To investigate hyperbolicity of the six-equation model (
2.16) at pressure equilibrium closed by (
2.17) and (
2.19), we first rewrite the system in terms of the primitive variables volume fraction
\(\alpha _1\), equilibrium pressure
p, velocities
\(v_k\) and densities
\(\rho _k\). Let
$$\begin{aligned} \mathbf {u}^\mathrm{T}=(u_1,u_2,u_3,u_4,u_5,u_6)=(\alpha _1\rho _1, \alpha _1\rho _1v_1,\alpha _1\rho _1E_1,\alpha _2\rho _2, \alpha _2\rho _2v_2,\alpha _2\rho _2E_2) \end{aligned}$$
denote the conserved quantities of the system (
2.16). Then, the primitive variables can be obtained as follows.
$$\begin{aligned}&v_1=\frac{u_2}{u_1}, \quad v_2=\frac{u_5}{u_4}, \quad \alpha _1\rho _1e_1=u_3-\frac{v_1u_2,}{2} \quad \alpha _2\rho _2e_2=u_6-\frac{v_2u_5}{2}, \end{aligned}$$
(3.1a)
$$\begin{aligned}&p=(\gamma _1-1)\alpha _1\rho _1e_1+(\gamma _2-1)\alpha _2\rho _2e_2, \quad {\alpha _1=\frac{p}{(\gamma _1-1)\rho _1e_1}} . \end{aligned}$$
(3.1b)
From (
2.1), we derive the evolution equations for the primitive variables:
$$\begin{aligned}&\frac{\partial \,\alpha _1}{\partial \,t}+ \frac{{1}}{G^0} \left( \left( \frac{z_2 v_2}{\alpha _2} + \frac{z_1 v_1}{\alpha _1}\right) \frac{\partial \,\alpha _1}{\partial \,x} + \left( v_1-v_2 \right) \frac{\partial \,p}{\partial \,x} + \rho _1 c_1^2 \frac{\partial \,v_1}{\partial \,x} - \rho _2 c_2^2 \frac{\partial \,v_2}{\partial \,x} \right) = \mathcal{A}, \end{aligned}$$
(3.2a)
$$\begin{aligned}&\frac{\partial \,p}{\partial \,t}+ \frac{1}{{\alpha _1\,\alpha _2\,}G^0} \left( z_1 z_2 \left( v_1 - v_2\right) \frac{\partial \,\alpha _1}{\partial \,x} + \left( \alpha _1 z_2 v_1 + \alpha _2 z_1 v_2 \right) \frac{\partial \,p}{\partial \,x} \right. \nonumber \\&\quad +\left. \alpha _1 z_2 \rho _1 c_1^2 \frac{\partial \,v_1}{\partial \,x} + \alpha _2 z_1 \rho _2 c_2^2 \frac{\partial \,v_2}{\partial \,x} \right) = \mathcal{P}, \end{aligned}$$
(3.2b)
$$\begin{aligned}&\frac{\partial \,v_1}{\partial \,t} - \frac{{P}-p}{\alpha _1\rho _1} \frac{\partial \,\alpha _1}{\partial \,x} + \tau _1 \frac{\partial \,p}{\partial \,x} + v_1 \frac{\partial \,v_1}{\partial \,x} = \mathcal{V}_1 \end{aligned}$$
(3.2c)
$$\begin{aligned}&\frac{\partial \,v_2}{\partial \,t} + \frac{{P}-p}{\alpha _2\rho _2} \frac{\partial \,\alpha _1}{\partial \,x} + \tau _2 \frac{\partial \,p}{\partial \,x} + v_2 \frac{\partial \,v_2}{\partial \,x} = \mathcal{V}_2 \end{aligned}$$
(3.2d)
$$\begin{aligned}&\frac{\partial \,\rho _1}{\partial \,t}+ \frac{1}{{\alpha _1\,\alpha _2\,}G^0} \left( \rho _1 z_2 \left( v_1 - v_2\right) \frac{\partial \,\alpha _1}{\partial \,x} - \rho _1 \alpha _2 \left( v_1 - v_2\right) \frac{\partial \,p}{\partial \,x} \right. \end{aligned}$$
(3.2e)
