Nonlinear fluid dynamics description of non-Newtonian fluids

Abstract

Nonlinear hydrodynamic equations for visco-elastic media are discussed. We start from the recently derived fully hydrodynamic nonlinear description of permanent elasticity that utilizes the (Eulerian) strain tensor. The reversible quadratic nonlinearities in the strain tensor dynamics are of the “lower convected” type, unambiguously. Replacing the (often neglected) strain diffusion by a relaxation of the strain as a minimal ingredient, a generalized hydrodynamic description of viscoelasticity is obtained. This can be used to get a nonlinear dynamic equation for the stress tensor (sometimes called constitutive equation) in terms of a power series in the variables. The form of this equation and in particular the form of the nonlinear convective term is not universal but depends on various material parameters. A comparison with existing phenomenological models is given. In particular we discuss how these ad-hoc models fit into the hydrodynamic description and where the various non-Newtonian contributions are coming from.

Appendices

Appendix A. Generalization of the phenomenological equations

In Eqs. (4) and (5) we have omitted possible reversible crosscouplings between flow and strain dynamics

$$ X^{{{\left( {ph} \right)}}}_{{ij}} = - \alpha _{{ijkl}} \Psi _{{kl}} - \beta _{{ijkl}} A_{{kl}} $$

(A1)

$$ \sigma ^{{{\left( {ph} \right)}}}_{{ij}} = - \nu _{{ijkl}} A_{{kl}} + \beta _{{klij}} \Psi _{{kl}} $$

(A2)

characterized by the tensor β _ijkl , which is of a slightly more complicated form than α _ijkl in Eq. (6). Being reversible it is not derived from a potential and lacks the ij–kl symmetry. This results in two different components β_4a δ _ij U _kl +β_4b δ _kl U _ij , where, however, the latter (as well as β_3,5) vanish due to incompressibility. Thus we are left with four parameters β_1,2,4a,6. Such terms, coming with probably small reactive transport parameters, have to compete with the parameter free, symmetry required terms already being part of Eqs. (1) and (2). These reversible crosscouplings are possible in the viscoelastic case only, and are absent for permanent elasticity. In the latter case only diffusion $X^{{{\left( {ph} \right)}}}_{{ij}}$=D∇ _k (∇ _i Ψ _jk +∇ _j Ψ _ik ) is present.

We also list here the five third-order static elastic contributions that follow from an elastic energy quartic in the strain tensor

$$ \begin{array}{*{20}l} {{K^{{{\left( 3 \right)}}}_{{ijkl}} } \hfill} & { = \hfill} & {{K_{6} \delta _{{ij}} \delta _{{kl}} U_{{pp}} U_{{qq}} + K_{7} {\left( {{\left[ {\delta _{{ik}} \delta _{{jl}} + \delta _{{il}} \delta _{{jk}} } \right]}U_{{pp}} U_{{qq}} + \delta _{{ij}} \delta _{{kl}} U_{{pq}} U_{{pg}} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + K_{s} {\left( {\delta _{{ij}} U_{{kp}} U_{{lp}} + \delta _{{kl}} U_{{ip}} U_{{jp}} + \frac{1} {4}{\left[ {\delta _{{ik}} U_{{jl}} + \delta _{{jk}} U_{{il}} + \delta _{{il}} U_{{jk}} + \delta _{{jl}} U_{{ik}} } \right]}U_{{pp}} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + K_{9} {\left( {\delta _{{ik}} \delta _{{jl}} + \delta _{{il}} \delta _{{jk}} } \right)}U_{{pq}} U_{{pq}} + K_{{10}} {\left( {\delta _{{ik}} U_{{jp}} U_{{lp}} + \delta _{{jk}} U_{{ip}} U_{{lp}} + \delta _{{il}} U_{{jp}} U_{{kp}} + \delta _{{jl}} U_{{ip}} U_{{kp}} } \right)}} \hfill} \\ \end{array} $$

(A3)

and that are a generalization of the first and second-order terms kept above in the main text. A necessary stability condition involving the second-order modulus K ₂ requires 27K ₁(K ₉+K ₁₀)>$2K^{2}_{2}$.

