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Large-scale longitudinal distortions of Marangoni wave patterns in the non-isothermal liquid layer covered by surfactant

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Abstract

In the present paper, we consider single traveling waves (STW) generated by the oscillatory instability of Marangoni convection in the thin non-isothermal liquid layer with deformable free surface. The layer is covered by insoluble surfactant that plays an active role in the pattern selection together with inhomogeneity of temperature along the interface and surface deformability. Using the weakly nonlinear analysis we derived the modified complex Ginzburg–Landau equation describing the large-scale distortions of STWs near the bifurcation point. Linear stability analysis reveals existence of two modulational modes: one is for the amplitude and another one for the phase (Benjamin–Feir). Numerically, we found that STWs are stable with respect to longitudinal modulations in the case without surfactant. In the presence of the insoluble surfactant both modulational modes are found. The stability maps for different values of the surfactant concentration are plotted.

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Data availability statement

Data sets generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

A.N. acknowledges financial support from the Israel Science Foundation (Grant N. 843/18).

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Author notes

  1. Alexander B. Mikishev and Alexander A. Nepomnyashchy contributed equally to this work.

Authors and Affiliations

  1. Department of Engineering Technology, Sam Houston State University, Huntsville, TX, 77341, USA

    Alexander B. Mikishev

  2. Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Alexander A. Nepomnyashchy

Authors
  1. Alexander B. Mikishev
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  2. Alexander A. Nepomnyashchy
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Correspondence to Alexander B. Mikishev.