$$\begin{aligned}&\quad + \left. \rho _1 \left( \alpha _1 \rho _2 c_2^2 + \left( {P}-p\right) {\overline{\kappa }}\right) \frac{\partial \,v_1}{\partial \,x} + \alpha _2 \rho _1 c_1^2 \frac{\partial \,v_2}{\partial \,x} \right) + v_1 \frac{\partial \,\rho _1}{\partial \,x} = \mathcal{R}_1 , \nonumber \\&\frac{\partial \,\rho _2}{\partial \,t}+ \frac{1}{{\alpha _1\,\alpha _2\,}G^0} \left( \rho _2 z_1 \left( v_1 - v_2\right) \frac{\partial \,\alpha _1}{\partial \,x} + \rho _2 \alpha _1 \left( v_1 - v_2\right) \frac{\partial \,p}{\partial \,x} \right. \nonumber \\&\quad + \left. \alpha _1 \rho _2 c_2^2 \frac{\partial \,v_1}{\partial \,x} + \rho _2 \left( \alpha _2 \rho _1 c_1^2 + \left( {P}-p\right) {\overline{\kappa }}\right) \frac{\partial \,v_2}{\partial \,x} \right) + v_2 \frac{\partial \,\rho _2}{\partial \,x} = \mathcal{R}_2 \end{aligned}$$
(3.2f)
with relaxation terms
$$\begin{aligned}&\mathcal{A}= \frac{\mathcal{C}}{\varrho } +\frac{{\mathbb E}_p^0}{G^0} , \end{aligned}$$
(3.3a)
$$\begin{aligned}&\mathcal{P}= \frac{1}{\alpha _1} \left( \kappa _1{\mathbb E}_{01}^0 - \frac{p}{\varrho } \mathcal{C}\right) + \frac{1}{\alpha _2} \left( \kappa _2{\mathbb E}_{02}^0 + \frac{p}{\varrho } \mathcal{C}\right) , \end{aligned}$$
(3.3b)
$$\begin{aligned}&\mathcal{V}_k = (-1)^k \frac{1}{\alpha _k \rho _k}\left( v_k\mathcal{C}-\mathcal{M}\right) , \end{aligned}$$
(3.3c)
$$\begin{aligned}&\mathcal{R}_k =(-1)^k \frac{\rho _k}{\alpha _k} \left( \frac{{\mathbb E}_p^0}{G^0} - \left( \frac{1}{\rho _k}-\frac{1}{\varrho }\right) \mathcal{C}\right) . \end{aligned}$$
(3.3d)
Here, we use the notation
$$\begin{aligned}&{\overline{\gamma }}:= \alpha _2 \gamma _1+ \alpha _1 \gamma _2 , \end{aligned}$$
(3.4a)
$$\begin{aligned}&{\overline{\kappa }}:= \alpha _2 \kappa _1 + \alpha _1 \kappa _2 , \end{aligned}$$
(3.4b)
$$\begin{aligned}&\varDelta := {P}-p , \end{aligned}$$
(3.4c)
$$\begin{aligned}&c^2_k:= \frac{\gamma _k p}{\rho _k} , \end{aligned}$$
(3.4d)
$$\begin{aligned}&z_k:= \gamma _k p + \varDelta \kappa _k . \end{aligned}$$
(3.4e)
Then, we can rewrite
\(G^0\) as
$$\begin{aligned} G^0 = \frac{p {\overline{\gamma }}+ \varDelta {\overline{\kappa }}}{{\alpha _1\,\alpha _2}} . \end{aligned}$$
In compact form, the system of primitive variables can be written in quasi-conservative form as
$$\begin{aligned} \frac{\partial \,{{{\varvec{w}}}}}{\partial \,t} + {{{\varvec{J}}}}({{{\varvec{w}}}}) \frac{\partial \,{{{\varvec{w}}}}}{\partial \,x} = {{{\varvec{S}}}}({{{\varvec{w}}}}) \end{aligned}$$
with
\({{{\varvec{w}}}}:=(\alpha _1,p,v_1,v_2,\rho _1,\rho _2)^\mathrm{T}\) and
\({{{\varvec{S}}}}:=(\mathcal{A},\mathcal{P},\mathcal{V}_1,\mathcal{V}_2,\mathcal{R}_1,\mathcal{R}_2)^\mathrm{T}\) the vector of primitive variables and relaxation terms, respectively. The matrix
\({{{\varvec{J}}}}\) is represented by a
\(2\times 2\)-block matrix