Appendix B. Influence of the tensor traces

Here we discuss the influence of those phenomenological parameters in Eqs. (6) and (7) that are connected with the traces of the tensors involved, α_3,4,5,6, ν₆, and K _3,4,5, as well as β_1,2,4a,6 Eqs. (A1) and (A2) neglected in the main text. The phenomenological parts of the dynamic and static equations then read

$$ \begin{array}{*{20}l} {{X^{{{\left( {ph} \right)}}}_{{ij}} } \hfill} & { = \hfill} & {{ - \alpha _{1} \Psi _{{ij}} - \alpha _{2} {\left( {U_{{ik}} \Psi _{{jk}} + U_{{jk}} \Psi _{{ik}} } \right)} - \alpha _{3} \delta _{{ij}} \Psi _{{kk}} - \alpha _{4} {\left( {\delta _{{ij}} U_{{kl}} \Psi _{{kl}} + U_{{ij}} \Psi _{{kk}} } \right)} - \alpha _{5} \delta _{{ij}} U_{{kk}} \Psi _{u} } \hfill} \\ {{} \hfill} & {{} \hfill} & {{ - 2\alpha _{6} U_{{kk}} \Psi _{{ij}} - \beta _{1} A_{{ij}} - \beta _{2} {\left( {U_{{ik}} A_{{jk}} + U_{{jk}} A_{{ik}} } \right)} - \beta _{{4a}} \delta _{{ij}} U_{{kl}} A_{{kl}} - 2\beta _{6} U_{{kk}} A_{{ij}} } \hfill} \\ \end{array} $$

(B1)

$$ \begin{array}{*{20}l} {{\sigma ^{{{\left( {ph} \right)}}}_{{ij}} } \hfill} & { = \hfill} & {{ - \nu _{1} A_{{ij}} - \nu _{2} {\left( {U_{{ik}} A_{{jk}} + U_{{jk}} A_{{ik}} } \right)} - 2\nu _{6} A_{{ij}} U_{{kk}} + \beta _{1} \Psi _{{ij}} + \beta _{2} {\left( {\Psi _{{ik}} U_{{jk}} + \Psi _{{jk}} U_{{ik}} } \right)}} \hfill} \\ {{} \hfill} & {{} \hfill} & {{ + \beta _{{4a}} \Psi _{{kk}} U_{{ij}} + 2\beta _{6} \Psi _{{ij}} U_{{ll}} } \hfill} \\ \end{array} $$

(B2)

$$ \Psi _{{ij}} = K_{1} U_{{ij}} + 2K_{2} U_{{ik}} U_{{jk}} + K_{3} \delta _{{ij}} U_{{kk}} + K_{4} {\left( {\delta _{{ij}} U_{{kl}} U_{{kl}} + 2U_{{ij}} U_{{kk}} } \right)} + K_{5} \delta _{{ij}} U_{{kk}} U_{{ll}} $$

(B3)

The strain relaxation and the stress take the form

$$ \begin{array}{*{20}l} {{\frac{{D_{{ - 1}} }} {{Dt}}U_{{ij}} - A_{{ij}} } \hfill} & { = \hfill} & {{ - \tau ^{{ - 1}}_{1} U_{{ij}} - \tau ^{{ - 1}}_{2} U_{{ik}} U_{{jk}} - \tau ^{{ - 1}}_{3} \delta _{{ij}} U_{{kk}} - \tau ^{{ - 1}}_{4} \delta _{{ij}} U_{{kl}} U_{{kl}} - \tau ^{{ - 1}}_{5} \delta _{{ij}} U_{{kk}} U_{{ll}} } \hfill} \\ {{} \hfill} & {{} \hfill} & {{ - \tau ^{{ - 1}}_{6} U_{{ij}} U_{{kk}} - \beta _{1} A_{{ij}} - \beta _{2} {\left( {U_{{ik}} A_{{jk}} + U_{{jk}} A_{{ik}} } \right)} - \beta _{{4a}} \delta _{{ij}} U_{{kl}} A_{{kl}} - 2\beta _{6} U_{{kk}} A_{{ij}} } \hfill} \\ \end{array} $$

