Presence Of Normal Stresses Research Articles

OBurnett Equations: Thermodynamically Consistent Continuum Theory Beyond the Navier–Stokes Regime

Abstract Rarefied gas flows are highly nonequilibrium flows whose flow physics cannot be discerned accurately within the framework of the Navier–Stokes equations. The Burnett equations and the Grad moment equations, which form a super-set of the Navier–Stokes equations, have been proposed in the literature to model such flows but not much success has been achieved because of some inherent limitations of these equations. In this review article, we mainly focus on the recently proposed Onsager-Burnett equations (Singh et al., 2017, “Derivation of stable Burnett equations for rarefied gas flows,” Phys. Rev. E 96, p. 013106) for rarefied gas flows, and the progress achieved so far by solving these equations for some benchmark flow problems. Like Burnett and Grad equations, the OBurnett equations form a super-set of the Navier–Stokes equations and belong to the class of higher order continuum transport equations. However, there are two fundamental aspects where the significance of the OBurnett equations is clearly visible. First, the OBurnett equations are unconditionally stable as well as thermodynamically consistent unlike the conventional Burnett and Grad moment equations. Second, the OBurnett constitutive relations for the stress tensor and the heat flux vector do not have any higher order derivatives of velocity, pressure, or temperature. This is quite significant since now the equations need the same number of boundary conditions as that of the Navier–Stokes equations. As such, the OBurnett equations form a complete theory, which cannot be said for the conventional Burnett equations. These two important aspects help to set the OBurnett equations apart from the rest of the higher order continuum theories. The results of the OBurnett equations are compiled for two benchmark rarefied flow problems: force-driven compressible Poiseuille flow and the normal shock wave flow problem. For force-driven compressible Poiseuille flow, the OBurnett equations successfully capture the nonequilibrium effects such as nonuniform pressure profile and presence of normal stresses and tangential heat flux in the flow. The accurate description of highly nonequilibrium internal structure of normal shocks has always been the stringent test for the higher order continuum theories. The results of the OBurnett equations for normal shocks show that there is no theoretical upper Mach number limit for the equations. Further, the equations predict smooth shock structures at all Mach numbers, existence of heteroclinic trajectory, positive entropy generation throughout the shock, and significant improvement over the results of the Navier–Stokes equations. Finally, the recently proposed Grad's second problem, which has the potential to become a benchmark problem, is discussed. The solution of Grad's second problem for different interaction potentials (Maxwell and hard-sphere molecules) within the Burnett hydrodynamics is also presented at length and some important remarks are made in this context.

Rod-climbing rheometry revisited.

The rod-climbing or "Weissenberg" effect in which the free surface of a complex fluid climbs a thin rotating rod is a popular and convincing experiment demonstrating the existence of elasticity in polymeric fluids. The interface shape and steady-state climbing height depend on the rotation rate, fluid elasticity (through the presence of normal stresses), surface tension, and inertia. By solving the equations of motion in the low rotation rate limit for a second-order fluid, a mathematical relationship between the interface deflection and the fluid material functions, specifically the first and second normal stress differences, emerges. This relationship has been used in the past to measure the climbing constant, a combination of the first (Ψ1,0) and second (Ψ2,0) normal stress difference coefficients from experimental observations of rod-climbing in the low shear rate limit. However, a quantitative reconciliation of such observations with the capabilities of modern-day torsional rheometers is lacking. To this end, we combine rod-climbing experiments with both small amplitude oscillatory shear (SAOS) flow measurements and steady shear measurements of the first normal stress difference from commercial rheometers to quantify the values of both Ψ1,0 and Ψ2,0 for a series of polymer solutions. Furthermore, by retaining the oft-neglected inertial terms, we show that the "climbing constant"  = 0.5Ψ1,0 + 2Ψ2,0 can be measured even when the fluids, in fact, experience rod descending. A climbing condition derived by considering the competition between elasticity and inertial effects accurately predicts whether a fluid will undergo rod-climbing or rod-descending. Our results suggest a more general description, "rotating rod rheometry" instead of "rod-climbing rheometry", to be more apt and less restrictive. The analysis and observations presented in this study establish rotating rod rheometry combined with SAOS measurements as a prime candidate for measuring normal stress differences in complex fluids at low shear rates that are often below commercial rheometers' sensitivity limits.

