A new closure approximation based on the Cohen/Padé spring law and a novel method for its implementation

Abstract

A new closure-approximated model based on the Cohen/Padé (CP) spring law is presented. Because it is inspired by the finitely extensible non-linear elastic Peterlin version (FENE-P), the model is titled CP-P. The rheological behaviors in simple and pressure-driven pipe shear flows in both the steady-state and time-dependent conditions are compared between the two models. To this end, a new approach based on the second moment Fokker–Planck equation instead of the Langevin equation is developed, which was confirmed by means of the existing analytical and stochastic data prior to its use. Application of this new approach is restricted to the closure approximated models but at a much less computational expense than the CONNFFESSIT-like simulations. In addition, the Brownian configuration field method is employed to make verification possible and more importantly, to shed light on the rheological performance of the CP model. According to the results, the CP-P model behaves similarly to the FENE-P model for the shear flows. The transient rheological behavior of the CP model is significantly different from that of the CP-P model, but both are the same in steady state.

Appendices

Appendix 1

According to Table 1, for the Hookean dumbbells h(σ^*, b) = 1 which in combination with Eq. 14 simply compiles to

$$ {\left\langle {\boldsymbol{Q}}^{*}{\boldsymbol{Q}}^{*}\right\rangle}_0={\boldsymbol{\sigma}}_0^{*}=\boldsymbol{\delta} $$

(A1)

The initial conformation tensor of FENE-P and CP-P can be obtained similarly, but it needs more complex calculations,

FENE-P

$$ {\left\langle {\boldsymbol{Q}}^{*}{\boldsymbol{Q}}^{*}\right\rangle}_0={\boldsymbol{\sigma}}_0^{*}=\left(1-Tr\left({\mathtt{\sigma}}_0^{*}\right)/b\right)\mathtt{\delta}; $$

(A2)

$$ \Rightarrow Tr\left({\mathtt{\sigma}}_0^{*}\right)=3\left(1-Tr\left({\mathtt{\sigma}}_0^{*}\right)/b\right); $$

(A3)

$$ \Rightarrow Tr\left({\mathtt{\sigma}}_0^{*}\right)=\frac{3b}{b+3}; $$

(A4)

$$ \Rightarrow {\left\langle {\boldsymbol{Q}}^{*}{\boldsymbol{Q}}^{*}\right\rangle}_0={\mathtt{\sigma}}_0^{*}=\frac{b}{b+3}\mathtt{\delta} $$

(A5)

CP-P

$$ {\left\langle {\boldsymbol{Q}}^{*}{\boldsymbol{Q}}^{*}\right\rangle}_0={\mathtt{\sigma}}_0^{*}=\frac{3\left(1-Tr\left({\mathtt{\sigma}}_0^{*}\right)/b\right)}{\left(3-Tr\left({\mathtt{\sigma}}_0^{*}\right)/b\right)}\mathtt{\delta}; $$

(A6)

$$ \Rightarrow Tr\left({\mathtt{\sigma}}_0^{*}\right)=\frac{9\left(1-Tr\left({\mathtt{\sigma}}_0^{*}\right)/b\right)}{\left(3-Tr\left({\mathtt{\sigma}}_0^{*}\right)/b\right)}; $$

(A7)

$$ \Rightarrow {\left(Tr\left({\mathtt{\sigma}}_0^{*}\right)\right)}^2-\left(3b+9\right)Tr\left({\mathtt{\sigma}}_0^{*}\right)+9b=0; $$

(A8)

$$ \Rightarrow Tr\left({\mathtt{\sigma}}_0^{*}\right)=\frac{3b+9-\sqrt{{\left(3b+9\right)}^2-36b}}{2}; $$

(A9)

$$ \Rightarrow {\left\langle {\boldsymbol{Q}}^{*}{\boldsymbol{Q}}^{*}\right\rangle}_0={\mathtt{\sigma}}_0^{*}=\frac{3\sqrt{{\left(b+3\right)}^2-4b}-b-9}{\sqrt{{\left(b+3\right)}^2-4b}+b-3}\mathtt{\delta} $$

(A10)

