A linear stability analysis is conducted to decipher how the thermal and surfactant Marangoni numbers affect primary instabilities when a two-dimensional incompressible viscous film flows down a uniformly heated flexible substrate under gravity. Since the viscous film is nonisothermal and a soluble surfactant is dissolved in it, we require additional heat- and mass-transfer equations along with the Navier-Stokes equations for describing this film flow. To explore linear stability, we formulate the Orr-Sommerfeld-type boundary value problem (OS BVP). The analytical long-wave series expansion and the numerical Chebyshev spectral collocation method are applied to solve the OS BVP. The long-wave results disclose that surface mode instability intensifies in the presence of the thermal Marangoni number but weakens as the surfactant Marangoni number rises, creating a surfactant-free isothermal film flowing over a flexible substrate when these two opposing impacts balance each other. Moreover, we identify a range of the inclination angle for which the surface mode becomes linearly stable, primarily because the stabilizing effects of the surfactant Marangoni stress and depthwise gravitational force dominate over the destabilizing effects of the thermal Marangoni stress and spring stiffness coefficient $C_K$. In the arbitrary wave-number regime, the numerical results reveal that surface mode, S-mode, and P-mode instabilities are amplified with rising values of the thermal Marangoni number but are weakened with rising values of the surfactant Marangoni number. Moreover, we observe a modal coalescence between the surface mode and the S-mode in the long-wave regime as the thermal Marangoni number increases. The base surface surfactant concentration parameter, ${\mathrm{\Gamma }}_E$, exhibits both stabilizing and destabilizing influences on all the surface mode, S-mode, and P-mode instabilities. However, the surfactant solubility parameter, ${\xi }_s$, shows only a stabilizing influence on all instability modes. If the magnitudes of the spring stiffness coefficient $C_K$, the longitudinal tension parameter $C_T$, and the flexural rigidity parameter $C_B$ are significantly lowered, unstable domains of the P-mode and the wall mode, which emerged separately in the finite wave-number regimes, coalesce at higher values of ${\mathrm{\Gamma }}_E$. Additionally, we observe a competition between the surface mode and the wall mode. When the damping coefficient, $C_D$, is equal to 0.32, the wall mode controls the primary instability in the finite wavelength regime. As soon as $C_D$ increases to 0.35, the surface mode governs the primary instability in the long-wave regime at low values of the surfactant Marangoni number. On the other hand, we see the existence of the unstable shear mode in the high Reynolds number regime when the inclination angle is small enough. The initiation of shear mode instability can be quickened by raising the thermal Marangoni number, while its initiation will be delayed if the surfactant Marangoni number goes up. Finally, we demonstrate that the primary instabilities induced by the surface mode and the wall mode may also appear in different ranges of $C_D$ due to the destabilizing impact of the thermal Marangoni number even when the Reynolds number approaches zero.

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• Flow instability
• Instability of free-surface flows
• Thermocapillarity