• Journal of the Korean Mathematical Society
• /
• v.60 no.2
• /
• pp.273-302
• /
• 2023

In this paper, a fully discrete numerical scheme for the viscoelastic Oldroyd flow is considered with an introduced auxiliary variable. Our scheme is based on the finite element approximation for the spatial discretization and the backward Euler scheme for the time discretization. The integral term is discretized by the right trapezoidal rule. Firstly, we present the corresponding equivalent form of the considered model, and show the relationship between the origin problem and its equivalent system in finite element discretization. Secondly, unconditional stability and optimal error estimates of fully discrete numerical solutions in various norms are established. Finally, some numerical results are provided to confirm the established theoretical analysis and show the performances of the considered numerical scheme.

• Korea-Australia Rheology Journal
• /
• v.21 no.1
• /
• pp.47-58
• /
• 2009

We have investigated the transient behavior of 1D fully developed Poiseuille viscoelastic flow under finite pressure gradient described by the Oldroyd-B and Leonov constitutive equations. For analysis we employ a simple $2^{nd}$ order discretization scheme such as central difference for space and the Crank-Nicolson for time approximation. For the analysis of the Oldroyd-B model, we also apply the analytical solution, which is obtained again in this work in terms of elementary solution procedure simpler than the previous one (Waters and King, 1970). Both models demonstrate qualitatively similar solutions, but their eventual steady flowrate exhibits noticeable difference due to the absence or presence of shear thinning behavior. In the inertialess flow, the flowrate instantaneously attains a large value corresponding to the Newtonian creeping flow and then decreases to its steady value when the applied pressure gradient is low. However with finite liquid density the flow field shows severe fluctuation even accompanying reversals of flow directions. As the assigned pressure gradient increases, the flowrate achieves its steady value significantly higher than its value during oscillations after quite long period of time. We have also illustrated comparison between 1D and 2D results and possible mechanism of complex 2D flow rearrangement employing a previous solution of [mite element computation. In addition, we discuss some mathematical points regarding missing boundary conditions in 2D modeling due to the change of the type of differential equations when varying from inertialess to inertial flow.

• The Korean Journal of Rheology
• /
• v.2 no.1
• /
• pp.33-45
• /
• 1990

This study analyzes the planar 4:1 contraction flow of viscoelastic fluids with retardation time using finite volume method. To consider separately the elasticity effect of the viscoelastic fluid without shear thinn-ing effect, Oldroyd B liquid model is adopted for the numerical simulation. Instead of the stream function-vorticity formulation, SIMPLER algorithm with staggered grid system which incorporates primitive variable has been introduced in discretizing the momentum equations. An upwind corrected scheme has been used in discetizing the constitutive equations for the non-Newtonian part of the stress. The size of the corner vortex is shown to be slightly influenced by the Weissenberg number. However as the Weissenberg number is increased the chang-ing of the vortex shape agrees qualitatively well with some experimental studies.

• Korea-Australia Rheology Journal
• /
• v.19 no.2
• /
• pp.67-73
• /
• 2007

A finite volume method (FVM) base on the SIMPLE algorithm as the pressure correction strategy and the traditional staggered mesh is used to investigate steady, fully developed flow of Oldroyd-3-constant fluids through a duct with square cross-section. Both effects of the two viscoelastic material parameters, We and ${\mu}$, on pattern and strength of the secondary flow are investigated. An amusing sixteen vortices pattern of the secondary flow, which has never been reported, is shown in the present work. The reason for the changes of the pattern and strength of the secondary flow is discussed carefully. We found that it is variation of second normal stress difference that causes the changes of the pattern and strength of the secondary flow.

• Proceedings of the Korean Society of Rheology Conference
• /
• 2001.09a
• /
• pp.158.4-158
• /
• 2001

• Proceedings of the Korean Society for Technology of Plasticity Conference
• /
• 2006.05a
• /
• pp.99-104
• /
• 2006

A semi-Lagrangian finite element scheme with objective time stepping algorithm for solving viscoelastic flow problem is presented. The convection terms in the momentum and constitutive equations are treated using a quasi-monotone semi-Lagrangian scheme, in which characteristic feet on a regular grid are traced backwards over a single time-step. Concerned with the generalized midpoint rule type of algorithms formulated to exactly preserve objectivity, we use the geometric transformation such as pull-back, push-forward operation. The method is applied to the 4:1 planar contraction problem for an Oldroyd B fluid for both creeping and inertial flow conditions.

• Korea-Australia Rheology Journal
• /
• v.18 no.2
• /
• pp.41-49
• /
• 2006

A modified Volume-of-Fluid (VOF) numerical method is used to predict the dynamics of a liquid drop of a low viscosity dilute polymer solution, forming in air from a circular nozzle. Viscoelastic effects are rep-resented using an Oldroyd-B model. Predicted drop shapes are compared with experimental observations. The main features, including the timing of the shape evolution and the 'bead-on-a-string' effect, are well reproduced by the simulations. The results confirm published conclusions of the third author, that the deformation is effectively Newtonian until near the time of Newtonian pinch-off and that the elastic stress becomes large in the pinch region due to the higher extensional flow there.

• Korea-Australia Rheology Journal
• /
• v.21 no.1
• /
• pp.59-69
• /
• 2009

The prediction of pressure drop for a droplet flow in a confined micro channel is presented using FE-FTM (Finite Element - Front Tracking Method). A single droplet is passing through 5:1:5 contraction - straight narrow channel - expansion flow domain. The pressure drop is investigated especially when the droplet flows in the straight narrow channel. We explore the effects of droplet size, capillary number (Ca), viscosity ratio ($\chi$) between droplet and medium, and fluid elasticity represented by the Oldroyd-B constitutive model on the excess pressure drop (${\Delta}p^+$) against single phase flow. The tightly fitted droplets in the narrow channel are mainly considered in the range of $0.001{\leq}Ca{\leq}1$ and $0.01{\leq}{\chi}{\leq}100$. In Newtonian droplet/Newtonian medium, two characteristic features are observed. First, an approximate relation ${\Delta}p^+{\sim}{\chi}$ observed for ${\chi}{\geq}1$. The excess pressure drop necessary for droplet flow is roughly proportional to $\chi$. Second, ${\Delta}p^+$ seems inversely proportional to Ca, which is represented as ${\Delta}p^+{\sim}Ca^m$ with negative m irrespective of $\chi$. In addition, we observe that the film thickness (${\delta}_f$) between droplet interface and channel wall decreases with decreasing Ca, showing ${\delta}_f{\sim}Ca^n$ Can with positive n independent of $\chi$. Consequently, the excess pressure drop (${\Delta}p^+$) is strongly dependent on the film thickness (${\delta}_f$). The droplets larger than the channel width show enhancement of ${\Delta}p^+$, whereas the smaller droplets show no significant change in ${\Delta}p^+$. Also, the droplet deformation in the narrow channel is affected by the flow history of the contraction flow at the entrance region, but rather surprisingly ${\Delta}p^+$ is not affected by this flow history. Instead, ${\Delta}p^+$ is more dependent on ${\delta}_f$ irrespective of the droplet shape. As for the effect of fluid elasticity, an increase in ${\delta}_f$ induced by the normal stress difference in viscoelastic medium results in a drastic reduction of ${\Delta}p^+$.