Systems analysis of hybrid, multi-scale complex flow simulations using Newton-GMRES

Abstract

We present a methodology to analyze the stationary states and stability of complex fluid flows by using hybrid, discrete, and/or continuum multi-scale simulations. Building on existing theories, our scheme extracts dynamical and equilibrium characteristics from carefully chosen time integrations of these multi-scale evolution equations. Two canonical problems are presented to demonstrate the ability and accuracy of the formalism. The first is an investigation of flow-induced transitions seen in homogeneous, hard- rod liquid crystal suspensions subjected to a linear shear flow. In the second problem, we study the phenomenon of draw resonance, a dynamical instability in an isothermal fiber-spinning process, by using a multi-scale hybrid simulation that incorporates both stochastic and continuum models.

Appendix: Numerical discretization

The partial differential equations for the fiber-spinning process are solved by discretizing the computational domain z ∈ [0, 1] into N elements and time domain t ∈ [0,T _f ] in N _T steps. We then use the finite element method to discretize spatially the mass and momentum balances and the equation for the velocity gradient G. Continuous, quadratic basis functions φ _i are used for A and v _z , whereas continuous linear basis functions ψ _i are used for G. In contrast, the polymer contributions to the stress tensor, τ _{p, zz} and τ _{p, rr} , and configuration fields, Q _{z, i} and Q _{r, i} , are discretized in space by using the discontinuous Galerkin method with local linear discontinuous basis functions $\psi_{\text{DG},\,l}$. For time discretization, all terms are treated explicitly to yield the following algorithm.

• 1. Given A ^k, $\tau_{p,\,zz}^k$, $\tau_{p,\,rr}^k$, and the initial guess $v_z^k$ and G ^k, update the velocity and velocity gradient by solving the following nonlinear problem for the weak formulation

  $$\begin{array}{lll} \int_0^1\left(\frac{\partial A^k}{\partial z}\left[-3\frac{\partial v_z^{k+1}}{\partial z}+3(1-\beta)G^{k+1}\right.\right. \\ \quad\qquad\qquad+\left.(\tau_{p,\,zz}^k-\tau_{p,\,rr}^k)\vphantom{\frac{\partial v_z^{k+1}}{\partial z}}\right] \\ \quad\quad+\;A^k\left[-3\frac{\partial^2v_z^{k+1}}{\partial z^2}+3(1-\beta)\frac{\partial G^{k+1}}{\partial z}\right. \\ \qquad\qquad\quad+\left.\left.\frac{\partial}{\partial z}\left[\tau_{p,\,zz}^k-\tau_{p,\,rr}^k\right]\vphantom{\frac{\partial^2v_z^{k+1}}{\partial z^2}}\right]\right)\phi_i\,\textrm{d}z = 0 \\ \int_0^1\Bigl(G^{k+1}-\frac{\partial v_z^{k+1}}{\partial z}\Bigr)\psi_i\,\textrm{d}z = 0 \end{array}$$

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• 2. Given the new velocity field update the area as follows

  $$\begin{array}{lll} \int_0^1A^{k+1}\phi_i\,\textrm{d}z \notag\\ \qquad\qquad= \int_0^1A^k\phi_i\,\textrm{d}z-\Delta t\int_0^1\left(v_z^{k+1}\frac{\partial A^k}{\partial z} +A^k\frac{\partial v_z^{k+1}}{\partial z}\right)\phi_i\,\textrm{d}z\end{array}$$

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• 3. Update the polymer contributions to the stress tensor for the Oldroyd-B model by using the following weak formulations over each element [z _j ,z _j + 1] of the domain

