1.3.2. Capillarity Height

Similar way also we derived the capillarity height in terms of the diameter of the capillarity tube.

Capillary height

h = (4 * σ * cos(θ)) / (ρ * g * d)

Where:

• h = capillary height
• σ = surface tension force
• θ = angle of contact
• d = diameter

Capillarity height is the height to which a liquid can rise or fall in a narrow tube due to surface tension. The equation provided expresses this height (h) in terms of various factors:

1. Surface tension (σ) creates a force that pulls the liquid upward.
2. The angle of contact (θ) indicates how wet the liquid is against the surface of the tube. A smaller angle improves capillary action.
3. The diameter (d) of the tube is important; a smaller diameter results in a greater capillary height because the surface area relative to the volume increases, enhancing the effect of surface tension.

Imagine a thin straw in a glass of water. When you place the straw in the water, the water rises up the straw due to capillarity. The narrower the straw is, the higher the water will rise. This phenomenon can be compared to a tiny sponge soaking up water. The narrower the space within the sponge, the more water it can absorb due to capillary action.

The surface tension force when we equate with a pressure difference and the area where it is acting it the force, both the side the force equating what you have done it the force the net force acting on this part is taking care of the force due to the surface tensions.

The concept of surface tension leads to a pressure difference across the liquid's surface in a capillary tube. When liquid rises due to surface tension, the upward pressure (from the liquid's internal forces) needs to balance the downward pressure caused by the weight of the liquid column. This equilibrium determines the height to which the liquid can rise. The balance of forces involves setting the force generated by surface tension equal to the hydrostatic pressure difference created by the height of the liquid column.

Think about a water droplet on a leaf: the droplet tends to hold its shape due to surface tension. If you have a tiny straw and place it in the droplet, the water inside the straw will rise up until the weight of the water column inside the straw balances the force from the surface tension pulling it up and the weight of the water itself.

As you know very basic things, when you consider z as a at the free surface level is zero. As z increases in the downwards the pressure will be P = ρgh.

In a static fluid, the pressure at a given depth (z) can be calculated using the formula P = ρgh, where:

• P is the pressure at depth z,
• ρ (rho) is the density of the fluid,
• g is the acceleration due to gravity,
• h is the height of the fluid column above that point. This relationship means that as you go deeper into a fluid, the pressure increases linearly with depth.

It's like diving into a swimming pool—if you're underwater, you can feel the pressure increasing as you go deeper. At the surface, your ears might feel normal, but as you descend, they may pop due to the increasing pressure, demonstrating how pressure builds with depth.

And very simple, what will be the pressure distribution in a fluid which is at the rest the static fluid will be the weight of the fluid divide by the area.

In a static fluid, to find the equilibrium state, it's essential to consider how the forces (like pressure and buoyancy) balance each other. In capillarity, the weight of the liquid column it counteracts the upward force due to surface tension. Essentially, when calculating pressures in a column of liquid, the pressure is determined by the weight of the liquid above divided by its cross-sectional area.

Imagine a tall tower of blocks. Each block adds weight, but the total weight is distributed over the area of the base. If you were to push down on the top of the blocks, the pressure at the bottom felt by the base would depend on how many blocks were stacked above and the area they cover.

Let me repeat these things that very simple things as we discuss the Newton laws of viscosity which establish the relationship between shear stress and the velocity gradient. Similar way we derive the capillarity height will be a functions of contact angles, surface tensions, and the diameter of the capillarity tube.

In summary, capillarity height is derived based on laws of physics-related forces acting on a fluid surface, specifically addressing the balance of surface tension, liquid cohesion, and density of the fluid. The established relationships allow for predicting how high a fluid will rise in a capillary tube based on its properties and the tube's characteristics.

If you've ever seen how a paper towel absorbs water, that's capillary action at work! The tiny spaces in the towel draw water upwards, demonstrating how surfaces allow liquids to climb when they are sufficiently narrow. This is the same principle applied in various engineering mechanisms, like how ink moves in a fountain pen.

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