$$\begin{aligned} {{{\varvec{J}}}}= \left( \begin{matrix} {{{\varvec{J}}}}_4 &{} {{\varvec{*}}}_{4\times 2} \\ {{{\varvec{0}}} }_{2\times 4} &{} {\text{ diag }}(v_1,v_2) \end{matrix} \right) , \end{aligned}$$
(3.5)
where the principal part
\({{{\varvec{J}}}}_4\) can be split into two parts separating terms depending and not depending on
\(\varDelta \)$$\begin{aligned} {{{\varvec{J}}}}_4 = \begin{pmatrix} j_{11} &{}\quad j_{12} &{}\quad j_{13} &{}\quad j_{14} \\ j_{21} &{}\quad j_{22} &{}\quad j_{23} &{}\quad j_{24} \\ 0 &{}\quad j_{32} &{}\quad v_{1} &{}\quad 0 \\ 0 &{}\quad j_{42} &{}\quad 0 &{}\quad v_{2} \\ \end{pmatrix} + f(\varDelta ) \begin{pmatrix} \tilde{j}_{11} &{}\quad \tilde{j}_{12} &{}\quad \tilde{j}_{13} &{}\quad \tilde{j}_{14} \\ \tilde{j}_{21} &{}\quad \tilde{j}_{22} &{}\quad \tilde{j}_{23} &{}\quad \tilde{j}_{24} \\ \tilde{j}_{31} &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \tilde{j}_{41} &{}\quad 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} \end{aligned}$$
with
$$\begin{aligned} f(\varDelta ):=\frac{\varDelta }{\varDelta {\overline{\kappa }}+ p {\overline{\gamma }}}. \end{aligned}$$
Introducing the convex parameters
$$\begin{aligned} \beta _1 := \frac{\alpha _1 \gamma _2 }{{\overline{\gamma }}},\quad \beta _2 := \frac{\alpha _2 \gamma _1 }{{\overline{\gamma }}},\quad \beta _1+\beta _2=1,\quad \beta _i\in [0,1], \quad i=1,2, \end{aligned}$$
(3.6)
the matrix entries can be written as
$$\begin{aligned}&j_{11} = \beta _2 v_1+\beta _1 v_2, \quad j_{12} = \frac{{\overline{\gamma }}}{p\gamma _1\gamma _2} \beta _1\beta _2(v_1-v_2) , \quad j_{13} = \frac{{\overline{\gamma }}}{\gamma _2} \beta _1\beta _2, \quad j_{14} = -\frac{{\overline{\gamma }}}{\gamma _1} \beta _1\beta _2, \\&j_{21} = \frac{p \gamma _1\gamma _2}{{\overline{\gamma }}} \left( v_{1} - v_{2} \right) , \quad j_{22} = \beta _1 v_1+\beta _2 v_2, \quad j_{23} = \beta _1 \gamma _1 p, \quad j_{24} = \beta _2 \gamma _2 p,\\&j_{32} = \frac{1}{\rho _1}, \quad j_{42} = \frac{1}{\rho _2}, \\&\tilde{j}_{11} = \left( \alpha _1\beta _2-\alpha _2\beta _1 \right) \left( v_1 - v_2 \right) , \quad \tilde{j}_{12} = - {\overline{\kappa }}\frac{{\overline{\gamma }}}{\gamma _1\gamma _2}\beta _1\beta _2 \left( v_1 - v_2 \right) , \quad \tilde{j}_{13} = - {\overline{\kappa }}\frac{{\overline{\gamma }}}{\gamma _2}\beta _1\beta _2,\\&\tilde{j}_{14} = {\overline{\kappa }}\frac{{\overline{\gamma }}}{\gamma _1}\beta _1\beta _2, \quad \tilde{j}_{21} = \left( v_1 - v_2 \right) \left( p \left( \beta _2\gamma _1\kappa _2+\beta _1\gamma _2\kappa _1\right) +\varDelta \kappa _1\kappa _2\right) , \\&\tilde{j}_{22} = - \frac{{\overline{\gamma }}}{\gamma _1\gamma _2} \beta _1 \beta _2 \left( \kappa _1-\kappa _2 \right) \left( v_1 - v_2 \right) , \quad \tilde{j}_{23} = - \left( \kappa _1 - \kappa _2 \right) \alpha _1 \beta _2 p, \\&\tilde{j}_{24} = \left( \kappa _1 - \kappa _2 \right) \alpha _2 \beta _1 p, \quad \tilde{j}_{31} = - \frac{\left( \varDelta \bar{\kappa } + p \bar{\gamma } \right) }{\alpha _1 \rho _1}, \quad \tilde{j}_{41} = \frac{\left( \varDelta \bar{\kappa } + p \bar{\gamma } \right) }{\alpha _2 \rho _2}. \end{aligned}$$