Numerical investigation on the effect of electrical force and periodic force on neutrally buoyant spherical particles in a Newtonian fluid at low Reynolds numbers

This topic examines a periodically forced neutrally buoyant spherical particle in an electrically driven dilute solution with the velocity at very long distances (infinity) at low Reynolds numbers. This is a novel problem since it considers the periodic force as well as the electric force operating on the spherical particle/s. When particle-laden suspensions are exposed to electric stress, electrophoresis and electrostriction occur. Consequently, these studies might be used to blood flow to perform electrophoresis on red blood cells and study their activity. We have used the formalism of Lovalenti and Brady [14] to obtain the necessary equations for our considered problem. All the results with U∞ = 0 and U∞ equals specific values are plotted in the form of phase plots between displacements of the particle versus the velocity of the particle. It is observed that the resistance to the change in motion of the particle is manifested in the phase plots and the periodic force is dominant over the considered DC electric field for the Reynolds numbers considered. The DC force translates the particles with different amplitudes of the periodic force. We also computed the only rheological parameter the normal stress difference using Batchelor's technique with respect to volume fraction and amplitude of periodic force with respect to time. The presence of normal stress indicates the non – Newtonian behaviour in this type of phenomena.

Modeling Shear Behavior of Woven Fabric Thermoplastic Composites for Crash Simulations

Woven fabric thermoplastic composites possess high specific strength and stiffness along with thermoformability. To utilize the full potential of these materials to achieve better crash-safe designs in automotive structural parts, the measurement of non-linear shear behavior and its material modeling for FEM simulations is required. The standard testing method was used to measure the pure shear behavior of woven fabric composites. These results were compared with the shear behavior of material in the presence of normal stresses along the fiber direction. Tensile and compression cyclic testing of ± 45° laminate were carried out to measure the stiffness degradation and hardening of the material in the presence of tensile normal and compression normal stress. A methodology is proposed for taking into account the differences in shear behavior under different loading directions in an FEM simulation. Based on the experimental evidence, improvements in the mathematical description of plasticity and damage in continuum damage mechanics models are proposed. The model was implemented as a user-defined material subroutine (VUMAT) for Abaqus. The experimental results from coupon tests were used to verify the results of a single element simulation. Finally, a three-point bending test was used to validate the predictions of the user material model.

Coupled interface-based modelling approach for the numerical analysis of curved masonry specimens strengthened by CFRP

Aim of the present paper is to numerically study the bond behavior of curved masonry specimens externally strengthened by Carbon Fiber Reinforced Polymer systems (CFRP). A simple 1D-modeling approach is presented to this aim, where the coupled behavior between shear and normal stresses developing at the reinforcement/masonry interface level is specifically introduced to properly account for the role played by the curvature radius. The model is indeed enriched by the introduction of shear stress-slip laws able to account for the beneficial friction effect, when compression normal stresses develop at the interface level and the reduction of the slip strength corresponding to the de-cohesion in presence of normal stresses in tension. Considering some case studies derived from the current literature, consisting of shear-lap bond tests of curved masonry specimens characterized by different curvatures of the bonded surface and different strengthening configurations, the validation of the proposed approach is carried out. In particular, two modeling strategies are considered and critically compared: the first one, denoted as approach (A), where the presence of the mortar joints is neglected, and the second one, denoted as approach (B), where mortar joints are specifically introduced in the model. Finally, the results obtained by using the proposed simple approach are compared with those obtained from both sophisticated FE numerical models and theoretical formulas deduced from the current literature.

Fracture structure near a longitudinal shear macrorupture

Fracture evolution the near a main longitudinal shear in the presence of normal stresses is studied. Experiments with model materials (gypsum, cheese) showed that a multiscale echelon structure of cracks feathering the main rupture is formed under the shear domination conditions. A system of small cracks in the initial echelon is replaced by an echelon of larger and sparser cracks. Intensive transverse compression along the normal to the shear plane, which imitates the initial stress concentrator, takes the fracture region away from the shear plane. A model of evolution development of the observed echelon structure along the main rupture front under the shear domination conditions is proposed.

