Fluid Constitutive Equation Research Articles

Periodic and chaotic acoustic oscillations of a bubble gas immersed in an Upper Convective Maxwell fluid

In the present work, the non-linear dynamics of a spherical gas bubble oscillating in a viscoelastic liquid is investigated numerically. The radial oscillations of the bubble are governed by a modified Rayleigh–Plesset equation due to the viscoelastic behavior of the liquid. For simplicity, the Upper Convective Maxwell model is chosen as the fluid constitutive equation. In addition, thermal dissipation effects within the gas bubble are taken into account and comparing with previous approaches, we consider the nearly isothermal model for the compression gas. The purely adiabatic case is also included. The resulting nondimensional governing equations depend on different dimensionless parameters; however, the rheological character is directly dictated by the Deborah number, De. The numerical results for the radial oscillations predict periodic solutions for values of De between 1 and 4 and showing a clear chaotic behavior for De ∼ 4.4, independently of the intensity of the thermal damping. In addition, the physical influence of other relevant parameters, like the characteristic Reynolds number, is also clarified.

Numerical Simulation of Unsteady Flow for Visco-Elastic Fluid in An Eccentric Annulus with Inner Rod Reciprocation

The equation governing the unsteady flow of viscoelastic fluids in an eccentric annulus was derived by using the common conversion Maxwell fluid constitutive equation and then discretized by the control volume method. The velocity distribution of flow field was computed by using the ADI method. The influences of the pressure gradient, eccentricity, stroke length, and stroke frequency on the velocity and flow capacity in the flow field was analyzed. The foundation for further research of the eccentric wear problem of the pumping rod in polymer-flood well was laid.

Creeping flow of viscoplastic materials through a planar expansion followed by a contraction

In this work, the creeping flow of a viscoplastic fluid through a planar channel with an expansion followed by a contraction is analyzed numerically. The solution of the conservation equations of mass and momentum is obtained via the finite volume method. In order to model the non-Newtonian behavior of the fluid, it was used the generalized Newtonian fluid constitutive equation. The viscosity function was the one proposed by Souza Mendes and Dutra [Souza Mendes, P.R., Dutra, E.S.S., 2004. Viscosity function for yield-stress liquids. Appl. Rheol. 14, 296–302]. The yielded and unyielded regions are obtained for several combinations of rheological parameters. The influence of these parameters on pressure drop through the cavity is also obtained and analyzed.

Tensor representations and solutions of constitutive equations for viscoelastic fluids

An approach is developed using tensor representations to assess and characterize both the transient behavior and equilibrium states of viscoelastic fluid constitutive equations in viscometric flows. The methodology is based on the replacement of the differential constitutive equation for the deviatoric part of the viscoelastic stress tensor by an equivalent and more tractable set of differential equations for the characteristic scalar invariants. In the case of planar flows, this equivalence leads to an explicit, closed-form analytic solution for the time evolution of the extra-stress tensor that is formally expressed as a second-order fluid relation, with time-dependent coefficients. As a validation of the approach, an analysis of the transient and equilibrium system characteristics of fluid flows described by the corotational Jeffreys model and general Oldroyd-type constitutive equations is presented.

α-NEM and model reduction

Numerical modeling of non-Newtonian flows typically involves the coupling between the equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an ad...

Natural element meshless simulation of flows involving short fiber suspensions

Numerical modeling of non-Newtonian flows typically involves the coupling between the equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. Thus, the numerical modeling of short fiber suspensions flows requires a description of the microstructural evolution (fiber orientation) which affects the flow kinematics and that is itself governed by this kinematics (coupled problem). Some industrial flows involve moving or free boundaries (injection, extrusion, …). Lagrangian descriptions allow an accurate description of the flow front tracking as well as an accurate integration of transport equations along the flow trajectories. However, Lagrangian techniques in the context of finite elements have the important drawback of requiring frequent remeshing in order to avoid large elements distortions. The natural element method (NEM) has the capabilities of Lagrangian models to describe the flow front tracking as well as to treat the convection terms related to the fiber orientation equation without the mesh quality requirement characteristics of the standard finite elements method.

A characteristics strategy for solving advection equations in 2D steady flows containing recirculating areas

Numerical modelling of non-Newtonian flows usually involves the coupling between equations of motion characterized by an elliptic character, and the fluid constitutive equation, which defines an advection problem linked to the fluid history. There are different numerical techniques to treat the hyperbolic character of advection equations. In non-recirculating flows, Eulerian discretisations can give an accurate mesh size dependent solution within a short computing time. However, the existence of steady recirculating flow areas induces additional difficulties. Actually, in these flows neither boundary conditions nor initial conditions are known. In a former paper we have proved that in such flows Eulerian techniques lead to solutions with significant deviations from the exact one. These deviations obviously decrease as the mesh density increases. In other paper, the authors have proved that some linear advection equations modelling non-Newtonian fluid behaviors have only one solution in steady recirculating flows. This solution is found imposing the solution periodicity along the closed streamlines, where the equation is integrated by the method of characteristics. In this paper we propose a characteristics algorithm for solving advection equations in general steady flows, which may contain recirculating areas.

