Tuesday, 30 June 2015

fluid dynamics - How much of the forces when entering water is related to surface tension?


When an object enters water with high velocity, (like in Why is jumping into water from high altitude fatal?), most of it's kinetic energy will be converted, eg to accelerate water, deform the object etc. -
What is the relevance of the surface tension to this?


Are the effects related to surface tension just a small part, or even the dominant part regarding the forces.



Answer



Unless I have made a conceptual mistake (which is very possible), surface tension plays essentially no role in the damping of the impact of a fast-moving object with a liquid surface.


To see this, a simple way to model it is to pretend that the water isn't there, but only its surface is, and see what happens when an object deforms this surface. Let there be a sphere of density $\rho=1.0\text{g/cm}^3$ and radius $r=1\text{ft}$ with velocity $v=200\text{mph}$, and let it collide with the interface and sink in halfways, stretching the interface over the surface of the sphere.


Before the collision, the surface energy of the patch of interface that the sphere collides with is $$E_i=\gamma A_1=\gamma\pi r^2$$ and after collision, the stretched surface has a surface energy of $$E_f=\gamma A_2=2\gamma\pi r^2$$ and so the energy loss by the sphere becomes $$\Delta E=E_f-E_i=\gamma\pi r^2$$ which in the case of water becomes (in Mathematica):


<< PhysicalConstants`

r = 1 Foot;
\[Gamma] = 72.8 Dyne/(Centi Meter);
Convert[\[Pi] r^2 \[Gamma], Joule]


0.0212477 Joule



Meanwhile, the kinetic energy of the ball is $$E_k=\frac{1}{2}\left(\frac{4}{3}\pi r^3\rho\right)v^2$$ which is:


\[Rho] = 1.0 Gram/(Centi Meter)^3;
v = 200 Mile/Hour;

Convert[1/2 (4/3 \[Pi] r^3 \[Rho]) v^2, Joule]


474085 Joule



and hence the surface tension provides less than one millionth of the slowdown associated with the collision of the sphere with the liquid surface. Thus the surface tension component is negligible.


I'd suspect that kinematic drag provides most of the actual energy loss (you're basically slamming into 200 pounds of water and shoving it out of the way when you collide), but I've never taken fluid dynamics so I'll await explanations from people with more experience.


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