Obviously, the matrix is well defined if and only if
$$\begin{aligned} \varDelta \ne -\frac{p {\overline{\gamma }}}{{\overline{\kappa }}} = -p - \frac{p}{{\overline{\kappa }}} {\quad \text{ or, } \text{ equivalently, }\quad P\ne -\frac{p}{{\overline{\kappa }}}}. \end{aligned}$$
(3.7)
This is a first constraint on the choice of the interfacial pressure at pressure equilibrium.
Obviously, the velocities
\(v_1\) and
\(v_2\) are eigenvalues of
\({{{\varvec{J}}}}\). In the following, we investigate the roots of the characteristic polynomial
\(\chi \). For this purpose, we introduce the transformation
$$\begin{aligned} x: = \left( \lambda -\frac{v_1+v_2}{2}\right) ,\quad \delta := \frac{v_1-v_2}{2} \end{aligned}$$
(3.8)
also applied in [
21]. Then, the characteristic polynomial can be written in the form
$$\begin{aligned} P_4(x) = x^4 + a_2 x^2 + a_1 x + a_0 \end{aligned}$$
(3.9)
with coefficients
$$\begin{aligned} a_2&:= -(A-f(\varDelta )D + 2\delta ^2) , \end{aligned}$$
(3.10a)
$$\begin{aligned} a_1&:= -2 \delta (B-f(\varDelta )C ) , \end{aligned}$$
(3.10b)
$$\begin{aligned} a_0&:= \delta ^4 - \delta ^2 (A-f(\varDelta )D ) - f(\varDelta ) c_1^2 c_2^2 \end{aligned}$$
(3.10c)
and linear combinations of the squares of the single-fluid sound speeds
$$\begin{aligned} A := \beta _1 c_1^2 + \beta _2 c_2^2,\quad B := \beta _1 c_1^2 - \beta _2 c_2^2,\quad C := \beta _2 c_1^2 - \beta _1 c_2^2,\quad D := \beta _1 c_2^2 + \beta _2 c_1^2. \end{aligned}$$
(3.11)
We emphasize that the sign of the terms
A and
D is independent of the enumeration of the fluids, whereas the terms
B and
C flip sign.
In the following, we derive sufficient and necessary conditions on
\(f(\varDelta )\) or, equivalently,
\(\varDelta \) for which the conditions (3.13) hold. First of all, we verify that
$$\begin{aligned} f(\varDelta ) \le 0 \quad \text{ or, } \text{ equiv., }\quad -\frac{{\overline{\gamma }}}{{\overline{\kappa }}} p \le \varDelta \le 0 {\quad \text{ or, } \text{ equiv., }\quad -\frac{p}{{\overline{\kappa }}} \le P \le p} . \end{aligned}$$
(3.14)
is a necessary condition on
\(f(\varDelta )\) to ensure the existence of real roots of the characteristic polynomial (
3.9) for
all admissible physical states
$$\begin{aligned} \mathcal{D}:= \left\{ (\alpha _1,\rho _1,\rho _2,v_1,v_2,p) {:}\, \alpha _1\in [0,1].\ v_k\in {\mathbb R},\ \rho _k\in {\mathbb R}_+,\ \rho \in {\mathbb R}_+ \right\} . \end{aligned}$$
Besides condition (
3.7), this is another constraint on the choice of the interfacial pressure
P at pressure equilibrium.
