What is the kinematic conditions for free-surface

[Chuck88]

When I am studying the Rayleigh-Taylor instability, I saw this equation:

[tex]
\frac{\partial \eta}{\partial t} + u' \frac{\partial \eta}{\partial x} = \omega ' (\eta)
[/tex]

I do not quite understand the meaning of this equation. Can some one provide me with some instructions and information.

The detailed information of Rayleigh-Taylor instability is presented below.

http://en.wikipedia.org/wiki/Rayleigh-Taylor_instability

 

[OldEngr63]

Without trying to wade through the meaning of your equation (I don't know what your symbols mean, etc), the basic kinematic condition for a free surface is that the velocity vector must be tangent to the surface, or stated differently, the component of velocity normal to the free surface must be zero.

 

[AlephZero]

the basic kinematic condition for a free surface is that the velocity vector must be tangent to the surface, or stated differently, the component of velocity normal to the free surface must be zero.

That is only true is the free surface is not moving. The general condition is that the normal components of the fluid velocity and the free surface velocity are equal.

 

[OldEngr63]

AlephZero is correct.

 

[alphy]

The kinematic conditions for a free-surface refer to the movement and deformation of a fluid interface, such as the interface between two different fluids or between a fluid and air. In order to study the Rayleigh-Taylor instability, it is important to understand the kinematic conditions that govern the behavior of the fluid interface.

The equation you have referenced is a simplified form of the Navier-Stokes equation, which describes the motion of a fluid. In this equation, η represents the height of the fluid interface, t represents time, x represents the position along the interface, u' represents the velocity of the interface, and ω' represents the vorticity or rotational motion of the fluid.

This equation describes the evolution of the interface over time, taking into account the velocity and rotational motion of the fluid. The Rayleigh-Taylor instability occurs when there is a density difference between two fluids, causing the heavier fluid to sink and the lighter fluid to rise. This can lead to the formation of complex structures and mixing patterns at the interface.

To fully understand the behavior of the Rayleigh-Taylor instability, it is important to also consider other factors such as the density and viscosity of the fluids, as well as the effects of surface tension and gravity. Further research and analysis is needed to fully understand the kinematic conditions and behavior of this phenomenon.

I recommend exploring the Wikipedia page on Rayleigh-Taylor instability for more detailed information and resources. Additionally, consulting with experts in the field or conducting further research may also provide a deeper understanding of this complex phenomenon.