On the solution of Fokker–Planck equations in steady recirculating flows involving short fiber suspensions

Numerical modeling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic character, and the fluid constitutive equation, which is an advection equation linked to the fluid history. Thus, the numerical modeling of short fiber suspensions flows requires a description of the microstructural evolution (fiber orientation) which affects the flow kinematics and that is itself governed by this kinematics. There are different ways to describe the microstructural state. The use of orientation tensors, whose evolution is governed by advection equations, is widely considered. Other possibility is to describe the fiber orientation state from its probability density whose evolution is described by the Fokker–Planck equation (which does not involve any closure relation). The numerical treatment of advection problems is not a simple matter. Moreover, many industrial flows show often steady recirculating areas which introduce some additional difficulties, because in these flows neither boundary conditions nor initial conditions are known. In some of our former papers, we have proposed accurate techniques to solve the linear and non-linear advection equations governing the evolution of the second-order orientation tensor, in steady recirculating flows. These techniques combine a standard treatment of the non-linearity with a more original treatment of the associated linear problems imposing the periodicity condition. In this paper, we generalize those numerical strategies to compute the steady solution of the Fokker–Planck equation (which governs the evolution of the fiber orientation probability distribution) in steady recirculating flows.

Short fibers suspension in steady recirculating flows

AbstractNumerical modeling of short fiber suspensions flows involves the coupling between motion equations, which definean elliptic problem, and the fluid constitutive equation, which introduces a non‐linear advection problem related to the fiber orientation (induced anisotropy). In a previous work these authors have proposed a numerical procedure to determine a steady solution of the fibers orientation in steady recirculating flows, taking into account that neither initial nor boundary conditions are given. This procedure may be used in the numerical simulation of SFRT flows involving recirculating parts as encountered in the simulationof industrial processes, as well as in inverse rheological identification using, for example, rotative rheometric devices.

Study on Thin Film Lubrication With Second-Order Fluid

The basic lubrication equations are deduced from the original second-order fluid constitutive equations. Two examples of lubrication, a plane inclined slider and a journal bearing, are calculated respectively. The Reynolds boundary conditions are used in the calculation of the journal bearing. In this calculation, it is found that the load carrying capacities of the slider and the journal bearing are of different tendencies with the increase of the Deborah number. Furthermore, the results show that with the decrease of the film thickness, the increase of the normal stress of second-order fluid is greater than that of Newtonian fluid. Finally, it is found that the distribution of the normal stress changes significantly at a certain thickness.

Further considerations on steady recirculating flows involving non-Newtonian fluids

Numerical modeling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic behavior, and the fluid constitutive equation, which is an advection equation linked to the fluid history in differential models. In a former paper, the authors have proved that linear advection equations have only one solution in steady recirculating flows. This solution is found imposing the solution periodicity along the closed streamlines of the flow. The non-linear character of constitutive equations was taken into account in a second paper. In this case, standard strategies, as Picard or Newton techniques for example, were generalized to impose the solution periodicity for each linearized problem found in the iterative procedures. In the present paper, we treat other questions related to steady recirculating flows of non-Newtonian fluids. We will prove that in some cases the steady solution of the extra-stress tensor cannot be found as the steady state associated with the evolution problem. We will also prove that only one steady solution exists for multi-mode linear differential models. Finally, we will present some results concerning integral viscoelastic models in general steady recirculating flows.

NMR measurements and simulations of particle migration in non-newtonian fluids

Shear-induced migration of particles is studied during the slow flow of suspensions of neutrally buoyant spheres, at 50% particle volume fraction, in an inelastic but shear-thinning, suspending fluid. The suspension is flowing in between a rotating inner cylinder and a stationary outer cylinder. The conditions are such that nonhydrodynamic effects are negligible. Nuclear magnetic resonance (NMR) imaging demonstrates that the movement of particles away from the high shear rate region is more pronounced than for a Newtonian suspending liquid. We test a continuum constitutive model for the evolution of particle concentration in a flowing suspension proposed by Phillips et al., but extended to shear-thinning, suspending fluids. The fluid constitutive equation is Carreau-like in its shear-thinning behavior but also varies with the local particle concentration. The model captures many of the trends found in the experimental data, but does not yet agree quantitatively. In fact, quantitative agreement with a diffusive flux constitutive equation would be impossible without the addition of another fitting parameter that may depend on the shear-thinning nature of the suspending fluid. Because of this, we feel that the Phillips model may be fundamentally inadequate for simulating flows of particles in non-Newtonian suspending fluids without the introduction of a normal stress correction or other augmenting terms.

On the steady solution of linear advection problems in steady recirculating flows

Numerical modelling of non-Newtonian flows typically involves the coupling between equations of motion characterized by an elliptic behaviour, and the fluid constitutive equation, which is an advection equation linked to the fluid history. In this paper we prove that linear steady advection problems in steady recirculating flows have only one solution when the kinematics differs from a rigid motion. We also give a numerical procedure to determine this steady solution. We will describe this numerical procedure for two linear models the first will be the SFRT flow model and the second will be a simplified linear formulation of the Pom–Pom viscoelastic model.