Appendix A

Appendix A

$$\begin{aligned}{} & {} \omega _1={\mathcal {D}}_{11}/{\mathcal {D}}_{10} \end{aligned}$$
(A1)
$$\begin{aligned} {\mathcal {D}}_{11}&=-i k \{3 K_c^2 M_0 (K_c^2 L+i \omega _0) [24 \alpha _1+3 G +6 K_c^2 S+4 (\alpha _1-1) M_o+4 \alpha _2 N] \\ & \quad -2 (K_c^2 (M_o+6)+6 (\beta +i \omega _0)) [G K_c^2 (4 L+N) +4 i G \omega _0+2 K_c^4 S (4 L+N) \\ & \quad +K_c^2 (6 (\alpha _1-1) L M_o+8 i S \omega _0) +6 i \omega _0 ((\alpha _1-1) M_o+\alpha _2 N)] +2 K_c^4 M_o N (G+2 K_c^2 S-6 \alpha _2 L)\}, \\{} & {} {\mathcal {D}}_{10}=6 \{(\alpha _2-2) K_c^4 M_o N -6 \alpha _1 K_c^2 M_o (K_c^2 L+i \omega _0)+(K_c^2 (M_o+6)\\ & \quad +6 (\beta +i \omega _0)) (K_c^2 (2 (L+N)-\alpha _2 N)+2 i \omega _0)\}. \\ a_{22} &=\frac{{\mathcal {D}}_{21}+\omega _c{\mathcal {D}}_{22}+\omega _c^2{\mathcal {D}}_{23}}{{\mathcal {D}}_{00} +{\mathcal {D}}_{01}\omega _c+{\mathcal {D}}_{02}\omega _c^2+36i\omega _c^3}, \end{aligned}$$
(A2)
$$\begin{aligned} {\mathcal {D}}_{23}&=-18 k^2 [2 G+2 (K_c^2 S+\alpha _2 N)+(\alpha _1-2) M_o], \\ {\mathcal {D}}_{22}&=3 i K_c^2 [6 \beta G+2 G K_c^2 (12 L-\alpha _1 M_o-3 \alpha _2 N +6 N+12)+2 K_c^4 S (12 L-\alpha _1 M_o-3 \alpha _2 N+6 N \\ & \quad +12)+K_c^2 (3 M_o (4 \alpha _1 (L+N+1)-8 L-4 N-7) +12 \alpha _2 N (2 L-\alpha _2 N+N+2)+((4-3 \alpha _1) \alpha _1-1) \\{} & {} \quad \times M_o^2+(13-15 \alpha _1) \alpha _2 M_o N+6 \beta S) +6 \beta ((\alpha _1-1) M_o+\alpha _2 N)], \\ {\mathcal {D}}_{21}&=3 K_c^4 [-4 G K_c^2 (L (\alpha _1 M_o-12) +3 (\alpha _2-2) N)+12 \beta G L-3 (\alpha _2-2) \beta G N \\{} & {} \quad -4 K_c^4 S (L (\alpha _1 M_o-12)+3 (\alpha _2-2) N) -2 K_c^2 L M_o ((\alpha _1-1) (3 \alpha _1-1) M_o+3 \alpha _1 (\alpha _2 N-4) \\{} & {} \quad -\alpha _2 N+21)+48 \alpha _2 K_c^2 L N+12 \beta K_c^2 L S -24 (\alpha _2-1) K_c^2 N ((\alpha _1-1) M_o+\alpha _2 N) -3 (\alpha _2-2) \beta K_c^2 N S \\{} & {} \quad +6 \beta (2 L-\alpha _2 N+N) ((\alpha _1-1)M_o+\alpha _2 N)], \\{\mathcal {D}}_{00}&=2 K_c^6 [G L (M_o-48)-12 G N+4 K_c^2 S (L (M_o-48)-12N) +72 L M_o]-6 \beta K_c^4 (4 L+N)(G+4K_c^2 S), \\ {\mathcal {D}}_{01}&=-i K_c^2 [K_c^2 (G (48 L-M_o+12 N+48) +4 K_c^2 S (48 L-M_o+12 N+48)-48 L (M_o-3) \\ & \quad -72 (M_o-2 N))+12 \beta (G+4 K_c^2 S+3 (L+N))], \\ {\mathcal {D}}_{02}&=24K_c^2(3+G+3L-M_{osc}+3N+4K_c^2S)+18\beta , \\ b_{22}&=\frac{{\mathcal {D}}_{31}+{\mathcal {D}}_{32}\omega _c+{\mathcal {D}}_{33}\omega _c^2-36i\alpha _1\omega _c^3}{2({\mathcal {D}}_{00}+{\mathcal {D}}_{01}\omega _c+{\mathcal {D}}_{02}\omega _c^2+36i\omega _c^3)}, \end{aligned}$$
(A3)
$$\begin{aligned} {\mathcal {D}}_{31}&=K_c^4 [G^2 K_c^2 (4 (4 \alpha _1-1) L+N (4 \alpha _1+\alpha _2-2)) +G (5 K_c^4 S (4 (4 \alpha _1-1) L+N (4 \alpha _1+\alpha _2-2)) \\ & \quad +2 K_c^2 (6 L (8 \alpha _1+\alpha _1 (2 \alpha _1-5) M_o+M_o +2 \alpha _1 \alpha _2 N-\alpha _2 N-2)+N (12 \alpha _1 \\{} & {} \quad +(3 \alpha _1+\alpha _2-2) ((\alpha _1-1)M_o+\alpha _2 N)-3)) +18 \beta (4 L-(\alpha _2-2) N))+4 K_c^6 S^2 (4 (4 \alpha _1-1) L \\ &\quad +N (4 \alpha _1+\alpha _2-2))+8 K_c^4 S (3 L (16 \alpha _1 +(\alpha _1 (4 \alpha _1-7)+2) M_o+4 \alpha _1 \alpha _2 N-2 \alpha _2 N-4) \\ & \quad +N (12 \alpha _1+(3 \alpha _1+\alpha _2-2) ((\alpha _1-1) M_o+\alpha _2 N)-3)) +6 K_c^2 (-2 L M_o (12 \alpha _1+3 \alpha _1^2 M_o \\ & \quad -4 \alpha _1 