(B4)

$$ \begin{array}{*{20}l} {{\sigma _{{ij}} } \hfill} & { = \hfill} & {{ - \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1} U_{{ij}} + \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{2} U_{{ik}} U_{{jk}} - \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{3} \delta _{{ij}} U_{{kk}} + \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{3} U_{{ij}} U_{{kk}} - \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{4} \delta _{{ij}} U_{{kl}} U_{{kl}} } \hfill} \\ {{} \hfill} & {{} \hfill} & {{ - \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{5} \delta _{{ij}} U_{{kk}} U_{{ll}} - \nu _{1} A_{{ij}} - \nu _{2} {\left( {U_{{ik}} A_{{jk}} + U_{{jk}} A_{{ik}} } \right)} - 2\nu _{6} A_{{ij}} U_{{kk}} } \hfill} \\ \end{array} $$

(B5)

with the new parameters $\tau ^{{ - 1}}_{3} = \alpha _{1} K_{3} + \alpha _{3} K_{1} + 3\alpha _{3} K_{3}$, $\tau ^{{ - 1}}_{4} = \alpha _{1} K_{4} + 2\alpha _{3} K_{2} + 3\alpha _{3} K_{4} + \alpha _{4} K_{1}$, $\tau ^{{ - 1}}_{5} = 2\alpha _{3} K_{4} + \alpha _{4} K_{3} + \alpha _{5} K_{1} + \alpha _{1} K_{5} + 3\alpha _{3} K_{5} + 3\alpha _{5} K_{3} + 2\alpha _{6} K_{3}$, $\tau ^{{ - 1}}_{6}$=2α₂ K ₃+α₄ K ₁+2α₁ K ₄+3α₄ K ₃+2α₆ K ₁, $\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1}$=K ₁(1−β₁), $\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{3}$=K ₃(1−β₁), $\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{4}$=K ₄(1–β₁), $\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{2} = K^{'}_{2}$+2β₂ K ₁+2β₁ K ₂, $\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{3} = 2 \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{3}$ (1+β₂+(3/2)β_4a )−2$\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{4}$+K ₁(β_4a +2β₆), and $\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}^{'}_{5}$=K ₅(1−β₁)−2K ₃β₆.

The procedure to switch from the strain relaxation equation to an effective dynamic equation for the stress tensor is the same as described above in the main text. Due to the many new terms, it is more involved, and it is complicated additionally by the fact that the traces and the traceless parts of σ _ij and U _ij are related by different parameters (even in the linear case) as can be seen from the modified Eq. (12):

$$ \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1} U^{{(lin)}}_{{ij}} = - \sigma _{{ij}} - \nu _{1} A_{{ij}} + \frac{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{3} }} {{3 \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{3} + \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1} }}\delta _{{ij}} \sigma _{{kk}} $$

(B6)

We refrain from writing down more details, but discuss the structure of the final result