Appendix 2

Applying Eq. 14 in matrix form when the fluid reaches steady state in the Couette flow yields

$$ \begin{array}{l}\left[\begin{array}{ccc}\hfill 0\hfill & \hfill -\frac{U}{L}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\cdot \left[\begin{array}{ccc}\hfill \left\langle {Q}_x^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_x^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_x^{*}{Q}_z^{*}\right\rangle \hfill \\ {}\hfill \left\langle {Q}_y^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_y^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_y^{*}{Q}_z^{*}\right\rangle \hfill \\ {}\hfill \left\langle {Q}_z^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_z^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_z^{*}{Q}_z^{*}\right\rangle \hfill \end{array}\right]+\left[\begin{array}{ccc}\hfill \left\langle {Q}_x^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_x^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_x^{*}{Q}_z^{*}\right\rangle \hfill \\ {}\hfill \left\langle {Q}_y^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_y^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_y^{*}{Q}_z^{*}\right\rangle \hfill \\ {}\hfill \left\langle {Q}_z^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_z^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_z^{*}{Q}_z^{*}\right\rangle \hfill \end{array}\right]\cdot \left[\begin{array}{ccc}\hfill 0\hfill & \hfill -\frac{U}{L}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\\ {}+\left[\begin{array}{ccc}\hfill \frac{1}{\lambda_{\mathrm{H}}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill \frac{1}{\lambda_{\mathrm{H}}}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1}{\lambda_{\mathrm{H}}}\hfill \end{array}\right]-\frac{h}{\lambda_{\mathrm{H}}}\left[\begin{array}{ccc}\hfill \left\langle {Q}_x^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_x^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_x^{*}{Q}_z^{*}\right\rangle \hfill \\ {}\hfill \left\langle {Q}_y^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_y^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_y^{*}{Q}_z^{*}\right\rangle \hfill \\ {}\hfill \left\langle {Q}_z^{*}{Q}_x^{*}\right\rangle \hfill & \hfill \left\langle {Q}_z^{*}{Q}_y^{*}\right\rangle \hfill & \hfill \left\langle {Q}_z^{*}{Q}_z^{*}\right\rangle \hfill \end{array}\right]=0\end{array} $$

(B1)

This provides nine equations as follow:

$$ -\frac{U}{L}\left\langle {Q}_y^{*}{Q}_x^{*}\right\rangle -\left\langle {Q}_x^{*}{Q}_y^{*}\right\rangle \frac{U}{L}+\frac{1}{\lambda_{\mathrm{H}}}-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_x^{*}{Q}_x^{*}\right\rangle =0 $$

(B2)

$$ -\frac{U}{L}\left\langle {Q}_y^{*}{Q}_y^{*}\right\rangle +0+0-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_x^{*}{Q}_y^{*}\right\rangle =0 $$

(B3)

$$ -\frac{U}{L}\left\langle {Q}_y^{*}{Q}_z^{*}\right\rangle +0+0-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_x^{*}{Q}_z^{*}\right\rangle =0 $$

(B4)

$$ 0-\left\langle {Q}_y^{*}{Q}_y^{*}\right\rangle \frac{U}{L}+0-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_y^{*}{Q}_x^{*}\right\rangle =0 $$

(B5)

$$ 0+0+\frac{1}{\lambda_{\mathrm{H}}}-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_y^{*}{Q}_y^{*}\right\rangle =0;\Rightarrow \left(\left\langle {Q}_y^{*}{Q}_y^{*}\right\rangle =\frac{1}{h}\right) $$

(B6)

$$ 0+0+0-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_y^{*}{Q}_z^{*}\right\rangle =0;\Rightarrow \left(\left\langle {Q}_y^{*}{Q}_z^{*}\right\rangle =0\right) $$

(B7)

$$ 0-\left\langle {Q}_z^{*}{Q}_y^{*}\right\rangle \frac{U}{L}+0-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_z^{*}{Q}_x^{*}\right\rangle =0 $$

(B8)

$$ 0+0+0-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_z^{*}{Q}_y^{*}\right\rangle =0;\Rightarrow \left(\left\langle {Q}_z^{*}{Q}_y^{*}\right\rangle =0\right) $$

(B9)

$$ 0+0+\frac{1}{\lambda_{\mathrm{H}}}-\frac{h}{\lambda_{\mathrm{H}}}\left\langle {Q}_z^{*}{Q}_z^{*}\right\rangle =0;\Rightarrow \left(\left\langle {Q}_z^{*}{Q}_z^{*}\right\rangle =\frac{1}{h}\right) $$

(B10)