M_o+M_o+(3 \alpha _1-1) \alpha _2 N-3)+12 \beta L S-3 (\alpha _2-2) \beta N S) \\ &\quad +36 \beta (2 L-\alpha _2 N+N) ((\alpha _1-1) M_o+\alpha _2 N)], \\ {\mathcal {D}}_{32}&=2 i K_c^2 [(4 \alpha _1-1) G^2 K_c^2+18 \beta G +5 (4 \alpha _1-1) G K_c^4 S+3 G K_c^2 (8 \alpha _1+4 (3 \alpha _1+2) L \\& \quad +\alpha _1 (2 \alpha _1-5) M_o+M_o+N (2 \alpha _1 (\alpha _2+3)-3 \alpha _2+4)-2) +4 (4 \alpha _1-1) K_c^6 S^2 \\ & \quad +12 (9 \alpha _1+2) K_c^4 L S+6 K_c^4 S (16 \alpha _1+(\alpha _1 (4 \alpha _ 1-7)+2) M_o +N (4 \alpha _1 \alpha _2+6 \alpha _1-3 \alpha _2+2)-4) \\ & \quad +3 K_c^2 (6 L (4 \alpha _1+((\alpha _1-2) \alpha _1-1) M_o +(\alpha _1+1) \alpha _2 N-1)+((4-3 \alpha _1) \alpha _1-1) M_o^2 \\ & \quad +M_o (-12 \alpha _1+N (\alpha _1 (6 \alpha _1-7 \alpha _2-4) +5 \alpha _2-2)+3)+2 N (3 \alpha _1 (\alpha _2 N+4) \\ & \quad +\alpha _2 (1-2 \alpha _2) N-3)+6 \beta S)+18 \beta ((\alpha _1-1) M_o+\alpha _2 N)], \\ {\mathcal {D}}_{33}&=6 K_c^2 [-2 (3 \alpha _1+2) G-2 (9 \alpha _1+2) K_c^2 S -3 (4 \alpha _1 (L+N+1)+((\alpha _1-2) \alpha _1-1) M_o \\ & \quad +(\alpha _1+1) \alpha _2 N)+3], \\ c_{22}&=\frac{{\mathcal {D}}_{41}+{\mathcal {D}}_{42}\omega _c+{\mathcal {D}}_{43}\omega _c^2}{2({\mathcal {D}}_{00}+{\mathcal {D}}_{01}\omega _c+{\mathcal {D}}_{02}\omega _c^2+36i\omega _c^3)}, \end{aligned}$$
(A4)
$$\begin{aligned} {\mathcal {D}}_{41}&=-K_c^4 (-12 (\alpha _2-1) \beta G^2 +G^2 K_c^2 (M_o (4 \alpha _1+\alpha _2-2)-48 (\alpha _2-1))\\ & \quad +5 G K_c^4 S (M_o (4 \alpha _1+\alpha _2-2)-48 (\alpha _2-1))+2 (\alpha _1-1) G K_c^2 M_o^2 (3 \alpha _1\\ & \quad +\alpha _2-2)+2 G K_c^2 M_o (3 \alpha _1 (\alpha _2 (N-16)+12) +\alpha _2 ((\alpha _2-2) N+84)-99)+48 \alpha _2 (1-2 \alpha _2) G K_c^2 N\\ & \quad -60 (\alpha _2-1) \beta G K_c^2 S +12 (1-2 \alpha _2) \beta G ((\alpha _1-1) M_o+\alpha _2 N)+4 K_c^6 S^2 (M_o (4 \alpha _1+\alpha _2-2)-48 (\alpha _2-1))\\ & \quad +8 (\alpha _1-1) K_c^4 M_o^2 S (3 \alpha _1+\alpha _2-2) +8 K_c^4 M_o S (3 \alpha _1 (\alpha _2 (N-16)+12)+\alpha _2 ((\alpha _2-2) N+57)-45)\\ & \quad +192 \alpha _2 (1-2 \alpha _2) K_c^4 N S -48 (\alpha _2-1) \beta K_c^4 S^2+48 K_c^2 ((\alpha _1-1) M_o\\ & \quad +\alpha _2 N) (3 (\alpha _2-1) M_o+(1-2 \alpha _2) \beta S)),\\ {\mathcal {D}}_{42}&=6 i K_c^2 [4 (\alpha _2-1) G^2 K_c^2+3 (\alpha _2+2) \beta G +2 G K_c^2 (2 (5 (\alpha _2-1) K_c^2 S+\alpha _2 (2 \alpha _2 N-N+3)+6)\\ & \quad +M_o (\alpha _1 (4 \alpha _2-5)-6 \alpha _2+6))+16 (\alpha _2-1) K_c^6 S^2 +4 K_c^4 S (M_o (8 \alpha _1 \alpha _2-7 \alpha _1-9 \alpha _2+6)\\ & \quad +\alpha _2 (8 \alpha _2 N-4 N+3)+6)+K_c^2 (-2 (\alpha _1-1) M_o^2 (3 \alpha _1 +4 \alpha _2-5)-2 M_o (\alpha _2 (3 \alpha _1 (N-4)-5 N+12)\\ & \quad +4 \alpha _2^2 N+9)+24 \alpha _2 (\alpha _2+1) N+3 (\alpha _2+2) \beta S) +6 (\alpha _2+1) \beta ((\alpha _1-1) M_o+\alpha _2 N)], \\ {\mathcal {D}}_{43}&=-36K_c^2[2N\alpha _2(\alpha _2+1) +(G+K_c^2S)(\alpha _2+2)+2M_{o}(\alpha _2(\alpha _1-1)-1)]. \end{aligned}$$

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Mikishev, A.B., Nepomnyashchy, A.A. Large-scale longitudinal distortions of Marangoni wave patterns in the non-isothermal liquid layer covered by surfactant. Eur. Phys. J. Spec. Top. (2024). https://doi.org/10.1140/epjs/s11734-024-01118-1

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