$$ \begin{array}{*{20}l} {{\tau _{1} \frac{{D_{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{a}}} }} {{Dt}}} \hfill} & { + \hfill} & {{\sigma _{{ij}} + B_{1} \delta _{{ij}} \sigma _{{kk}} = - \ifmmode\expandafter\tilde\else\expandafter\~\fi{\nu }_{\infty } A_{{ij}} - \nu _{1} \tau _{1} \frac{{D_{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{b}}} }} {{Dt}}A_{{ij}} + \frac{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{r}}} {{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1} }}\sigma _{{ik}} \sigma _{{jk}} + B_{2} \sigma _{{kk}} \frac{\partial } {{\partial t}}A_{{ij}} } \hfill} \\ {{} \hfill} & { + \hfill} & {{\frac{{\tau _{1} \nu _{2} }} {{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1} }}{\left( {{\left[ {\sigma _{{jk}} + B_{3} \delta _{{jk}} \sigma _{{pp}} + \nu _{1} A_{{jk}} } \right]}\frac{\partial } {{\partial t}}A_{{ik}} + {\left[ {\sigma _{{ik}} + B_{3} \delta _{{ik}} \sigma _{{qq}} + \nu _{1} A_{{ik}} } \right]}\frac{\partial } {{\partial t}}A_{{jk}} } \right)}} \hfill} \\ {{} \hfill} & { + \hfill} & {{\sigma _{{kk}} {\left( {B_{4} \sigma _{{ij}} + B_{5} A_{{ij}} } \right)} + \delta _{{ij}} {\left( {B_{6} \sigma _{{kl}} + B_{7} \sigma _{{kl}} A_{{kl}} + B_{8} A_{{kl}} A_{{kl}} + B_{9} \sigma _{{kk}} \sigma _{u} } \right)} + O{\left( 3 \right)}} \hfill} \\ \end{array} $$

(B7)

with $\ifmmode\expandafter\tilde\else\expandafter\~\fi{\nu }_{\infty }$=ν₁+K ₁τ₁(1−β₁)² and the tilded numbers ã, $\ifmmode\expandafter\tilde\else\expandafter\~\fi{b}$, and $\ifmmode\expandafter\tilde\else\expandafter\~\fi{r}$ being much more complicated than the untilded ones (Eqs. 16 and 17). There are structurally new terms related to σ _kk , a linear one (with B ₁=$\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{3}$/(3$\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{3}$+$\ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1}$)+τ₁/τ₃), and eight quadratic ones characterized by coefficients B _2,...,9 . Nevertheless, the trace of the stress tensor and its time derivative are of second-order and do not influence the dynamics of the traceless part in O(2). Thus, for the deviator, $\sigma ^{0}_{{ij}}$=σ _ij −(1/3)δ _ij σ _kk , the dynamic equation has exactly the form of Eq. (14), but with the tilded numbers and parameters instead of the untilded ones. For the trace we find to second-order

$$ \begin{array}{*{20}l} {{\tau _{1} \frac{d} {{dt}}\sigma _{{kk}} + {\left( {1 + 3B_{1} } \right)}\sigma _{{kk}} } \hfill} & { = \hfill} & {{2 \ifmmode\expandafter\tilde\else\expandafter\~\fi{b}\nu _{1} \tau _{1} A_{{kl}} A_{{kl}} + \frac{{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{r}}} {{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1} }}\sigma ^{0}_{{kl}} \sigma ^{0}_{{kl}} + \frac{{2\tau _{1} \nu _{2} }} {{ \ifmmode\expandafter\tilde\else\expandafter\~\fi{K}_{1} }}{\left( {\sigma ^{0}_{{kl}} + \nu _{1} A_{{kl}} } \right)}\frac{\partial } {{\partial t}}A_{{kl}} } \hfill} \\ {{} \hfill} & { + \hfill} & {{3{\left( {B_{6} \sigma ^{0}_{{kl}} \sigma ^{0}_{{kl}} + {\left[ {B_{7} + \frac{2} {3}\tau _{1} \ifmmode\expandafter\tilde\else\expandafter\~\fi{a}} \right]}\sigma ^{0}_{{kl}} A_{{kl}} + B_{8} A_{{kl}} A_{{kl}} } \right)} + O{\left( 3 \right)}} \hfill} \\ \end{array} $$

(B8)

indicating that its relaxational dynamics is completely given by the flow A _ij and the traceless part $\sigma ^{0}_{{ij}}$.