# Effects of Interfacial Tension Gradients

Since surface tension depends on temperature, a permanent nonuniformity of temperature or concentration (for a multicomponent system) at a liquid-vapor interface causes a surface tension gradient. The interfacial area with small surface tension expands at the expense of an area with greater surface tension, which in turn establishes a steady flow pattern in the liquid; this flow caused by the surface tension gradient along the liquid-vapor (gas) interface is referred to as the Marangoni effect. The surface tension of a multicomponent liquid that is in equilibrium with the vapor is a function of temperature and composition of the mixture, i.e.,

 $\sigma =\sigma (T,{{x}_{1}},{{x}_{2}},\cdots ,{{x}_{N-1}})$ (1)

where xi is the molar fraction of the ith component in the liquid phase and N is the total number of components in the liquid phase. The change of surface tension can be caused by either change of temperature or composition, i.e.,

 $d\sigma ={{\left( \frac{\partial \sigma }{\partial T} \right)}_{{{x}_{i}}}}dT+\sum\limits_{i=1}^{N-1}{{{\left( \frac{\partial \sigma }{\partial {{x}_{i}}} \right)}_{T,{{x}_{j\ne i}}}}d{{x}_{i}}}$ (2)

Recall from thermodynamics that as the critical temperature for a given fluid is approached, the properties of the liquid and vapor phases of the fluid become identical, i.e., σ vanishes. It therefore follows that the surface tension decreases with temperature, i.e., [${{(\partial \sigma /\partial T)}_{{{x}_{i}}}}<0$]. For a pure substance, surface tension is a function of temperature only. The curve-fit equations for surface tension are almost linear for most fluids and can have the form

 σ = C0 − C1T (3)

where C0 and C1 are empirical constants that vary between substances. For water, ${{C}_{0}}=75.83\times {{10}^{-3}}\text{N/m}$ and ${{C}_{1}}=0.1477\times {{10}^{-3}}\text{N/m-}{}^{\text{o}}\text{C}$. The unit of the temperature T in eq. (3) is °C.

Figure 1 Marangoni effect: cellular flow driven by surface tension gradient.

Since surface tension varies with temperature, the surface tension will not be uniform if the temperature, along the liquid-vapor interface, is nonuniform. Consequently, the liquid in the lower surface tension region near the interface will be pulled toward the region with higher surface tension. Furthermore, because the surface tension usually decreases with increasing temperature, the flow driven by surface tension moves away from the interface with high temperature and towards the interface with low temperature. As noted before, this motion of a liquid caused by a surface tension gradient at the interface is referred to as the Marangoni effect. The most well-known example of surface-tension-driven flow is Bernard cellular flow, which occurs in a thin horizontal liquid layer heated from below. Figure 1 illustrates steady cellular flow driven by the Marangoni effect. Once a steady Marangoni flow is established, the liquid velocity is upward at point A and downward at point B. Since the liquid at surface point A comes directly from the hot surface, the temperature at point A is higher than that at point B. Consequently, the surface tension at point A is smaller than that at point B, and the fluid at point A is pulled toward point B. Although a temperature gradient exists in the vertical direction, the actual driving force is the surface tension gradient in the horizontal direction. The liquid surface loses heat to the gas phase at temperature, TG, through convection with a heat transfer coefficient of hδ.

An important boundary condition at the interface explains the liquid flow field caused by surface tension variation along the interface:

 ${{\left. {{\tau }_{yx}} \right|}_{y=\delta }}={{\mu }_{\ell }}{{\left( \frac{\partial u}{\partial y} \right)}_{y=\delta }}=\left( \frac{\partial \sigma }{\partial T} \right){{\left( \frac{\partial T}{\partial x} \right)}_{y=\delta }}$ (5)

Bernard cellular flow resulting from the Marangoni effect can occur only if certain conditions are met. The conditions for the onset of cellular motion can be predicted using linear stability analysis as presented by Carey (1992). The local temperature is assumed to equal the sum of the basic temperature $\bar{T}$ – given by the initial linear profile – and a small sinusoidal fluctuation T' that represents a Fourier component of random disturbances, i.e.,