Proof
We consider each of the conditions in (3.13) separately:
Condition (
3.13a) This condition only holds if
r defined by (
3.12) is not positive. We now derive a constraint on
\(f(\varDelta )\) that ensures the correct sign of
r for all physical admissible states in
\(\mathcal{D}\). For this purpose, we consider the term
\(-3r\) as a quadratic polynomial in
\(f\equiv f(\varDelta )\) depending on the parameter
\(y\equiv \delta ^2\):
$$\begin{aligned} G(f;y) := 12 a_0 + a_2^2 = D^2 f^2 - 2 \left( A D - 4 D y + 6 c_1^2 c_2^2 \right) f + (A-4y)^2 , \end{aligned}$$
where we suppress the dependency on the other physical quantities
\(\alpha _1\),
\(\rho _k\),
\(v_k\) and
p. We now verify for which
f this polynomial is nonnegative for
all velocity differences
\(\delta ^2\).
Since
\(D>0\) according to (
3.10), we may factorize the polynomial
$$\begin{aligned} G(f;y) = D^2(f-\hat{f}_-)(f-\hat{f}_+) \end{aligned}$$
(3.15)
with
$$\begin{aligned} \hat{f}_\pm := \frac{1}{D^2} \left( 6 c_1^2 c_2^2 + A D - 4 D y \pm \sqrt{(6 c_1^2c_2^2 + A D - 4 D y)^2 - D^2(A-4 y)^2} \right) . \end{aligned}$$
Herein, the discriminant can be written as
$$\begin{aligned} \hat{g}(y) = 12 c_1^2c_2^2 \left( 3 c_1^2 c_2^2 + A D - 4 D y \right) . \end{aligned}$$
Case 1a Obviously, if
$$\begin{aligned} \delta ^2 > \frac{A D + 3 c_1^2 c_2^2 }{4D} =:\hat{\delta }_{crit}^2, \end{aligned}$$
(3.16)
the discriminant is negative and, thus,
\(G(f;\delta ^2)\) must be positive because
\(D^2> 0\), i.e., in this case there is no constraint on
\(f(\varDelta )\).
Case 1b On the other hand, if
\(0\le \delta ^2\le \hat{\delta }_{crit}^2\), i.e.,
\(-3 c_1^2 c_2^2 \le AD\), then the roots
\(\hat{f}_\pm \) are real and we conclude from the factorization (
3.15) that
\(G(f;\delta ^2)\) is nonnegative if
$$\begin{aligned} f\le \hat{f}_- \quad \text{ or }\quad f \ge \hat{f}_+ . \end{aligned}$$
(3.17)
By definition of
\(\hat{f}_\pm \) and the positivity of the single-fluid sound speeds we check that
$$\begin{aligned} 0 \le \hat{f}_- \le \frac{(A-4\delta ^2)D+6 c_1^2 c_2^2}{D^2} \le \hat{f}_+ \le 2 \frac{(A-4\delta ^2)D+6 c_1^2 c_2^2}{D^2} = \hat{f}_-+\hat{f}_+. \end{aligned}$$
(3.18)
Note that for
\(0< \delta ^2 = A/4 < \hat{\delta }_{crit}^2\) the minimum and the maximum are attained, i.e.,
\(\hat{f}_-=0\) and
\(\hat{f}_+=12 c_1^2 c_2^2\).
To summarize the findings of the above investigations for the different cases, we conclude that
G is nonnegative for all physical states if the condition (
3.14) holds.
