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When a fluid is moving with velocity V, what becomes of its surface tension, does it increase or decrease in value

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Surface tension does not change with fluid velocity. Typically surface tension is only affected by solutes, surfactants, or temperature gradients. The subsequent effect of gradients in surface tension are referred to as Marangoni effects.

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Surface tension is not influenced by velocity, but it may become negligible compared to inertial forces which arise due to velocity.

Inertial forces scale as $\rho V^2L^2$ and surface tension forces scale as $\sigma L$. The ratio of these is the Weber number $We$:

$$ We = \frac{\rho V^2 L}{\sigma}$$

When $We >> 1$, or when the velocity $V$ or length scale $L$ are very large, surface tension becomes less important compared to inertia; you will not see the effects of surface tension for high $We$ but they still exist. The velocity by itself, however, does not affect surface tension.

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