1.3. Formulas and Derivations

Newton's laws of viscosities established the relationship between shear stress and the velocity gradient.

Newton's Law of Viscosity
$\tau = \mu \frac{du}{dy}$
where:

• $\tau$ = shear stress
• $\mu$ = dynamic viscosity
• $\frac{du}{dy}$ = velocity gradient

Newton’s law of viscosity describes how fluid flows when a force is applied. Shear stress refers to the force per unit area that causes the fluid layers to slide past each other. The dynamic viscosity (μ) is a measure of a fluid's resistance to flow. When a fluid is in motion, the velocity of the fluid changes with distance from a surface. The velocity gradient (du/dy) shows how quickly velocity changes in relation to distance across the fluid. Therefore, the law essentially connects how hard you push (shear stress) and how quickly the fluid flows (velocity gradient).

Imagine stirring honey with a spoon. The harder you stir (greater shear stress), the faster the honey moves (higher velocity gradient). If you stirred very gently (less shear stress), the honey flows slowly, demonstrating how viscosity affects flow.

We derived the capillarity height in terms of the diameter of the capillary tube:

Capillary Height
$h = \frac{4\sigma \cos(\theta)}{\rho g d}$
where:

• $h$ = capillary height
• $\sigma$ = surface tension force
• $\theta$ = angle of contact
• $d$ = diameter of the capillary tube

Capillary rise occurs when a liquid rises in a narrow tube due to the adhesive forces between the liquid and the tube material. The height to which the liquid rises (h) depends on several factors: surface tension (σ) which attracts the liquid upwards, the contact angle (θ) which describes how the liquid interacts with the tube walls, the density of the liquid (ρ), gravitational acceleration (g), and the tube's diameter (d). This equation allows us to predict how high the liquid will rise based on these parameters.

Think of a straw in a drink. When you place the straw in the liquid, the liquid rises due to the interaction between the straw and the liquid. A thinner straw (smaller diameter) causes the liquid to rise higher compared to a wider straw, illustrating capillarity. The surface tension acts like tiny 'fingers' pulling the liquid upward.

When considering a fluid at rest, the pressure increases linearly with depth:

$P = P_0 + \rho g h$
where:

• $P$ = pressure at depth h
• $P_0$ = atmospheric pressure
• $\rho$ = density
• $g$ = acceleration due to gravity
• $h$ = depth

In a static fluid, the pressure at any point below the surface is determined by the weight of the fluid above it. This weight increases with depth, meaning that the pressure will also increase. The equation shows that the pressure at a given depth is the sum of the atmospheric pressure above the fluid and the hydrostatic pressure due to the fluid column above that depth. Essentially, the deeper you go, the more fluid weight is pushing down due to gravity, leading to higher pressure.

Picture yourself at the beach. When you dive underwater, you feel the pressure on your ears increase as you go deeper. This sensation is due to the weight of the water above you pushing down more as you swim closer to the ocean floor.

To analyze the stability of floating bodies, we look at the distance between the center of buoyancy (BM) and the center of gravity (BG):

1. Stable Condition: If BM is above BG
2. Unstable Condition: If BM is below BG

The stability of a floating object, like a boat or a buoy, is determined by the positions of its center of buoyancy (BM) and center of gravity (BG). If the center of buoyancy is above the center of gravity, the object will be stable; it tends to return to an upright position if tilted. However, if the center of buoyancy is below the center of gravity, the object will be unstable; it will tip over easily. Therefore, understanding these positions helps engineers design boats and devices to ensure they remain upright in water.

Consider a tall, thin lamp. If it's knocked, it will easily tip over because its center of gravity is higher than its support base, making it unstable. Now imagine a low, wide vase; it’s tougher to tip because the center of gravity is closer to ground level, increasing stability.

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