Non-Newtonian viscous oscillating free surface jets, and a new strain-rate dependent viscosity form for flows experiencing low strain rates

A model for oscillating free surface jet flow of a fluid from an elliptical orifice, together with experimental measurements, can be exploited to characterize the elongational viscosity of non-Newtonian inelastic fluids. The oscillating jet flow is predominantly elongational, with a small strain that oscillates rapidly between large and zero strain rates. We find that to reproduce the experimentally observed steady oscillating jet flow in model simulations, the assumed form of the non-Newtonian viscosity as a function of strain rate must have zero gradient, i.e., be Newtonian, at zero strain rate (a behavior exhibited, in general, by real inelastic fluids). We demonstrate that the Cross, Carreau, Prandtl-Eyring, and Powell-Eyring forms, although they have finite viscosity at zero strain rate, have either nonzero or even unbounded gradient at zero, and hence are unable to model oscillating jet behavior. We propose a new non-Newtonian viscous form which has all of the desirable features of existing forms (high and low strain rate plateaus, with adjustable location and steepness of the transition) and the additional feature of Newtonian behavior at low strain rates. Received: 7 February 2000 Accepted: 31 October 2000     

An incompressible SPH method for simulation of unsteady viscoelastic free-surface flows

In this paper, an incompressible smoothed particle hydrodynamics (SPH) method is presented to solve unsteady free-surface flows. Both Newtonian and viscoelastic fluids are considered. In the case of viscoelastic fluids, both the Maxwell and Oldroyd-B models are investigated. The proposed SPH method uses a Poisson pressure equation to satisfy the incompressibility constraints. The solution algorithm is an explicit predictor-corrector scheme and employs an adaptive smoothing length based on density variations. To alleviate the numerical difficulties encountered when fluid is highly stretched, an artificial stress term is incorporated into the momentum equation which reduces the risk of unrealistic fractures in the material. Two challenging test cases, the impacting drop and the jet buckling problems, are solved to demonstrate the capability of the proposed scheme in handling viscoelastic flows with complex free surfaces. The jet buckling test case was solved for a wide range of Weissenberg numbers. It was shown that in all cases the method is stable and fairly accurate and agrees well with the available data.     

Unsteady flow of viscoelastic fluid between two cylinders using fractional Maxwell model

The unsteady flow of an incompressible fractional Maxwell fluid between two infinite coaxial cylinders is studied by means of integral transforms.The motion of the fluid is due to the inner cylinder that applies a time dependent torsional shear to the fluid.The exact solutions for velocity and shear stress are presented in series form in terms of some generalized functions.They can easily be particularized to give similar solutions for Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is highlighted by graphical illustrations.     

A note on the unsteady flow of a fractional Maxwell fluid through a circular cylinder

In this note the velocity field and the adequate shear stress corresponding to the unsteady flow of a fractional Maxwell fluid due to a constantly accelerating circular cylinder have been determined by means of the Laplace and finite Hankel transforms.The obtained solutions satisfy all imposed initial and boundary conditions.They can easily be reduced to give similar solutions for ordinary Maxwell and Newtonian fluids.Finally,the influence of pertinent parameters on the fluid motion,as well as a comparison between models,is underlined by graphical illustrations.     

Simulation of container filling process with two inlets by improved smoothed particle hydrodynamics (SPH) method

In this article, an improved smoothed particle hydrodynamics (SPH) method is proposed to simulate the filling process with two inlets. Improvements are achieved by deriving a corrected kernel gradient of SPH and a density re-initialisation. In addition, a new treatment of solid wall boundaries is presented. Thus, the improved SPH method has higher accuracy and better stability, and conserves both linear and angular momentums. The validity of the new boundary treatment is shown by simulating the spin-down problem. The bench tests are also presented to demonstrate the performance of the improved SPH method. Then the filling process with a single inlet is simulated to show the ability to capture complex-free surface of the proposed method. Finally, the filling process with two inlets is numerically investigated. The numerical results show that the filling patterns are affected significantly by Reynolds number, aspect ratio of the container and the location of the inlets.     

Surface tension driven jet break up of strain-hardening polymer solutions

Experimental studies attempting to ascertain the influence of viscoelasticity on the atomization of polymer solution are often hindered by the inability to decouple the effect of shear thinning from the effect of extensional hardening. Here, the influence of viscoelasticity on the jet break up of a series of non-shear-thinning viscoelastic fluids is quantified. Previous characterization using an opposed-nozzle rheometer identified the critical extensional rates for strain hardening of these model fluids. The strain hardening fluids exhibit a beads-on-string structure with reduction or elimination of satellite drops. Capillary instabilities grow on the filaments connecting the spheres and eventually break the filaments up into a string of very small drops about one order of magnitude smaller than the satellite drops formed by a Newtonian fluid with the same shear viscosity, surface tension, and density. These results confirm that strain hardening is the key rheological property in jet break up and that the critical extensional rate of a fluid is pertinent in determining the final characteristics of break up. Results suggest that the opposed-nozzle rheometer does probe extensional behavior in the range of extensional rates that are relevant to jet break up, providing a tool to roughly predict jet break up.     