  $$\begin{array}{lll} \int_{z_j}^{z_{j+1}}\text{De}\tau^{k+1}_{p,\,zz}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \qquad= \int_{z_j}^{z_{j+1}}\text{De}\tau^k_{p,\,zz}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \qquad\qquad +\;\Delta t\int_{z_j}^{z_{j+1}}\left(-2(1-\beta)G^{k+1} -\tau^k_{p,\,zz}\vphantom{\frac{\partial\tau^k_{p,\,zz}}{\partial z}}\right.\\ \qquad\qquad\qquad +\;2\text{De}G^{k+1}\tau^k_{p,\,zz}\\ \qquad\qquad\qquad-\left.v_z^{k+1}\frac{\partial\tau^k_{p,\,zz}}{\partial z}\right)\psi_{\text{DG},\,l}\,\:\textrm{d}z\:+ {\mathcal{Z}}_{1}\\ \qquad\times\begin{cases} {\mathrm{For}}\:\:\: v_z^{k+1}\Bigr\lvert_{z_j}\:\:\:\:>0 \\[6pt] {\mathcal{Z}}_{1} = \Delta t\:\text{De}\left({\psi_{\text{DG},\,l}}v_z^{k+1}(\tau_{p,\,zz}^{k,\text{ex}}-\tau_{p,\,zz}^{k,\text{in}})\right)\Bigr\lvert_{z_j}\\[6pt] {\mathrm{For}}\:\:\: v_z^{k+1}\Bigr\lvert_{z_{j+1}}<0\\[6pt] {\mathcal{Z}}_{1} = -\Delta t\:\text{De}\left({\psi_{\text{DG},\,l}}v_z^{k+1}(\tau_{p,\,zz}^{k,\text{ex}}-\tau_{p,\,zz}^{k,\text{in}})\right)\Bigr\lvert_{z_{j+1}} \end{cases}\\ \end{array}$$

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  $$\begin{array}{lll} \int_{z_j}^{z_{j+1}}\text{De}\tau^{k+1}_{p,\,rr}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \;\;= \int_{z_j}^{z_{j+1}}\text{De}\tau^k_{p,\,rr}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \qquad +\Delta t\int_{z_j}^{z_{j+1}}\left((1-\beta)G^{k+1} -\tau^k_{p,\,rr}-\text{De}G^{k+1}\tau^k_{p,\,rr}\vphantom{\frac{\partial\tau^k_{p,\,rr}}{\partial z}}\right.\\ \qquad\qquad\qquad-\left.v_z^{k+1}\frac{\partial\tau^k_{p,\,rr}}{\partial z}\right)\psi_{\text{DG},\,l}\,\textrm{d}z\:+\:{\mathcal{Z}}_{2}\\ \qquad\times\begin{cases} {\mathrm{For}} \:\:\: v_z^{k+1}\Bigr\lvert_{z_j}\:\:\:\:>0 \\[4pt] {\mathcal{Z}}_{2} = \Delta t\:\text{De}\left({\psi_{\text{DG},\,l}}v_z^{k+1}(\tau_{p,\,rr}^{k,\text{ex}}-\tau_{p,\,rr}^{k,\text{in}})\right)\Bigr\lvert_{z_j}\\[4pt] {\mathrm{For}}\:\:\:v_z^{k+1}\Bigr\lvert_{z_{j+1}}<0 \\[4pt] {\mathcal{Z}}_{2} = -\Delta t\:\text{De}\left({\psi_{\text{DG},\,l}}v_z^{k+1}(\tau_{p,\,rr}^{k,\text{ex}}-\tau_{p,\,rr}^{k,\text{in}})\right)\Bigr\lvert_{z_{j+1}} \end{cases} \end{array}$$

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  For the Hookean dumbbell model, instead, we lift $\tau_{p,\,zz}^k$ and $\tau_{p,\,rr}^k$ to a consistent ensemble of molecular configurations $\{Q^k_{z,\,i},Q^k_{r,\,i}\}$ that are then evolved using the following weak formulations

  $$\begin{array}{lll} \int_{z_j}^{z_{j+1}}Q^{k+1}_{z,\,i}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \;\;=\int_{z_j}^{z_{j+1}}Q^{k}_{z,\,i}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \quad+\;\Delta t\int_{z_j}^{z_{j+1}}\left[-v_z^{k+1}\frac{\partial Q^k_{z,\,i}}{\partial z}+G^{k+1}Q^k_{z,\,i} \right.\\ \qquad\qquad\qquad-\left.\frac{1}{2\text{De}}Q^k_{z,\,i}\vphantom{\frac{\partial Q^k_{z,\,i}}{\partial z}}\right]\psi_{\text{DG},\,l}\,\textrm{d}z \\ \quad+\Delta W_{z,\,i}\int_{z_j}^{z_{j+1}}\psi_{\text{DG},\,l}\,\textrm{d}z\:+\:{\mathcal{Z}}_{3}\\ \quad\times\begin{cases} {\mathrm{For}}\:\:\:v_z^{k+1}\Bigr\lvert_{z_j}>0 \\ {\mathcal{Z}}_{3} = \Delta t\:\left({\psi_{\text{DG},\,l}}v_z^{k+1}(Q^{k,\text{ex}}_{z,\,i}-Q^{k,\text{in}}_{z,\,i})\right)\Bigr\lvert_{z_j}\\ {\mathrm{For}}\:\:\:v_z^{k+1}\Bigr\lvert_{z_{j+1}}<0 \\ {\mathcal{Z}}_{3} = -\Delta t\:\left({\psi_{\text{DG},\,l}}v_z^{k+1}(Q^{k,\text{ex}}_{z,\,i}-Q^{k,\text{in}}_{z,\,i})\right)\Bigr\lvert_{z_{j+1}} \end{cases} \end{array}$$

