A material point method for simulation of viscoelastic flows

Abstract

A novel variant of the Material Point Method is proposed to solve flow problems for incompressible viscoelastic fluids. As in the original spirit of the method, material points possess all necessary information about the material constitutive behavior and move with the flow. They are additionally employed as integration points, whose weights are computed as the system evolves in order to ensure exact integration of the discretized conservation equations. The method is shown to achieve quadratic convergence for transient start-up flow of the Oldroyd-B fluid. In addition, we apply the method to flow of a Giesekus fluid in concentric rotating cylinders, and in flow through an abrupt 4:1 contraction, comparing favorably to available analytical, experimental, and simulation results as appropriate.

Appendix

1.1 Analytical solution for Giesekus fluid in Poiseuille flow

The analytical solutions utilized in enforcing consistent velocity and material stress profiles at the entrance of an inflow channel were derived by Yoo et al. and are summarized here. The solutions correspond to a fluid flowing through a channel of height H at an average flow rate of $\langle v_x \rangle$. The material properties are defined relaxation time $\lambda$, polymeric viscosity $\eta _{p}$, and mobility parameter $\alpha$, and the flow is characterized by a Weissenberg number $Wi = \frac{\lambda \langle v_{x} \rangle }{H}$. The coordinate y in the expressions is non-dimensionalized by the H. The solutions are

$$\begin{aligned} v_{x}(y)= & {} c_2 \left[ (1-2c_{0}^2) \ln \left( \frac{c_0 + \sqrt{1-\phi ^{2}y^{2}}}{c_0 + c_3}\right) \right. \nonumber \\&\left. +\, c_4\left( \frac{1}{c_0 +\sqrt{1-\phi ^{2}y^{2}}} - \frac{1}{c_0 + c_3} \right) \right] \end{aligned}$$

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$$\begin{aligned} \sigma _{xx}(y)= & {} s_0 \left[ \frac{(1-\alpha )(1 - \sqrt{1 - \phi ^{2}y^{2}}) + \frac{1}{2}\phi ^{2}y^{2}}{c_0 + \sqrt{1 - \phi ^{2}y^{2}}} \right] \end{aligned}$$

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$$\begin{aligned} \sigma _{yy}(y)= & {} \frac{s_0}{2} \left( \sqrt{1 - \phi ^{2}y^{2}} - 1 \right) \end{aligned}$$

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$$\begin{aligned} \sigma _{xy}(y)= & {} -\frac{s_0}{2} \phi y \end{aligned}$$

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The constants $\phi , c_0, c_2, c_3, c_4$ and $s_0$ in the above expressions are

$$\begin{aligned} \phi= & {} 2\alpha Wi (-\frac{\partial p}{\partial x})\end{aligned}$$

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$$\begin{aligned} c_0= & {} 2\alpha - 1 \end{aligned}$$

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$$\begin{aligned} c_{2}= & {} \frac{\langle v_{x}\rangle }{\phi Wi} \end{aligned}$$

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$$\begin{aligned} c_{3}= & {} \sqrt{1 - \phi ^{2}} \end{aligned}$$

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$$\begin{aligned} c_{4}= & {} 4\alpha c_{0} (1-\alpha )\end{aligned}$$

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$$\begin{aligned} s_{0}= & {} \frac{\eta _{p}}{\lambda \alpha } \end{aligned}$$

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Full specification of the solution requires solving for the pressure gradient, $\frac{\partial p}{\partial x}$ (or $\alpha$) for a given value of the average flow rate. This is accomplished by solving the nonlinear equation for a given input flow rate,

$$\begin{aligned} Wi \left( 1 - \frac{1}{\langle v_{x} \rangle } \int _{0}^{1} v_{x}(y;\alpha ) \right) = 0 \end{aligned}$$

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