Break-up of a viscoelastic liquid jet

Applying Green's continuum theory of a slender body, the process of liquid jet break-up is analysed for a viscoelastic upper-convected Jeffreys fluid. In contrast to a Newtonian liquid an enforced growth of the perturbation is received from a linear analysis. A non-linear numerical analysis shows the viscosity-dependent filament formation between growing droplets of the viscoelastic liquid. The radius of these filaments decreases in an uniaxial extensional flow.     

Breakup of Newtonian and non-Newtonian fluids in air jets

The breakup of droplets of non-Newtonian fluids has been investigated by high speed photography and impaction following preliminary results of Newtonian fluids, which confirmed the suitability of the measurement techniques. Single droplets with diameters from 2.4 to 3.3 mm, were arranged to fall under gravity into a jet of air with velocities up to 36o m/s. The droplets of Newtonian fluids, water and Diesel oil, were atomised in the expected manner within three main regimes characterised by the Weber number of the droplet and air jet conditions, while similar droplets of non-Newtonian fluids were found not to atomise but to develop under shear and stretching into ligaments of fluid separated from a local region of their surface; these ligaments were elongated until breakup occurred, though not into small droplets as with the Newtonian fluids. Some of the non-Newtonian fluids (TEP with 7.5% and lo% K125, with and without water) were found not to break up at the maximum speed of the tests and they will be re-examined at higher jet velocities. Increase in the concentration of K125 in TEP resulted in higher critical speed for a given droplet diameter.The authors would like to thank Prof. J. H. Whitelaw of Imperial College for many discussions and useful suggestions during the course of this work, and Dr. G. Cambray of CBDE for his valuable administrative support     

Exact solution for the unsteady flow of a semi-infinite micropolar fluid

The unsteady motion of an incompressible micropolar fluid filling a half-space bounded by a horizontal infinite plate that started to move suddenly is considered. Laplace transform techniques are used. The solution in the Laplace transform domain is obtained by using a direct approach. The inverse Laplace transforms are obtained in an exact manner using the complex inversion formula of the transform together with contour integration techniques. The solution in the case of classical viscous fluids is recovered as a special case of this work when the micropolarity coecient is assumed to be zero. Numerical computations are carried out and represented graphically.     

Numerical simulations of bouncing jets

Bouncing jets are fascinating phenomenon occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non‐Newtonian fluids when the jets fall in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier–Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is carried out with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing. Copyright © 2015 John Wiley & Sons, Ltd.     

A modified SPH method to model the coalescence of colliding non-Newtonian liquid droplets

This paper develops a modified smoothed particle hydrodynamics (SPH) method to model the coalescence of colliding non-Newtonian liquid droplets. In the present SPH, a van der Waals (vdW) equation of state is particularly used to represent the gas-to-liquid phase transition similar to that of a real fluid. To remove the unphysical behavior of the particle clustering, also known as tensile instability, an optimized particle shifting technique is implemented in the simulations. To validate the numerical method, the formation of a Newtonian vdW droplet is first tested, and it clearly demonstrates that the tensile instability can be effectively removed. The method is then extended to simulate the head-on binary collision of vdW liquid droplets. Both Newtonian and non-Newtonian fluid flows are considered. The effect of Reynolds number on the coalescence process of droplets is analyzed. It is observed that the time up to the completion of the first oscillation period does not always increase as the Reynolds number increases. Results for the off-center binary collision of non-Newtonian vdW liquid droplets are lastly presented. All the results enrich the simulations of the droplet dynamics and deepen understandings of flow physics. Also, the present SPH is able to model the coalescence of colliding non-Newtonian liquid droplets without tensile instability.     

Production of superfine fibers by the electrospinning method

Weakly conducting fluid flows in an electric field in the presence of phase interfaces are investigated theoretically and experimentally. The object of the investigation is an axisymmetric jet. The analysis is carried out within the framework of electrohydrodynamics (EHD) with allowance for surface charge transfer on mobile interfaces. The jet shape is calculated at large distances from the outflow point. The theoretical and experimental data are compared for Newtonian and polymeric fluids.     

