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In learning about fluid mechanics and surface tension, we learned that in particularly large pressure differences between two fluids and a relatively small diameter, we see either a concave or convex shape of the fluid.

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I am just wondering at the atomic level how this occurs and why this shape stays (as opposed to completely flat).

My idea is that as particles are always bouncing and colliding, the higher-pressure fluid continues to collide and some particles stick on the boundary surface (past the intuitive flat line). The boundary surface has greater intermolecular stickiness (surface of tube) compared to the lower stickiness of the two fluids (middle of the tube). I think this is why naturally it would result in concave or convex but if there's any intuition or insights, id be grateful to hear them. Thanks in advance.

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  • $\begingroup$ So, a heavier fluid should be concave right? Check out mercury. $\endgroup$
    – Solar Mike
    Dec 27, 2023 at 8:31
  • $\begingroup$ Water is convex like mercury until the surface is wetted. After that, it sticks to the wall so you have a concave surface. Mercury does not wet glass so it remains convex. $\endgroup$ Dec 27, 2023 at 21:01

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The concept to read up on is contact angle, which quantifies the qualitative concepts of hydrophobic vs hydrophilic. The closely related concepts are surface tension which results in the curvature we see.

At macro-scale, the two-phase boundary between the liquid and gas (the curved area), tends to a constant curvature, but this curvature depends on two things. One of them is the surface tension, which is a property of liquid and gas (and temperature etc). The other is pressure. There is a 3-way relation between surface-tension, curvature, and pressure, called the Young-Laplace equation.

Now we introduce the walls of the tube. Where the liquid-gas boundary surface meets the solid of the wall, is the three-phase contact line (a circle, i.e. where the curved liquid-gas boundary surface intersects the cylindrical walls if talking about a tube).

A cross section view tends to produce a characteristic value for the angle of the liquid-gas surface vs the wall, and this is called the contact angle. It is a property of all three materials!

Some solid materials will "grip" the same liquid-gas boundary more, resulting in a sharper contact angle. Literally, the solid exerts an opposing force on the liquid-gas boundary, if there is pressure attempting to push or stretch the liquid-gas boundary surface. This force is then reflected in more pressure difference on the liquid-gas boundary surface. Since the surface tension is the same (it's just a property of the liquid and gas), the resulting curvature will be more when the solid+liquid+gas combination has a sharper contact angle.

Roughly speaking, this curvature is then maintained, because if more pressure is applied, the contact line "gives way" and the entire liquid boundary moves forward, lets say, while keeping the same curvature.

Looking more closely, the next phenomenon is that the contact angle tends to have hysteresis. If you're pushing the fluid in one direction, then change it and begin to pull back the fluid, in terms of flow, then the contact angle isn't maintained exactly, but will allow some reduction before it gives way again in the opposite direction.

Looking even closer, contact angle phenomena, in practice, are affected by micro-scale geometry- i.e. surface roughness. The contact line won't give way all at once all around its circumference, but will jump in one spot, then in another, following any micro-contours of the wall. Might get caught a bit on a scratch in the glass, for instance.

Last but not least, the finest surface contamination (e.g. anything oily or soapy) is quite significant.

All this just sets up a detailed way of quantifying what happens, but doesn't answer the question for atomic scale unfortunately. Arguably into "physics" territory rather than "engineering". From what I don't actually understand, it can be approached from the direction of thermodynamics.

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Start with a horizontal, flat solid surface. A liquid on the surface meets the solid at a contact point. The angle formed into the liquid at the contact point is called the contact angle. We say that a liquid wets the surface when the contact angle is $< 90^\circ$. Otherwise, the liquid is non-wetting (non spreading) on the solid surface. For water, we use the specific terms hydrophilic (water loving or wetting) and hydrophobic (water hating or non-wetting).

Contact angle can be viewed from a microscopic (molecular) perspective as a result of differences for the attractive interactions between the liquid molecules, the solid surface molecules (or atoms), and the liquid-solid systems. The attractive interactions give rise to surface tension. Surface tension is the force that we would have to apply within the surface plane to stretch the molecules (or atoms) apart elastically. In a macroscopic thermodynamic sense, systems prefer to have interfaces that have the lowest surface tension. Hence, a liquid will prefer to wet (or spread) on a solid to a greater degree when the liquid has a lower surface tension than the underlying solid.

Now turn the horizontal surface to a vertical plane. Surface tension still exists. The liquid still "desires" to wet (or not wet) the solid. But gravity acts in a different direction. So, rather than a drop that is spread out on a horizontal surface, we obtain a meniscus that "rises up" (is concave) on the vertical wall. Alternatively, rather than a drop that is a bead (non-wetting) on a horizontal surface, we obtain a meniscus that is convex.

When we use wide tubes, the concave or convex boundary flattens out by the center of the tube. As we use smaller diameter tubes, the boundary shows up as a fully formed concave or convex meniscus.

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