deviatoric stress tensors, the constitutive theory for the equilibrium stress tensor is deterministic from the entropy inequality. However, for the deviatoric stress tensor, the entropy inequality requires a set of inequalities to be satisfied but does not provide a mechanism for deriving a constitutive theory. In the present work, we utilize the theory of generators and invariants to derive rate constitutive theories for thermoviscoelastic solids with memory. This is based on axioms and principles of continuum mechanics. However, we keep in mind that these constitutive theories must satisfy the inequalities resulting from the second law of thermodynamics. The constitutive theories for heat vector q are derived: (i) strictly using conditions resulting from the entropy inequality; (ii) using the theory of generators and invariants with admissible argument tensors that are consistent with the stress tensor as well as the theories in which simplifying assumptions are employed which yield much simplified theories. It is shown that the rate theories presented here describe thermoviscoelastic solids with memory. Mechanisms of dissipation and memory are demonstrated and discussed, and the derivation of memory modulus is presented. It is shown that simplified forms of the general theories presented here result in constitutive models that may resemble currently used constitutive models but are not the same. The work presented here is not to be viewed as extension of the current constitutive models; rather, it is a general framework for rate constitutive theories for thermoviscoelastic solids with memory based on the physics and derivations that are consistent within the framework of continuum mechanics and thermodynamics. The purpose of the simplified theories presented in the paper is to illustrate possible simplest theories within the consistent framework presented here.

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process.The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form.In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation.As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor.The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration.Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways.An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), "not closed" stress trajectories and nonzero stress work for a closed deformation trajectory.In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor.When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations.The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case -elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures.Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various

Application of the kinetic theory of gases to granular flows has greatly increased our understanding of ‘rapid’ granular flows. One of the underlying assumptions is that particles interact only through binary collisions. For a given set of material and flow parameters, as the concentration increases, the transition from a binary collision mode to other modes of interaction occurs. Kinetic theory can no longer be applied. A numerical model is utilized to simulate the mechanical behaviour of a small assembly of uniform, inelastic, frictional, deformable disks in a simple shear flow. There are two objectives: to obtain the ‘empirical’ constitutive law and to gain insight into the mechanisms that operate in the transitional and quasi-static regimes. In a simple shear flow, spatially and temporally averaged dimensionless stresses $\tau^*_{ij} = \tau_{ij}/(\rho_{\rm s}D^2\dot{\gamma}^2)$ are functions of the concentration C, the dimensionless shear rate $B =\dot{\gamma}/(K_n/m)^{\frac{1}{2}}$, and material parameters ζ n , K s /K n and μ. Here $\dot{\gamma}$ is the shear rate, K n is the normal stiffness of an assumed viscoelastic contact force model, K s /K n is the ratio of tangential to normal stiffness, ζ n is the normal damping coefficient, μ is the friction coefficient, and ρs, D and m are the particle density, diameter and mass, respectively. The range of B from 0.001 to 0.0707 was investigated for C ranging from 0.5 to 0.9, with material constants fixed as ζ n = 0.0709 (corresponding to the restitution coefficient e = 0.8 in binary impacts), K s /K n = 0.8 and μ = 0.5. It is found that for lower concentrations (C < 0.75) dimensionless stresses τ* ij are nearly independent of B, while for higher concentrations (C > 0.75) τ* ij monotonically decreases as B increases. Moreover, their relationship in this regime is well approximated by power law: τ* ij ∝ B−n(C). The powers n ij range from nearly zero for C = 0.775 (corresponding to the familiar square power dependency of dimensional stresses on the shear rate in the rapid flow regime), to nearly two for C = 0.9 (corresponding to shear-rate independence in quasi-static regime). The intermediate concentration range corresponds to transition. Distinct mechanisms that govern transitional and quasi-static regimes are observed and discussed.