Condition (
3.13b) For the investigation of this condition, we consider the term
\(a_2^2-4 a_0\) as a quadratic polynomial in
\(f\equiv f(\varDelta )\) depending on the parameter
\(y\equiv \delta ^2\):
$$\begin{aligned} F(f;y) :=a_2^2-4 a_0 = D^2 f^2 - 2 \left( A D + 4 D y -2 c_1^2 c_2^2 \right) f + A (A+8y) , \end{aligned}$$
where we suppress the dependency on the other physical quantities
\(\alpha _1\),
\(\rho _k\),
\(v_k\) and
p. We now verify for which
f this polynomial is nonnegative for
all velocity differences
\(\delta ^2\).
Since
\(D>0\) according to (
3.10), we may factorize the polynomial
$$\begin{aligned} F(f;y) = D^2\left( f(\varDelta ) - f_- \right) \left( f(\varDelta ) - f_+ \right) \end{aligned}$$
(3.19)
with
$$\begin{aligned} f_\pm := \frac{1}{D^2} \left( A D + 4 D y - 2 c_1^2 c_2^2 \pm 2 \sqrt{(2Dy-c_1^2c_2^2)^2 - c_1^2c_2^2 A D} \right) . \end{aligned}$$
Herein, the discriminant can be factorized by
$$\begin{aligned} g(y) = 4D^2 \left( y-y_- \right) \left( y-y+ \right) ,\quad y_\pm :=\frac{c_1c_2 \left( c_1 c_2 \pm \sqrt{A D} \right) }{2D} . \end{aligned}$$
Since the product
AD is positive due to the positivity of the single-fluid sound speeds and (
3.11), the roots
\(y_\pm \) are real numbers.
Case 1 Obviously, the discriminant
g becomes negative if
\(y_-< y < y_+\). Since
\(y\equiv \delta ^2\), the admissible regime reduces to
$$\begin{aligned} \max (0,\delta ^2_{crit,-})< \delta ^2 < \delta ^2_{crit,+},\quad \delta ^2_{crit,\pm }:=y_\pm . \end{aligned}$$
Thus, the polynomial
\(F(f;\delta ^2)\) must be positive because
\(D^2> 0\), i.e., in this case there is no constraint on
\(f(\varDelta )\).
Case 2 On the other hand, if
\(\delta ^2\ge \delta _{crit,+}^2\) or
\(\delta ^2 \le \max (0,\delta ^2_{crit,-})\) , then the discriminant
g is nonnegative and the roots
\(f_\pm \) are real. According to the factorization (
3.19) the polynomial
\(F(f;\delta ^2)\) is nonnegative if and only if
$$\begin{aligned} f\le f_- \quad \text{ or }\quad f \ge f_+ . \end{aligned}$$
(3.20)
If
\(\delta ^2\ge \delta _{crit,+}^2\), then we conclude from the definition of the roots
\(f_\pm \):
$$\begin{aligned} 0 \le f_- \le \frac{A + 4 \delta ^2 D- 2 c_1^2 c_2^2}{D^2} \le f_+ \le 2\frac{A + 4 \delta ^2 D- 2 c_1^2 c_2^2}{D^2} = f_-+f_+ . \end{aligned}$$
For the other option, we first note that the interval
\(\delta ^2 \le \max (0,\delta ^2_{crit,-})\) is non-empty if and only if
\(AD \le c_1^2 c_2^2\). Since by definition (
3.11) of
A and
D it holds
\(AD \ge c_1^2 c_2^2\), the only choice is
\(AD = c_1^2 c_2^2\). This is possible only for a pure fluid, i.e.,
\(\alpha _1=1\) or
\(\alpha _2=1\). Then, the roots coincide, i.e.,
\(f_- = f_+ = -c_1^2 c_2^2/D^2\). Thus, for this case the polynomial
\(F(f;\delta ^2)\) is nonnegative, i.e., no constraint is imposed on
\(f(\varDelta )\).
Condition (
3.13c) Obviously, this condition holds by positivity of
A and
D due to the positivity of the single-fluid sound speeds.