Dynamics of gas bubbles in viscoelastic fluids. I. Linear viscoelasticity

The nonlinear oscillations of spherical gas bubbles in linear viscoelastic fluids are studied. A novel approach is implemented to derive a governing system of nonlinear ordinary differential equations. The linear Maxwell and Jeffreys models are chosen as the fluid constitutive equations. An advantage of this new formulation is that, when compared with previous approaches, it facilitates perturbation methods and numerical investigations. Analytical solutions are obtained using a multiple scale perturbation method and compared with the Newtonian results for various Deborah numbers. Numerical analysis of the full equations supports the perturbation analysis, and further reveals significant differences between the viscoelastic and Newtonian cases. Differences in the oscillation phase and harmonic structure characterize some of the viscoelastic effects. Subharmonic excitations at particular fluid parameters lead to a discrete group modulation of the radial excursions; this appears to be a unique, previously undiscovered phenomenon. Implications for medical ultrasound applications are discussed in light of these current findings.

BINGHAM PLASTIC ANALYSIS OF ER VALVE FLOW

Yield shear stresses and shear rates at a valve wall are derived from experimental results on a series of concentric cylinder valves. The fluid constitutive equation used for this purpose is that of a Bingham Plastic. Valve plates (which are such that the radial gap is small compared to its mean pitch) are taken to be parallel so far as the derivation of the flow ν's pressure ν's geometry model is concerned. A range of electrode separations from 0.5 to 1.0mm are used with flow velocities being limited to the region where the viscous pressure drop component is below that caused by the electro stress. Results show that (away from the region of low shear rates and high voltages) the wall stresses for equivalent conditions are comparable for different valves, for a range of applied field strengths and mean flow velocities. Thus, provided the hysteretic region is avoided the fluid can be treated as a Bingham continuum with some stated reservations. However, this is only applied with precision for the truly corresponding situations defined in the paper.

NOIMISOTHERMAL MODEL OF SINGLE SCREW EXTRUSION OF GENERALIZED NEWTONIAN FLUIDS

Abstract A nonisothermal model of Ike single screw extrusion processing of generalized Newtonian fluids is presented. Various temperature dependent forms of a generalized Newtonian fluid constitutive equation representing the Herschel-Bulkley fluid and its simplifications, including Bingham plastic, power law of Ostwald-de Waele, and Newtonian fluids, are applicable. The model includes the generally ignored transverse convection terms of the equation of energy. The importance of keeping the transverse convection terms in the analysis is demonstrated by applying the model and comparing findings to experimental results involving the transverse flow temperature distributions in single screw extruders, available in the literature. The numerical instabilities, arising principally from the convection terms, generally encountered in high-Peclet-number extrusion flows, could be eliminated by the use of the streamline upwind / Petrov-Galerkin formulation. The model is sufficiently general to accommodate Navier's w...

An investigation into the flow of elastico-viscous fluids past a circular cylinder

The stress-relaxation effect is analyzed in a boundary layer flow of elastico-viscous fluids past a circular cylinder. The fluid's constitutive equation employs a time derivative which rotates together with the principal axes of the strain-rate tensor and permits a separate parametric description of the relaxation process and the normal stresses. The results show that the relaxation process retards the separation point and reduces the skin friction, which is in a qualitative agreement with previous experiments.

Significance of deformation rate softening of memory in viscoelastic fluid mechanics and polymer processing

AbstractIt is pointed out that in viscoelastic fluid constitutive equations, non‐linear response to deformation rate and strain beyond the second order is usually interpreted in terms of the deformation rate or strain dependence of the memory function. These non‐linearities act to decrease the extent of the memory. The dependence may be characterized by one or more dimensionless material parameters. A new dimensionless group based on the primary material parameters describing the intensity of the deformation rate dependence of the memory function is introduced and its significance is discussed. This is called the Yamamoto number. The solutions of viscoelastic fluid mechanics problems are considered to depend upon both the Weissenberg and Yamamoto numbers. Such problems include topics of interest in polymer melt processing such as uniaxial elouigatioiial flow, fiber spinning and vortex development in extrusion through a die entry.

Experimental investigation of the influence of molecular weight distribution on the rheological properties of polypropylene melts

AbstractAn experimental study of steady shear and elongational flow Theological properties of a series of polypropylene melts of varying molecular weight and distribution is reported. Broadening the molecular weight distribution increases the non‐Newtonian character of the shear viscosity function and increases the principal normal stress differences at fixed shear stress. The behavior is compared to earlier rheological property‐molecular weight studies. Correlations are developed for these properties in terms of molecular structure. Elongational flow studies indicate that for commercial and broader molecular weight distribution samples, ready failure by neck development occurs and the elongational viscosity appears to decrease with increasing elongation rate. For narrower molecular weight distribution samples, the elongational viscosity is an increasing function of elongation rate, The implication of these experimental results to viscoelastic fluid constitutive equations and polymer melt processing is developed.