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  $$\begin{array}{lll} \int_{z_j}^{z_{j+1}}Q^{k+1}_{r,\,i}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \;\;=\int_{z_j}^{z_{j+1}}Q^{k}_{r,\,i}\psi_{\text{DG},\,l}\,\textrm{d}z\\ \quad +\Delta t\int_{z_j}^{z_{j+1}}\left[-v_z^{k+1}\frac{\partial Q^k_{r,\,i}}{\partial z}-\frac{G^{k+1}}{2}Q^k_{r,\,i} \right.\\ \qquad\qquad\qquad-\left.\frac{1}{2\text{De}}Q^k_{r,\,i}\vphantom{\frac{\partial Q^k_{r,\,i}}{\partial z}}\right]\psi_{\text{DG},\,l}\,\textrm{d}z\\ \quad +\Delta W_{r,\,i}\int_{z_j}^{z_{j+1}}\psi_{\text{DG},\,l}\,\textrm{d}z\:+\: {\mathcal{Z}}_{4}\\ \quad\times\begin{cases} {\mathrm{For}}\:\:\:v_z^{k+1}\Bigr\lvert_{z_j}>0 \\[6pt] {\mathcal{Z}}_{4} = \Delta t \:\left({\psi_{\text{DG},\,l}}v_z^{k+1}(Q^{k,\text{ex}}_{r,\,i}-Q^{k,\text{in}}_{r,\,i})\right) \Bigr\lvert_{z_j}\\[6pt] {\mathrm{For}}\:\:\:v_z^{k+1}\Bigr\lvert_{z_{j+1}}<0 \\[6pt] {\mathcal{Z}}_{4} = -\Delta t\:\left({\psi_{\text{DG},\,l}}v_z^{k+1}(Q^{k,\text{ex}}_{r,\,i}-Q^{k,\text{in}}_{r,\,i})\right)\Bigr\lvert_{z_{j+1}} \end{cases}\\ \end{array}$$

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  For the Hookean dumbbell simulation, another ensemble of equilibrium configurations $\{\hat{Q}^k_{z,\,i},\hat{Q}^k_{r,\,i}\}$ are also initialized that have the same initial configuration as $\{Q^k_{z,\,i},Q^k_{r,\,i}\}$. These are then subjected to the same Wiener process and evolved according to a similar weak form with $v_z^{k+1}$ and G ^k + 1 equal to zero. The evolved configurations for the flow field and quiescent state are then used to evaluate the updated stress field, $\tau_{p,\,zz}^{k+1}$ and $\tau_{p,\,rr}^{k+1}$.

• 4. Repeat steps 1–3 with A ^k + 1, $\tau_{p,\,zz}^{k+1}$, $\tau_{p,\,rr}^{k+1}$, $v_z^{k+1}$, and G ^k + 1 until t = T _f .

In the discontinuous Galerkin method, the weak form of the advection term has been replaced with the sum of an integral over the element [z _j ,z _j + 1] and a jump term that is only evaluated at inflow boundaries. Specifically, terms labeled $\tau_{p,\,rr}^{k,\text{ex}}$ are evaluated upstream of a given element, while terms labeled $\tau_{p,\,rr}^{k,\text{in}}$ are evaluated within the element itself. If a boundary, z _j or z _j + 1 is not an inflow boundary, the jump term at that boundary is equal to zero. Also, in the weak formulation for the Hookean dumbbell configuration, the same Wiener process (ΔW _{z, i} or ΔW _{r, i} ) is applied to all points in space. In writing the weak formulation for the dumbbell configuration, the Wiener process has been scaled by $\sqrt{\Delta t/\text{De}}$.