1.6. Capillarity and Surface Tension

Capillary height is given by the formula:

$h = \frac{4\sigma \cos(\phi)}{\rho g d}$\n where:

• $h$ = capillary height
• $\sigma$ = surface tension force
• $\phi$ = angle of contact
• $d$ = diameter
• $\rho$ = density
• $g$ = acceleration due to gravity.

Capillarity is the ability of a liquid to flow in narrow spaces without the assistance of external forces. The height to which a liquid can rise in a capillary tube is determined by the balance between adhesive forces (the attraction between the liquid molecules and the tube wall) and cohesive forces (the attraction between liquid molecules themselves). The given formula quantifies this relationship, showing that capillary height depends on the surface tension of the liquid, the angle of contact with the tube, the fluid's density, gravity, and the diameter of the tube.

Think of capillarity like using a straw to drink juice. When you suck on the straw, the juice moves up against gravity, much like water rises in a narrow tube due to capillary action. The narrower the straw (smaller diameter), the higher the juice can rise because of the stronger adhesive forces acting between the liquid and the straw's wall.

The surface tension force can be equated with pressure difference across a liquid surface:

$F_{S} = \sigma \, A$\n This relates to the net force acting on the liquid due to surface tension.

Surface tension is the elastic tendency of a fluid surface that allows it to acquire the least surface area. The formula relates the force exerted by surface tension to the area of the liquid surface. When a liquid meets a gas (like air), the difference in pressure on either side of the liquid surface creates a net force. This net force can be influenced by the surface tension and acts to minimize the surface area.

Consider a soap bubble. Its surface is under tension, which keeps it spherical because this shape has the least surface area. If you poke the bubble, you disrupt this surface tension and the bubble pops, demonstrating how surface tension works to maintain its shape against external forces.

The pressure in a fluid at rest varies with depth:

$P = P_{0} + \rho g z$\n where:

• $P$ = fluid pressure at depth $z$
• $P_{0}$ = atmospheric pressure at the fluid surface
• $\rho$ = density of the fluid
• $g$ = acceleration due to gravity.

In a static fluid, the pressure increases as you go deeper due to the weight of the fluid above. This relationship can be represented with the formula where pressure at any depth is the sum of atmospheric pressure and the weight of the fluid above that point. The deeper you go into the fluid, the greater the pressure due to the additional weight of the fluid above.

Imagine you're at the beach and swimming. As you dive deeper underwater, you might feel the weight of the water pressing down on you—this is what is happening with pressure in a static fluid. The deeper you go, the more water (weight) is above you, increasing the pressure you feel.

For floating bodies, the stability can be analyzed using the concepts of the center of buoyancy (B) and the center of gravity (G). The distance between B and G determines stability.

A floating object is stable if the center of buoyancy (the centroid of the volume of displaced fluid) is vertically above the center of gravity (the point where the weight of the object is considered to act). If G is below B, the object will tend to return to an upright position when tilted. However, if G is above B, the object may capsize or tip over because the buoyant force won't be able to bring it back to its equilibrium state.

Think of a seesaw. If a heavier person sits closer to the middle, the seesaw remains balanced and stable. However, if a lighter person sits far from the middle, the seesaw may tip over, similar to how a floating object is stable or unstable depending on the position of its buoyancy and gravity centers.

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