We investigate aspects of low-magnetic-Reynolds-number flow between two parallel, perfectly insulating walls in the presence of an imposed magnetic field parallel to the bounding walls. We find a functional basis to describe the flow, well adapted to the problem of finding the attractor dimension and which is also used in subsequent direct numerical simulation of these flows. For given Reynolds and Hartmann numbers, we obtain an upper bound for the dimension of the attractor by means of known bounds on the nonlinear inertial term and this functional basis for the flow. Three distinct flow regimes emerge: a quasi-isotropic three-dimensional (3D) flow, a nonisotropic 3D flow, and a 2D flow. We find the transition curves between these regimes in the space parametrized by Hartmann number Ha and attractor dimension d(att). We find how the attractor dimension scales as a function of Reynolds and Hartmann numbers (Re and Ha) in each regime. We also investigate the thickness of the boundary layer along the bounding wall and find that in all regimes this scales as 1/Re, independently of the value of Ha, unlike Hartmann boundary layers found when the field is normal to the channel. The structure of the set of least dissipative modes is indeed quite different between these two cases but the properties of turbulence far from the walls (smallest scales and number of degrees of freedom) are found to be very similar.

This work investigates the interdependence of the phase behaviour, viscosity, temperature, molecular weight and shear rates of thermotropic liquid crystalline polyethers. The viscosity of the isotropic and nematic phases are quantitatively compared; a positive variation in viscosity with respect to temperature is found, with the isotropic viscosity being about an order of magnitude higher than the liquid crystalline viscosity. The dependence of viscosity upon molecular weight of well defined fractions is investigated in both the liquid-crystal and isotropic phases. In the liquid crystalline state the viscosity scales with M3.5–5. Variations in the viscosity due to temperature changes affect the isotropic phase more than the liquid crystal phase. No evidence for a negative first normal stress difference is seen. Finally, it is shown how the phase diagram of the material can be altered by shearing the material in the isotropic phase. This is evident by the onset of a shear thinning region at temperatures slightly above Ti, which can be attributed to the formation of shear induced liquid crystallinity.

Synergistic effects in flows of mixtures of wormlike micelles and hydroxyethyl celluloses with or without hydrophobic modifications

A numerical method is utilized to examine the steady and transient mass/heat transfer processes that involve a neutrally buoyant liquid sphere suspended in simple shear flow at low Reynolds numbers is described. By making use of the known Stokes velocity field, the convection‐diffusion equations are solved in the three‐dimensional spherical coordinates system. For the mass transfer either outside or inside a liquid sphere, Sherwood number Sh approaches an asymptotic value for a given viscosity ratio at sufficiently high Peclet number Pe. In terms of the numerical results obtained in this work, two new correlations are derived to predict Sh at finite Pe for various viscosity ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 343–352, 2014

The time-dependent rheological behavior of the thixotropic fluids is presented in various industrial fields (cosmetics, food, oil, etc.). Usually, a couple of equations define constitutive model for thixotropic substances: a constitutive equation based on linear viscoelastic models and a rate equation (an equation related to the micro-structural evolution of the substance). Many constitutive models do not take into account the micro-structural dependence of the shear modulus and viscosity in the dynamic principles from which are developed. The modified Jeffreys model (considering only one single micro-structure type) does not show this incoherence in its formulation. In this chapter, a constitutive model for thixotropic fluids, based on modified Jeffreys model, is presented with the addition of one more micro-structure type, besides of comments on some possible generalizations. The rheological coherence of this constitutive model and thermodynamic consistency are analyzed too. This model takes into account a simple isothermal laminar shear flows, and the micro-structures dynamics are relate to Brownian motion and de Gennes Reptation model via the Smoluchowski™s coagulation theory.

Four different microstructure constitutive equations (CEs) for discotic nematic liquid crystals based on Doi‘s modified nematodynamics theory are formulated. Their dynamic and steady state responses under simple shear flows are computed and analyzed in terms of the tensor order parameter Q, the orientation director triad (n, m, l), and the uniaxial S and biaxial P alignments. A unit sphere description of the director triad is used to characterize and classify the various predicted stable orientation states, and to discuss and analyze their multi-stabilities as a function of dimensionles shear rate. Various attractors, steady and periodic, are also identified and their stability is discussed in detail for all the CEs. A validation procedure based on the predicted microstructural response along with bifurcation diagrams of the individual CE and representative experimental observations as well as theoretical results is implemented, and used to select the most appropriate CE. The selected CE predicts, under shear, the simultaneous presence of stable in-plane (steady and periodic) states and out-of-plane steady state, and the classical transition among the in-plane periodic and steady states with increasing shear rate. The excellent performance of the selected CE in shear flows strongly suggests that it is a reliable contribution towards the formulation of a process model for mesophase pitch spinning.