\(\square \)
To derive sufficient conditions on
\(f(\varDelta )\) in the non-equilibrium case, the sign of
\(D_1\) needs to be further investigated. For this purpose, we first note that by rescaling of
\(D_1\) we may equivalently consider the sign of
\({\tilde{D}}_1:={\tilde{q}}^2 - 4 {\tilde{r}}^3\) with
\({\tilde{r}}:= - 3r = 12 a_0 + a_2^2\) and
\({\tilde{q}}:= - 27q = 12 a_2^3 + 27 a_1^2 -72 a_0 a_2\). To investigate the sign of
\({\tilde{D}}_1\), we split this term into two parts
$$\begin{aligned} {\tilde{D}}_1 = 432 c_1^2 c_2^2 \left( {\tilde{D}}_1^s + {\tilde{D}}_1^t \right) \end{aligned}$$
(3.21)
with
$$\begin{aligned}&{\tilde{D}}_1^s= f(\varDelta ) \left( 256 {\tilde{r}}_8^s \delta ^8 + 64 {\tilde{r}}_6^s \delta ^6 + 16 {\tilde{r}}_4^s \delta ^4 + 4 {\tilde{r}}_2^s \delta ^2 + {\tilde{r}}_0^s\right) , \end{aligned}$$
(3.22a)
$$\begin{aligned}&{\tilde{D}}_1^t = -\beta _1 \beta _2 (f(\varDelta )+1)^2 \delta ^2 \left( 256 {\tilde{r}}_8^t \delta ^6 + 64 {\tilde{r}}_6^t \delta ^4 + 16 {\tilde{r}}_4^t \delta ^2 + 4 {\tilde{r}}_2^t \right) \end{aligned}$$
(3.22b)
and coefficients
$$\begin{aligned}&{\tilde{r}}_8^s:= 1 , \end{aligned}$$
(3.23a)
$$\begin{aligned}&{\tilde{r}}_6^s:= 4 \left( D f(\varDelta )-A\right) , \end{aligned}$$
(3.23b)
$$\begin{aligned}&{\tilde{r}}_4^s:= 8 c_1^2 c_2^2 f(\varDelta ) + 6 (D f(\varDelta )-A)^2 , \end{aligned}$$
(3.23c)
$$\begin{aligned}&{\tilde{r}}_2^s:= 4 (D f(\varDelta )-A) \left( (D f(\varDelta )-A)^2 + 4 c_1^2 c_2^2 f(\varDelta ) \right) , \end{aligned}$$
(3.23d)
$$\begin{aligned}&{\tilde{r}}_0^s:=\left( 4 f(\varDelta ) c_1^2 c_2^2+(D f(\varDelta )-A)^2\right) ^2 \end{aligned}$$
(3.23e)
and
$$\begin{aligned}&{\tilde{r}}_8^t:= 1 , \end{aligned}$$
(3.24a)
$$\begin{aligned}&{\tilde{r}}_6^t:= 3 \left( D f(\varDelta )-A\right) , \end{aligned}$$
(3.24b)
$$\begin{aligned}&{\tilde{r}}_4^t:=3 (D f(\varDelta )-A)^2 + 9 c_1^2 c_2^2 \left( 4 f(\varDelta ) -3 \beta _1 \beta _2 (f(\varDelta )+1)^2 \right) , \end{aligned}$$
(3.24c)
$$\begin{aligned}&{\tilde{r}}_2^t:= (D f(\varDelta )-A) \left( (D f(\varDelta )-A)^2 +36 c_1^2 c_2^2 f(\varDelta ) \right) . \end{aligned}$$
(3.24d)
Here, we have applied the relations
$$\begin{aligned} C^2-D^2 =B^2- A^2=-4\beta _1\beta _2c_1^2 c_2^2,\ AD-CB= B^2-A^2 + 2 c_1^2 c_2^2 \end{aligned}$$
that hold by (
3.11). The derivation of this particular splitting is tedious work collecting appropriate terms. It is motivated by the observation that for local single-fluid the sign of
\({\tilde{D}}_1\) can be easily checked, see Remark
4.