 $T=\bar{T}+{T}'=({{T}_{w}}-\zeta y)+{T}'$ (6)

where ζ = (TwTδ) / δ. Since all base flow velocities are zero, the velocities are

 $v=\bar{v}+{v}'={v}'$ (7)
 $u=\bar{u}+{u}'={u}'$ (8)

Substituting eqs. (6) – (8) into the continuity, momentum, and energy equations, and subtracting the corresponding base flow equations, the resulting equations are solved assuming the following forms for T' and v':

 ${T}'=\theta (y){{e}^{i\alpha \,x+\beta \ t}}$ (9)
 ${v}'=V(y){{e}^{i\alpha \,x+\beta \ t}}$ (10)

where α = 2π / λ is the wave number and β can be a complex number with its real part representing the amplification factor, and its imaginary part representing the temporal frequency. The resulting stationary wave solutions (for β = 0), which are not amplified or dampened with time, are therefore interpreted as stable perturbations. The conditions for which these solutions are obtained are assumed to correspond to the onset of instability, which leads to cellular convection. The case of marginal stability corresponds to

 $\text{Ma}=\frac{8\hat{\alpha }(\hat{\alpha }\cosh \hat{\alpha }+Bi\sinh \hat{\alpha })(\hat{\alpha }-\sinh \hat{\alpha }\cosh \hat{\alpha })}{{{{\hat{\alpha }}}^{3}}\cosh \hat{\alpha }-{{\sinh }^{3}}\hat{\alpha }-(8Cr{{{\hat{\alpha }}}^{5}}\cosh \hat{\alpha })/(\text{Bo}+{{{\hat{\alpha }}}^{2}})}$ (11)

where Ma is the Marangoni number:

 $\text{Ma}=\frac{\zeta (d\sigma /dT){{\delta }^{2}}}{{{\alpha }_{\ell }}{{\mu }_{\ell }}}$ (12)

and ${{\alpha }_{\ell }}$ is the thermal diffusivity of the liquid.

The other dimensionless numbers in eq. (11) are the wave number $\hat{\alpha }$, the Biot number Bi, the Bond number Bo, and the Crispation number Cr. Their definitions are

 $\hat{\alpha }=\alpha \delta$ (13)
 $\text{Bi}=\frac{{{h}_{\delta }}\delta }{{{k}_{\ell }}}$ (14)
 $\text{Bo}=\frac{({{\rho }_{\ell }}-{{\rho }_{g}})g{{\delta }^{2}}}{\sigma }$ (15)
 $\text{Cr}=\frac{{{\mu }_{\ell }}{{\alpha }_{\ell }}}{\sigma \delta }$ (16)

The liquid film is stable if the Marangoni number is below that predicted by eq. (11). However, the liquid film is not stable if its Marangoni number is above that obtained by eq. (11). The marginal stability predicted by eq. (11) for some typical combinations of parameters is illustrated in Fig. 2. Since the Fourier components of all wavelengths can be contained in a random disturbance, the system becomes unstable when it is unstable at any wavelength. For a system with Cr < 10 − 4 and $\text{Bi}\to 0,\text{ Bo}\to 0$, Fig 2 indicates that the critical Marangoni number is about 80 and the associated dimensionless wave number $\hat{\alpha }=2.$

Figure 2 Stability plane for the onset of cellular motion (Carey, 1992; Reproduced by permission of Routledge/Taylor & Francis Group, LLC).

The Marangoni effect can have an important influence on heat and mass transfer processes, including the evaporation of a falling film, as well as causing vapor bubbles in a liquid with a temperature gradient to move toward the high temperature region during boiling.

## References

Carey, V.P., and Wemhoff, A. P., 2005, “Disjoining Pressure Effects in Ultra-Thin Liquid Films in Micropassages – Comparison of Thermodynamic Theory with Predictions of Molecular Dynamics Simulations,” IMECE2005-80234, Proceedings of 2005 ASME International Mechanical Engineering Congress and Exposition, Orlando, FL.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.