Displacement of yield-stress fluids in a fracture

We consider a displacement of several yield-stress fluids in a Hele-Shaw cell. The topic is relevant to the development of a model for the flow of multiple phases inside a narrow fracture with application to hydraulically fracturing a hydrocarbon-bearing underground formation. Existing models for fracturing flows include only pure power-law models without yield stress, and the present work is aimed at filling this gap. The fluids are assumed to be immiscible and incompressible. We consider fluid advection in a plane channel in the presence of density gradients. Gravity is taken into account, so that there can be slumping and gravitational convection. We use the lubrication approximation so that governing equations are reduced to a 2D width-averaged system formed by the quasi-linear elliptic equation for pressure and transport equations for volume concentrations of fluids. The numerical solution is obtained using a finite-difference method. The pressure equation is solved using an iterative algorithm and the Multigrid method, while the transport equations are solved using a second-order TVD flux-limiting scheme with the superbee limiter. This numerical model is validated against three different sets of experiments: (i) gravitational slumping of fluids in a closed Hele-Shaw cell, (ii) viscous fingering of fluids with a high viscosity contrast due to the Saffman–Taylor (S–T) instability in a Hele-Shaw cell at microgravity conditions, (iii) displacement of Bingham fluids in a Hele-Shaw cell with the development of fingers due to the S–T instability. Good agreement is observed between simulations and laboratory data. The model is then used to investigate the joint effect of fingering and slumping. Numerical simulations show that the slumping rate of yield-stress fluid is significantly less pronounced than that of a Newtonian fluid with the same density and viscosity. If a low-viscosity Newtonian fluid is injected after a yield-stress one, the S–T instability at the interface leads to the development of fingers. As a result, fingers penetrating into a fluid with a finite yield stress locally decrease the pressure gradient and unyielded zones develop as a consequence.     

Modeling and simulation of the viscoelastic fluid mold filling process by level set method

A model for unifying a viscoelastic fluid and a Newtonian fluid is established, in which the governing equations for the viscoelastic fluid and the Newtonian fluid are successfully united into a system of generalized Navier–Stokes equations. A level set method is set up to solve the model for capturing the moving interface in the mold filling process. The physical governing equations are solved by the finite volume method on a non-staggered grid and the interpolation technique on the collocated grid is used for the pressure-velocity and the stress-velocity decoupling problems. The level set and its reinitialization equation are solved by the finite difference method, in which the spatial derivatives are discretized by the 5th-order Weighted Essentially Non-Oscillatory (WENO) scheme, and the temporal derivatives are discretized by the 3rd-order Total Variation Diminishing Runge–Kutta (TVD-R–K) scheme. The validity of the method is verified by some benchmark problems. Then a simulation of viscoelastic fluid mold filling process is pursued with the method. The moving interface and all the information of the physical quantities during the injection process are captured. The die swelling phenomenon is found in the simulation. The influences of elasticity and viscosity on the physical quantities such as stresses etc. in the mold filling process are analyzed. Numerical results show that elastic characteristics such as the stretch and die swelling etc. reinforce accordingly as Weissenberg number increases. Pressures increase continuously in the mold filling process and the pressure maintains the maximum value at the inlet. Injection velocity is proportional to injection pressure. A higher viscosity leads to a higher pressure distribution, that is, the pressure decreases as Reynolds number increases.     

Extension of the Darcy�CForchheimer Law for Shear-Thinning Fluids and Validation via Pore-Scale Flow Simulations

Flow of non-Newtonian fluids through porous media at high Reynolds numbers is often encountered in chemical, pharmaceutical and food, as well as petroleum and groundwater engineering, and in many other industrial applications. Under the majority of operating conditions typically explored, the dependence of pressure drops on flow rate is non-linear and the development of models capable of describing accurately this dependence, in conjunction with non-trivial rheological behaviors, is of paramount importance. In this work, pore-scale single-phase flow simulations conducted on synthetic two-dimensional porous media are performed via computational fluid dynamics for both Newtonian and non-Newtonian fluids and the results are used for the extension and validation of the Darcy?CForchheimer law, herein proposed for shear-thinning fluid models of Cross, Ellis and Carreau. The inertial parameter ?? is demonstrated to be independent of the viscous properties of the fluids. The results of flow simulations show the superposition of two contributions to pressure drops: one, strictly related to the non-Newtonian properties of the fluid, dominates at low Reynolds numbers, while a quadratic one, arising at higher Reynolds numbers, is dependent on the porous medium properties. The use of pore-scale flow simulations on limited portions of the porous medium is here proposed for the determination of the macroscale-averaged parameters (permeability K, inertial coefficient ?? and shift factor ??), which are required for the estimation of pressure drops via the extended Darcy?CForchheimer law. The method can be applied for those fluids which would lead to critical conditions (high pressures for low permeability media and/or high flow rates) in laboratory tests.     