Study of the Crack Deviation Angle in Adhesive Ruptures of Ceramic Adhesively-Bonded Assemblies

Mechanical tests are commonly used to characterize structural adhesively-bonded assemblies since they give rise to typical values such as rupture load, stress, strain or energy. It is shown herewith, that for a single lap compression-shear test, it can also be of great interest to analyse the bonded assembly after rupture. The investigated fracture surfaces can reveal the presence of normal stresses in the joint and a valuation of their intensity can be done, from the crack deviation angle value. Actually, for adhesive ruptures, a transition zone exists where the crack propagates from one interface to the other. It is demonstrated from the failure occurrence that the deviation angle value does not depend on the adhesive itself but on distributions in the joint of different kinds of stresses (shear and normal ones). In addition to experimental angle measurements, numerical modelling was performed with the finite element computer code ANSYS. Experimental values match numerical values.

Theory of the nonequilibrium structure of dense simple fluids. II. High-shear-rate effects.

The integral equation for the dynamic pair correlation function derived previously from the modified moment method is solved for a non-Newtonian soft-sphere fluid undergoing a plane Couette flow. In this approach, the dynamic pair correlation function is a functional of the shear stress, which determines the macroscopic state of the nonequilibrium fluid. The distortions of the structure factor at high shear rates are examined for a non-Newtonian fluid. Normal stress effects are included in the study presented. It is found that as the shear rate increases, the structure factor fractures into domains of high intensity in the xy (shear), xy, and yz planes in the wave-vector space. At packing fractions in the range of \ensuremath{\eta}=0.3--0.45, the structure factor exhibits a plateau behavior at reduced shear rates \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\ge}100, and this behavior is seen to be directly related to the thixotropy of the fluid. In all planes considered, the structure factor suggests, essentially, a distorted-fluid-like structure, although there is evidence indicating that the approach to an ordered phase is also a probable interpretation. These effects are in qualitative agreement with the observations made on colloidal suspensions and low-shear-rate molecular-dynamics simulation results. In the presence of normal stresses, the structure factor in the xy plane is rotated in the direction of the shear flow. The angle of rotation is determined by the magnitude of the primary normal stress difference and also by the thermodynamic state of the fluid. The structure factor in the yz plane is similar to the case without normal stresses, and the xz plane is undistorted.

Dynamics of colloidal particles in sheared, non-Newtonian fluids

The dynamics of colloidal, axisymmetric particles suspended in non-Newtonian, polymeric liquids are examined by using optical techniques that are capable of following their average orientation. Polarimetry measurements of birefringence and dichroism are used on dispersions of colloidal iron oxide spheroids and small-angle light scattering is used on hardened, prolate human red blood cells. A variety of suspending liquids are used including Boger fluids designed to have some of the attributes of a second order fluid under steady state flow conditions, and concentrated polystyrene solutions with rheological properties similar to more commonly encountered polymer liquids. As a result of the non-Newtonian properties of the polymeric suspending fluids, and in particular the presence of normal stresses, the particles were observed to drift out of the orbits associated with their motion in Newtonian fluids. As predicted by Cohen and coworkers, in flows where the elastic forces on the particles dominate Brownian forces, the particles tend to drift away from orientations favoring the flow direction and towards the vorticity axis. When Brownian forces are large compared with the elastic forces, alignments in the flow direction are more probable. This trend, however, was only observed when the Boger fluids were used as suspending fluids. Although the polystyrene solutions possessed shear viscosities and first normal stress difference coefficients that were comparable in magnitude to those of the Boger fluids, only orientations along the flow direction were found.

Hyperostosis triangularis ilii -- a sacro-iliac stress phenomenon Part. 2 (Incidence, prognosis, pathogenesis, aetiology, tracer studies, differential diagnosis) (author's transl)

The anterior iliac margin adjacent to the sacro-iliac joint contains a physiological zone of hyperostosis. Its shape is that of a triangular pyramid. Because of its small size, it is radiological invisible, but is nevertheless an indication of the presence of normal stresses passing from the mass of the trunk to the lower extremities. If for any reason, such as is described in this paper, the stresses on the hyperostosis zone are increased, the latter becomes enlarged, while still keeping its shape. It then appears on the radiograph as hyperostosis triangularis ilii. In approximately 50% of cases there is a simultaneous but variable sacral hyperostosis. Observations on 40 patients, together with two autopsies and radionucleide investigations suggest that hyperostosis triangularis ilii represents a potentially reversible bone reaction of the physiological hyperostosis of the anterior iliac margin. The clinical differential diagnosis raised by the radiological finding of a "triangular ilium hyperostosis" is discussed.