Low Reynolds number flow through nozzle-diffuser elements in valveless micropumps

Introduction A number of problems occur in the fields of drilling and rock mechanics for which consideration must be given to the interaction of fluid flow and rock deformation. Such problems include those of borehole stability, chip removal from under a drill bit, drilling in the presence of a fluid pressure gradient between the drilling fluid and formation fluid, and drilling by use of hydraulic jets. We have recently developed a general theory of the influence of fluid pressure gradients and gravity on the plasticity of porous media. The solution of the problem considered here serves as an example of the application of that theory. The illustrative problem is to determine the load required on a flat problem is to determine the load required on a flat axially symmetric punch for incipient plasticity of the porous medium under the punch when fluid flows through the bottom face of the punch. The rock is assumed to behave as a Coulomb plastic material under the influence of body forces plastic material under the influence of body forces due to fluid pressure gradients and gravity. Numerical methods that have been used by Cox et al. for analyzing axially symmetric plastic deformation in soils with gravity force are applied to the problem considered here. Involved is an iterative process for determining the slip lines. The fluid flow field ‘used for calculating the fluid pressure gradient is based upon the work by Ham pressure gradient is based upon the work by Ham in his study of the potential distribution ahead of the bit in rotary drilling. The effective stresses in the porous rock and the punch force for incipient plasticity are computed in terms of the fluid plasticity are computed in terms of the fluid pressure and the cohesive strength and internal pressure and the cohesive strength and internal friction of the rock. PLASTICITY OF POROUS MEDIA PLASTICITY OF POROUS MEDIA A recently developed general theory of plasticity of porous media under the influence of fluid flow is summarized in this section. The equation of motion for the porous solid for the case of incipient plastic deformation reduces to the following equilibrium equation:(1) where Ts is the partial stress tensor of the solid; Fs is the body force acting on the solid per unit volume of the solid material; P is the interaction force between the solid and the fluid; and is the porosity, which is defined as the ratio of the pore porosity, which is defined as the ratio of the pore volume to the total volume of the solid-fluid mixture. The partial stress tensor Ts can be considered as the effective stress tensor that is used in sod mechanics. With the acceptance of the effective stress principle defined in Ref. 5, the yield function, f, in the following form is satisfied for plastic deformation of the porous medium. plastic deformation of the porous medium.(2) where EP is the plastic strain tensor and K and the work-hardening parameter. From the equation of motion for the fluid, the interaction force P can be expressed in the form(3) where is the inertial force of the fluid per unit volume of the mixture and F is the body force acting on the fluid per unit volume of fluid. For the case of incipient plastic deformation the solid can be considered static (velocities of the solid particles are zero), and the problem of determining particles are zero), and the problem of determining the fluid flow field is the one usually analyzed in petroleum engineering. petroleum engineering. Consider a flow of be fluid such that the inertial forces of the fluid can be neglected and assume that Darcy's law is applicable. SPEJ P. 271