The indentation of a Newtonian fluid in a finite cylindrical container by a right circular cylinder

The indentation of the free surface of a Newtonian fluid in a finite cylindrical container by a right circular cylinder is considered. It is assumed that weight and inertia effects are negligible compared to viscous effects. A finite difference technique is used to obtain approximate values for initial velocities, pressures, and stresses at any point in the fluid as well as an estimate of the force required to indent the fluid with a given velocity. The solution obtained forms the basis for a primary indenter viscometer for very viscous fluids which have viscosities in the range of 10⁴–10¹⁰ poises.     

An adaptive boundary element approach to transient free surface flow as applied to injection molding

An adaptive (Lagrangian) boundary element approach is proposed for the general two‐dimensional simulation of confined moving‐boundary flow of viscous incompressible fluids. Only the quasi‐steady creeping (Stokes) flow of a Newtonian fluid is examined. The method is stable as it includes remeshing capabilities of the deforming moving boundary, and thus it can handle large deformations. An algorithm is developed for mesh refinement of the deforming moving‐boundary mesh. Several flow problems are presented to illustrate the utility of the approach, with particular emphasis on cavity filling and viscous fingering, as applied to conventional and gas‐assisted injection molding. The accuracy of the method is assessed through the problem of jet flow and the transient fountain flow between two flat plates. Copyright © 2000 John Wiley & Sons, Ltd.     

On axisymmetric flow of Sisko fluid over a radially stretching sheet

This study presents an analysis of the axisymmetric flow of a non-Newtonian fluid over a radially stretching sheet. The momentum equations for two-dimensional flow are first modeled for Sisko fluid constitutive model, which is a combination of power-law and Newtonian fluids. The general momentum equations are then simplified by invoking the boundary layer analysis. Then a non-linear ordinary differential equation governing the axisymmetric boundary layer flow of Sisko fluid over a radially stretching sheet is obtained by introducing new suitable similarity transformations. The resulting non-linear ordinary differential equation is solved analytically via the homotopy analysis method (HAM). Closed form exact solution is then also obtained for the cases n=0 and 1. Analytical results are presented for the velocity profiles for some values of governing parameters such as power-law index, material parameter and stretching parameter. In addition, the local skin friction coefficient for several sets of the values of physical parameter is tabulated and analyzed. It is shown that the results presented in this study for the axisymmetric flow over a radially non-linear stretching sheet of Sisko fluid are quite general so that the corresponding results for the Newtonian fluid and the power-law fluid can be obtained as two limiting cases.     

An enhanced treatment of boundary conditions in implicit smoothed particle hydrodynamics简

In this work, an enhanced treatment of the solid boundaries is proposed for smoothed particle hydrodynamics with implicit time integration scheme （Implicit SPH）. Three types of virtual particles, i.e., boundary particles, image particles and mirror particles, are used to impose boundary conditions. Boundary particles are fixed on the solid boundary, and each boundary particle is associated with two fixed image particles inside the fluid domain and two fixed mirror particles outside the fluid domain. The image particles take the flow properties through fluid particles with moving least squares （MLS） interpolation and the properties of mirror particles can be obtained by the corresponding image particles. A repulsive force is also applied for boundary particles to prevent fluid particles from unphysical penetra- tion through solid boundaries. The new boundary treatment method has been validated with five numerical examples. All the numerical results show that Implicit SPH with this new boundary-treatment method can obtain accurate results for non-Newtonian fluids as well as Newtonian fluids, and this method is suitable for complex solid boundaries and can be easily extended to 3D problems.     

Front‐tracking by the level‐set and the volume penalization methods in a two‐phase microfluidic network

Two‐phase immiscible fluids in a two‐dimensional micro‐channels network are considered. The incompressible Stokes equations are used to describe the Newtonian fluid flow, while the Oldroyd‐B rheological model is used to capture the viscoelastic behavior. In order to perform numerical simulations in a complex geometry like a micro‐channels network, the volume penalization method is implemented. To follow the interface between the two fluids, the level‐set method is used, and the dynamics of the contact line is modeled by Cox law. Numerical results show the ability of the method to simulate two‐phase flows and to follow properly the contact line between the two immiscible fluids. Finally, simulations with realistic parameters are performed to show the difference when a Newtonian fluid is pushed by a viscoelastic fluid instead of a Newtonian one. Copyright © 2015 John Wiley & Sons, Ltd.