# Rayleigh-Taylor Instability

A given physical state is said to be stable if it can withstand a disturbance and still return to its original state. Otherwise, the particular state is unstable. The objective of stability analysis is to analyze the effect of a particular disturbance on the physical state. If φ represents a basic solution, a disturbance φ' is added to this basic solution and φ + φ' will be substituted into the governing equations. The governing equations with φ as a dependent variable are then subtracted from the governing equation with φ + φ' as dependent variables to yield disturbance equation for φ'. If the disturbance φ' damps out, φ is stable; otherwise, if the disturbance φ' grows with increasing time, φ is unstable.

Interface morphology is important for heat and mass transfer. In a horizontal co-current flow system where a dense liquid phase overlays a less dense vapor phase (see Fig. 1), both phases are incompressible, inviscid, and immiscible; the interface may become unstable if there is a disturbance $\delta \left( x,t \right).$ This instability is referred to as Rayleigh-Taylor instability. On the other hand, if the gravity is parallel to the directions of the liquid-vapor co-current flow (see Fig. 2), the instability is referred to as the Kelvin-Helmholtz instability. The conditions for which the interfaces are stable with respect to an arbitrary perturbation $\delta \left( x,t \right)$ will be presented in this subsection (Carey, 1992). Assuming that the liquid and vapor flows are two-dimensional, the governing equations for configuration in Fig. 1 are $\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1) $\rho \left[ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} \right]=-\frac{\partial p}{\partial x}$ (2) $\rho \left[ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y} \right]=-\frac{\partial p}{\partial y}-\rho g$ (3)

The velocities and the pressure are decomposed as follows into base flow and perturbed components: $u=\bar{u}+{u}'\begin{matrix} , & v=\bar{v}+{v}' & p=\bar{p}+{p}' \\ \end{matrix}$ (4)

Substituting eq. (4) into eqs. (1) – (3) and considering the fact that the base flow should also satisfy eqs. (1) – (3), the equations for the flow simplify to $\frac{\partial {u}'}{\partial x}+\frac{\partial {v}'}{\partial y}=0$ (5) $\rho \left[ \frac{\partial {u}'}{\partial t}+\bar{u}\left( \frac{\partial {u}'}{\partial x} \right) \right]=-\frac{\partial {p}'}{\partial x}$ (6) $\rho \left[ \frac{\partial {v}'}{\partial t}+\bar{u}\left( \frac{\partial {v}'}{\partial x} \right) \right]=-\frac{\partial {p}'}{\partial y}$ (7)

In arriving at eqs. (5) – (7), the products of perturbation (primed) terms are neglected, and we recognize that $\partial \bar{u}/\partial x=\partial \bar{u}/\partial y=\bar{v}=0$ . Differentiating eqs. (6) and (7) with respect to x and y, respectively, then summing them and substituting the continuity equation, yields the Laplace equation for the pressure perturbation field: $\frac{{{\partial }^{2}}{p}'}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}{p}'}{\partial {{y}^{2}}}=0$ (8)

The shape of the interface at time t can be described by

 δ(x,t) = Aeiαz + βt (9)

and the perturbation quantities v' and p' can be postulated to have the following forms: ${v}'(x,y,t)=\hat{v}{{e}^{i\alpha x+\beta t}}$ (10) ${p}'(x,y,t)=\hat{p}{{e}^{i\alpha x+\beta t}}$ (11)

where $\hat{v}\text{ and }\hat{p}$ are the magnitudes of the perturbation. Employing the Young-Laplace equation and the equation for the curvature of the liquid film, ${{p}_{v}}-{{p}_{\ell }}=\sigma /R,$ Carey (1992) used the perturbation analysis to obtain the following condition for an unstable interface ${{\left| {{{\bar{u}}}_{\ell }}-{{{\bar{u}}}_{v}} \right|}^{2}}>\frac{\left[ \sigma \alpha +\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g/\alpha \right]\left( {{\rho }_{\ell }}+{{\rho }_{v}} \right)}{{{\rho }_{\ell }}{{\rho }_{v}}}$ (12)

where α = 2π / λ is the wave number.

Surface tension and gravity tend to stabilize the interface (for this configuration only). The right side of this inequality has a minimum when the wave number is equal to a critical wave number αcrit: ${{\alpha }_{crit}}={{\left[ \frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{\sigma } \right]}^{1/2}}$ (13)

It follows from eq. (12) that ${{\left| {{{\bar{u}}}_{\ell }}-{{{\bar{u}}}_{v}} \right|}_{\text{crit}}}={{\left[ \frac{2\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)}{{{\rho }_{\ell }}} \right]}^{1/2}}{{\left[ \frac{\sigma \left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{\rho _{v}^{2}} \right]}^{1/4}}$ (14)

For motionless liquid over motionless vapor ( ${{\bar{u}}_{\ell }}={{\bar{u}}_{v}}=0$), we obtain from eq. (14) $\alpha >{{\alpha }_{crit}}={{\left[ \frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{\sigma } \right]}^{1/2}}$ (16)

The critical wavelength corresponding to the critical wave number is ${{\lambda }_{c}}=2\pi {{\left[ \frac{\sigma }{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{1/2}}$ (16)

A perturbation with a wavelength greater than λc will grow and result in instability. If the length of the interface in the x-direction is less than λc, the interface is stable because a perturbation of wavelength greater than λc cannot arise. A specific value of α exists where β in eqs. (9) – (11) is at its maximum. Its value is ${{\alpha }_{\max }}={{\left[ \frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{3\sigma } \right]}^{1/2}}$ (17)

The disturbance wavelength corresponding to αmax is referred to as the most dangerous wavelength, λD ${{\lambda }_{D}}=2\pi {{\left[ \frac{3\sigma }{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{1/2}}=\sqrt{3}{{\lambda }_{c}}$ (18)

which has many applications, including derivation of the critical heat flux for pool boiling.

When the direction of the gravity is parallel to the direction of the liquid-vapor co-current flow as shown in Fig. 1, the gravity will not have a significant effect on the pressures in the liquid and vapor phases. Equation (13) becomes ${{\left| {{{\bar{u}}}_{\ell }}-{{{\bar{u}}}_{v}} \right|}^{2}}>\frac{\sigma \alpha \left( {{\rho }_{\ell }}+{{\rho }_{v}} \right)}{{{\rho }_{\ell }}{{\rho }_{v}}}$ (19)

which is the condition for an unstable vertical interface, and this condition is termed the Kelvin-Helmholtz instability.

## References

Carey, V.P., 1992, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corp., Washington, D. C.

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