Comparison of two numerical methods for the solution of non-Newtonian flow in ducts

Two numerical methods, the Galerkin finite element method (FEM) and the boundary-fitted co-ordinate transformation method (BFCTM), have been applied to solve inelastic non-Newtonian fluid flow in ducts of irregular cross-section. Three representative fluid models, namely the power-law, the Ellis and the Bingham models, have been analysed. The application of the FEM is straightforward, while for the BFCTM the accurate estimation of viscosity on the duct boundary and the proper mesh adjustment appear to be critical for generating convergent solutions. A detailed comparison of the two numerical methods in terms of volumetric flow rate, axial velocity, shear rate, viscosity and CPU time is given. Both methods can generate accurate solutions of velocity over a wide range of variables, but the FEM requires much less computing time to reach the same level of accuracy. Only the BFCTM can be used to approximate shear rate and viscosity with reasonable accuracy.     

Simulation of non-Newtonian ink transfer between two separating plates for gravure-offset printing

The inks used in gravure-offset printing are non-Newtonian fluids with higher viscosities and lower surface tensions compared to Newtonian fluids. This paper examines the transfer of a non-Newtonian ink between two parallel plates when the top plate is moved upward with a constant velocity while the bottom plate is held fixed. Numerical simulations were carried out using the Carreau model to explore the behavior of a non-Newtonian ink in gravure-offset printing. The volume of fluid (VOF) model was adopted to demonstrate the stretching and break-up behaviors of the ink. The results indicate that the ink transfer ratio is greatly influenced by the contact angle, especially the contact angle at the upper plate (α). For lower values of α, oscillatory or unstable behavior of the position of minimum thickness of the ink between the two parallel plates during the stretching period is observed. This oscillation gradually diminishes as the contact angle at the upper plate is increased. Moreover, the number of satellite droplets increases as the velocity of the upper plate is increased. The surface tension of the conductive ink shows a positive impact on the ink transfer ratio to the upper plate. Indeed, the velocity of the upper plate has a significant influence on the ink transfer in gravure-offset printing when the Capillary number (Ca) is greater than 1 and the surface tension dominates over the ink transfer process when Ca is less than 1.     

A boundary integral method for two-dimensional (non)-Newtonian drops in slow viscous flow

A boundary integral method for the simulation of the time-dependent deformation of Newtonian or non-Newtonian drops suspended in a Newtonian fluid is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. The implementation of the boundary conditions is facilitated by rewriting the domain integral by means of the Gauss divergence theorem. To apply the divergence theorem smoothness assumptions are made concerning the non-Newtonian stress tensor. The correctness of these assumptions in actual simulations is checked with a numerical validation procedure. The method appears mathematically correct and the numerical algorithm is second order accurate. Besides this validation we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for Newtonian and non-Newtonian drops in two dimensions indicate that the steady state deformation is quite independent of the drop-fluid. The deformation process, however, appears to be strongly dependent on the drop-fluid. For the non-Newtonian drop a mechanical model is developed to describe the time-dependent deformation of the cylinder for small capillary numbers.     

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.     

Inertia effects in circular squeeze film bearing using Herschel–Bulkley lubricants

Recent engineering trends in lubrication emphasize that in order to analyze the performance of bearings adequately, it is necessary to take into account the combined effects of fluid inertia forces and non-Newtonian characteristics of lubricants. In the present work, the effects of fluid inertia forces in the circular squeeze film bearing lubricated with Herschel–Bulkley fluids with constant squeeze motion have been investigated. Herschel–Bulkley fluids are characterized by an yield value which leads to the formation of a rigid core in the flow region. The shape and extent of the core formation along the radial direction is determined numerically for various values of Herschel–Bulkley number and power-law index. The bearing performances such as pressure distribution and load capacity for different values of Herschel–Bulkley number, Reynolds number, power-law index have been computed. The effects of fluid inertia and non-Newtonian characteristics on the bearing performances have been discussed.     

Time dependent viscosity of concentrated alumina suspensions

Viscometric investigations of concentrated aqueous alumina suspensions with particles smaller than 5 μm have been performed. Experimental flow curves indicate thixotropy in the shear rate interval between =20 and 640 s⁻¹. In the range smaller than =200 s⁻¹ we found pseudoplastic flow behavior, in the higher range the material shows dilatancy. The non-Newtonian behavior results from a small content of sodium aluminum oxide in the alumina suspension. This gives rise to interparticle forces that can drive the suspension into a gel-like state. The time scale of this process is some days. On the short-time scale of some hours the material ages slowly increasing moderately the apparent viscosity. Studying the relaxation process after a shear rate jump, the shear stress time dependency at constant shear rate follows an exponential law. There is a single particular relaxation time for each shear rate. The relaxation towards a steady state occurs asymptotically over some 10³ s. Flow curves calculated from steady state data after relaxation processes are below the experimental flow curves which were measured during some 100 s. The flow curves follow the Herschel–Bulkley formula. The shape of the viscosity curves indicates changes of suspension structure at ca. =200 and 400 s⁻¹. At constant shear rates in the interval between =400 and 450 s⁻¹ the apparent viscosity of the alumina suspension fluctuates periodically in time in the same manner found for other suspensions. This effect is interpreted as periodic organization of agglomeration and deglomeration processes. Supposing, that the stabilisation energy of agglomerates is of the order of the energy introduced by the mechanical shear field, the observation of oscillations at =400 s⁻¹ is in agreement with the drastic slope change in the viscosity curves.     

Nonlinear steady states of hyperelastic membrane tubes conveying a viscous non-Newtonian fluid

We study possible steady states of an infinitely long tube made of a hyperelastic membrane and conveying either an inviscid, or a viscous fluid with power-law rheology. The tube model is geometrically and physically nonlinear; the fluid model is limited to smooth changes in the tube’s radius. For the inviscid case, we analyse the tube’s stretch and flow velocity range at which standing solitary waves of both the swelling and the necking type exist. For the viscous case, we first analyse the tube’s upstream and downstream limit states that are balanced by infinitely growing upstream (and decreasing downstream) fluid pressure and axial stress caused by fluid viscosity. Then we investigate conditions that can connect these limit states by a single solution. We show that such a solution exists only for sufficiently small flow speeds and that it has a form of a kink wave; solitary waves do not exist. For the case of a semi-infinite tube (infinite either upstream or downstream), there exist both kink and solitary wave solutions. For finite-length tubes, there exist solutions of any kind, i.e. in the form of pieces of kink waves, solitary waves, and periodic waves.     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

A thin-film equation for viscoelastic liquids of Jeffreys type

We derive a novel thin-film equation for linear viscoelastic media describable by generalized Maxwell or Jeffreys models. As a first application of this equation we discuss the shape of a liquid rim near a dewetting front. Although the dynamics of the liquid is equivalent to that of a phenomenological model recently proposed by Herminghaus et al. (S. Herminghaus, R. Seemann, K. Jacobs, Phys. Rev. Lett. 89, 056101 (2002)), the liquid rim profile in our model always shows oscillatory behaviour, contrary to that obtained in the former. This difference in behaviour is attributed to a different treatment of slip in both models.