*4.2. Locally Variable Surface Tension Due to Temperature*

Surface tension gradient in lamella due to temperature can induce flow in an unfavorable direction which is opposite to the drainage one (Marangoni effect); this affects the foam stability. The situation arises when the temperature on the top surface is lower than bottom, leading to higher surface tension on the top than in the bottom. Hence, flow advection can occur in the direction opposing the gravity, as shown in Figure 7 [6].

**Figure 7.** (**a**) Bubble with lamella showing the drainage direction and Marangoni convection, after Poulain et al. [6]. (**b**) Representation of surface tension variation due to temperature gradient.

A relation between surface tension and temperature can be written as:

$$\gamma(T) = \gamma\_{ref} - \frac{\partial \gamma}{\partial T} \left( T - T\_{ref} \right) \tag{28}$$

where *γref* is a reference surface tension value at a reference temperature, *Tref* = 25 °C , for example. A proposed way to back-couple the effect of temperature on the flow field, specifically on the surface tension, can be carried out through the SC interaction parameter. As a first approximation, Equation (28) can thereby be translated to the interaction parameter as shown in Equation (29). The relation can be argued to be mathematically plausible, since, practically, the relation between *<sup>G</sup>σ*,*<sup>σ</sup>* and *<sup>γ</sup>* is almost linear. Additionally, the wide range of surface tension which can be reached as shown is Figures 2 and 3 offers the possibility of surface tension tuning even for high gradients ( *∂γ <sup>∂</sup><sup>T</sup>* ).

$$G\_{\sigma\overline{\nu}}(T) = G\_{\sigma\overline{\sigma}\_{nf}} - \frac{\partial G\_{\sigma\overline{\nu}}}{\partial T} \left( T - T\_{ref} \right) \tag{29}$$

Samples from two simulations are shown in Figure 8, with a spatially variable surface tension and with constant surface tension. The domain is assumed to have linear temperature variation following the schematic on Figure 7b, with a bubble rising from bottom to top until reaching the interface. The case with variable surface tension showed a slightly delayed coalescence, though uninhibited, without using the mid-range interaction model. Flow vectors show flow in a direction opposing the drainage one, as shown in Figure 8a when compared to Figure 8b. Generally, when imposing the surface tension gradient on the bubble, it tends to have slower rising velocity. This can explain, for this situation, the direction of the Marangoni convection.

**Figure 8.** (**a**) Bubble rise simulation case with locally variable surface tension. (**b**) Bubble rise simulation with constant surface tension.
