A finite element - discrete element model
for viscoelastic suspension

M.Henry, J.Lambrechts, V.Legat

www.migflow.be

An Unresolved DEM-FEM model

  • Solid phase
     Discrete element
     Lagrangian representation
     Insight in contacts physics
  • Fluid phase
     Continuous medium
     Eulerian representation
     Computational convenience
  • Fluid-grains interaction
     Empirical momentum transfer

Parametrization are required

  • Averaged Navier-Stokes equations, $\mathbf{u} = (1-\phi) \mathbf{v}$, $\mathbf{u}^s = \phi \mathbf{v}^s$
    • \[\begin{aligned} \nabla \cdot \mathbf{u} + \nabla \cdot \mathbf{u}^s &= 0\\ \rho\left(\frac{\partial \mathbf{u}}{\partial t} + \nabla \cdot \frac{\mathbf{u u}}{1-\phi}\right) &= \nabla \cdot \left(-p\mathbf{I} + 2\eta^*\mathbf{D}\right) + \mathbf{f} + (1-\phi) \rho \mathbf{g}\\ \end{aligned} \]
  • Drag parametrization
    • \[ \mathbf{f} = g(\phi) C_d \pi r^2 \frac{\rho}{2} \left\lVert \mathbf{v^s} - \mathbf{v} \right\rVert \left( \mathbf{v}^s - \mathbf{v} \right) \]
  • Subgrid scale viscous model
    • \[ \eta^* = \eta\ \underbrace{(1+2.5\phi + 5\phi^2)}_{\text{Batchelor model}} \]

An implicit method for the contacts

  • Newton's second law
    • \[ \frac{d (m_k \mathbf{u}_k)}{d t} = m_k\mathbf{g} - \mathbf{f}_k \]
  • Dynamics equation
    • \[ \mathbf{M}\Delta \mathbf{u} = \mathbf{p} \]
  • Local Frame
    • \[ \mathbf{U} = \mathbf{H}\mathbf{u} \hspace{5mm} \mathbf{P} = \mathbf{H} \mathbf{p} \]
  • Solve for $\mathbf{P}$
    • \[ \begin{pmatrix} \Delta U_n \\ \Delta U_t \end{pmatrix} = \underbrace{\begin{pmatrix} w_{nn} & 0 \\ 0 & w_{tt} \end{pmatrix}}_{ \mathbf{H}\mathbf{M}^{-1}\mathbf{H}^T} \begin{pmatrix} P_n \\ P_t \end{pmatrix} \]
  • Normal and tangential components
    • \[ \begin{aligned} P_n &= \frac{1}{w_{nn}}\text{max}(0, U_n - \frac{\delta}{\Delta t})\\ P_t &= \frac{1}{w_{tt}}\text{min}(|U_t|, \frac{w_{tt}}{w_{nn}}\mu |U_n|)\frac{U_t}{|U_t|} \end{aligned} \]
  • Update velocities
    • \[ \mathit{u} \rightarrow \mathit{u} + \mathbf{M}^{-1}\mathbf{H}^T \mathbf{P} \]

Objectives

Is unresolved approach suitable
to study visco-elastic properties ?
How to model cohesive interactions
and clustering for colloidal suspension ?

Effective viscosity under a simple shear flow

Effective viscosity follows Krieger curves

\[ \eta_r = \left(1 - \frac{\phi}{\phi_c}\right)^\alpha \]

Contact dynamics allows to study cohesive materials

    Forces can be added into a particle-particle contact to study agglomeration phenomenon.
    Force derivated from Morse's potential : \[U_{morse}(r) = D_e \left[1 - e^{-\beta d \left(\frac{r}{d}-1\right)}\right]^2\]
Constitutive law proposed by A.Yazdani[2017]

An over-simplified model
for platelet interaction

Thrombus formation by Simon Yans
Which particle-particle law should we used ?
How to deal with parameters uncertainty ?
How to activate cohesive law based on a unresolved model ?

Predefined bond for particle chains

    Each chain is pre-generated contacts with a cohesive spring force based on a FENE model : \[ \begin{aligned} F_{s} = \frac{k \Delta \mathbf{x}}{1 - \left(\frac{\Delta\mathbf{x}}{Q}\right)^2} \end{aligned} \]

Conclusion

  • Unresolved multi-scale model is suitable to study visco-elastic properties of suspension

  • Cohesive particles and elongated chain can be modelled by the contact dynamics

  • Is this toy model suitable to study hard sphere colloids ?