Voluminous research exists on the effect of combination of low Reynolds number flow over low aspect ratio flat plate wings for usage in micro-aerial vehicles and UAVs, as they also fly under similar conditions and combination of low aspect ratio and low Reynolds number. It has been well established by many researchers (Torres and Mueller in AIAA 42(5):865–873, 2004; Okamoto and Azuma in AIAA 49(6):1135–1150, 2011; Mizoguchi and Itoh in AIAA 51(7):7–52, 2013; Shields and Mohseni in AIAA 50(1):85–99, 2012; Mizoguchi and Kajikawa in Trans Jpn Soc Aero Sp Sci 59(2):56–63, 2016; Pelletier and Mueller in J Aircr 37(5):825–832, 2000; Liu and Hsiao in J Mech 28(01):77–89, 2012; Taira and Colonius in J Fluid Mech 623:187–207, 2009) that low aspect ratio wings in low Reynolds number flow gives a higher lift coefficient and stall angle is also increased which is quite different from the high aspect ratio wing. Also, it is a well-established fact in the existing literature that linear theories cannot be applied as in case of high aspect ratio wings because of non-linear increase in lift coefficient. The reason for such increase in lift and stall angle has been attributed to wingtip vortices that become prominent in low aspect ratio wings called the vortical lift which keeps on increasing up to a certain high angle of attack till it interacts with the separation bubble. In the past, very limited work has been done to study the vortices and their contribution in enhancement of aerodynamic characteristics. In the present work, flat plate model with corrugations at an aspect ratio (AR) of 1.0 are studied and compared with flat plate wing of AR 1.0; also flow visualization in water tunnel has been carried out at Reynolds number of 102 to ascertain the flow physics involved.

The viscous constitutive relations of partially molten rocks developed here show that an anisotropy in grain‐scale melt distribution can lead to a strong anisotropy in the viscosity of the solid framework. With anisotropic viscosity, a direct coupling between shear and isotropic components of stress occurs, and hence, the role of shear deformation in melt migration significantly increases. We demonstrate the significant effects of viscous anisotropy on melt migration dynamics by solving the solid‐liquid two‐phase dynamics for two simple cases. First, in rotary shear deformation, an anisotropy creates a driving force for melt migration up stress gradients in the solid matrix, which does not exist with isotropic viscosity. Second, in uniform simple shear deformation, melt segregates spontaneously into low‐angle bands due to anisotropic melt alignment, in close agreement with experimental observations. Our results indicate that stress‐induced melt alignment at the grain scale drives further melt redistribution over distances much longer than the grain scale. The development of such “multiscale anisotropy” is demonstrated by “forward” or ab initio approaches based on the equations of two‐phase dynamics, in which the viscous constitutive relation is based on observed microstructure and realistic rheology. When applied to a simple approximation of flow in the mantle beneath ridge and subduction zone, the anisotropic constitutive relation significantly affects melt migration patterns.

Numerical solution of the Fokker–Planck equation for fiber suspensions: Application to the Folgar–Tucker–Lipscomb model

The forces due to surface tension and momentum change during evaporation, in conjunction with the forces due to viscous shear and inertia, govern the two-phase flow patterns and the heat transfer characteristics during flow boiling in microchannels. These forces are analyzed in this paper, and two new nondimensional groups, K1 and K2, relevant to flow boiling phenomenon are derived. These groups are able to represent some of the key flow boiling characteristics, including the CHF. In addition, a mechanistic description of the flow boiling phenomenon is presented. The small hydraulic dimensions of microchannel flow passages present a large frictional pressure drop in single-phase and two-phase flows. The small hydraulic diameter also leads to low Reynolds numbers, in the range 100–1000, or even lower for smaller diameter channels. Such low Reynolds numbers are rarely employed during flow boiling in conventional channels. In these low Reynolds number flows, nucleate boiling systematically emerges as the dominant mode of heat transfer. The high degree of wall superheat required to initiate nucleation in microchannels leads to rapid evaporation and flow instabilities, often resulting in flow reversal in multiple parallel channel configuration. Aided by strong evaporation rates, the bubbles nucleating on the wall grow rapidly and fill the entire channel. The contact line between the bubble base and the channel wall surface now becomes the entire perimeter at both ends of the vapor slug. Evaporation occurs at the moving contact line of the expanding vapor slug as well as over the channel wall covered with a thin evaporating film surrounding the vapor core. The usual nucleate boiling heat transfer mechanisms, including liquid film evaporation and transient heat conduction in the liquid adjacent to the contact line region, play an important role. The liquid film under the large vapor slug evaporates completely at downstream locations thus presenting a dryout condition periodically with the passage of each large vapor slug. The experimental data and high speed visual observations confirm some of the key features presented in this paper.