Fundamental Fluid Properties and Transport Phenomena

Capillary Action and Surface Tension

Capillary action describes the phenomenon where the level of a liquid inside a narrow tube (relative to its container) is either raised or lowered. This height difference is maintained by surface tension forces. The direction and magnitude of this change depend on the liquid's surface tension and its interaction with the tube material (wettability).

The vertical component of the surface tension force acting on the tube walls must balance the weight of the liquid column of height h. Horizontal forces typically cancel out.

The capillary height h can be determined by balancing these forces:

• Surface tension force (vertical component): Fv = γ · 2πR · cosθ
• Weight of liquid column: P = ρ · g · πR2h

Equating these forces (Fv = P) yields Jurin's Law:

h = (2γ · cosθ) / (ρ · g · R)

Where:

• h is the capillary height
• γ (gamma) is the surface tension of the liquid
• θ (theta) is the contact angle between the liquid and the tube wall
• ρ (rho) is the density of the liquid
• g is the acceleration due to gravity
• R is the radius of the tube

If the contact angle (θ) is 90°, then h = 0, meaning no capillary rise or fall. This formula demonstrates that the larger the tube radius (R), the lower the capillary height (h).

Droplet Formation and Cohesion

In the absence of external forces, a volume of liquid naturally tends to assume a spherical shape. This is because a sphere minimizes the surface area for a given volume, thereby achieving the lowest possible surface energy.

When a liquid flows slowly through a pipe of small diameter, it often forms individual droplets. For a droplet to detach, the weight of the drop must overcome the vertical force produced by surface tension holding it to the nozzle. At the moment of detachment, there is a critical balance between the surface tension force and the drop's weight (gravitational force).

• Surface tension force (acting along the circumference of the nozzle): F = γ · L = γ · 2πr
• Weight of the drop: P = m · g = ρ · V · g

Where:

• γ (gamma) is the surface tension
• r is the radius of the nozzle or the neck of the drop at detachment
• ρ (rho) is the liquid density
• V is the volume of the drop
• g is the acceleration due to gravity

By equating these forces (F = P), the volume or mass of the detaching drop can be determined.

Vapor Pressure and Boiling Point

Vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. More commonly, it refers to the pressure at which a liquid boils and is in equilibrium with its own vapor.

If a liquid is heated, its vapor pressure increases. When this vapor pressure becomes equal to the value of the external (ambient) pressure, the liquid begins to boil.

Vapor pressure is a direct function of temperature:

Pvapor = f(T)

Understanding Cavitation

Cavitation occurs when the pressure within a liquid drops to or below its vapor pressure at a constant temperature. When this happens, vapor bubbles (cavities) rapidly form within the liquid. These bubbles can then collapse violently when they move into regions of higher pressure, generating shockwaves. This phenomenon can cause significant damage to pumps, propellers, and other fluid machinery.

While not 'boiling at room temperature' in the conventional sense of heating, it is the formation of vapor bubbles due to pressure reduction, similar to boiling due to temperature increase.

Thermal Expansion Coefficient

The thermal expansion coefficient (specifically, the volumetric thermal expansion coefficient, β) quantifies the change in volume of a substance in response to a change in temperature. This effect is generally small but significant for both liquids and gases.

It is given by the formula:

β = (1/V) * (ΔV/ΔT)

Where:

• β (beta) is the volumetric thermal expansion coefficient
• V is the initial volume
• ΔV is the change in volume
• ΔT is the change in temperature

Fluid Diffusivity: Viscous and Thermal

When different miscible fluid species come into contact, they will intermix through a process called diffusion, occurring at a specific diffusion rate (Vd). Beyond mass diffusion, fluids also exhibit viscous and thermal diffusivity.

These coefficients represent the efficiency with which certain properties are transported within a fluid:

Viscous Diffusivity (Kinematic Viscosity)

The coefficient of viscous diffusivity, also known as kinematic viscosity (ν), describes the rate at which momentum is transported through a fluid. It is a measure of a fluid's resistance to shear flow under gravity.

Thermal Diffusivity

The thermal diffusivity coefficient (α) quantifies the rate at which heat is transported through a fluid by conduction. It indicates how quickly temperature changes propagate within a material.

Modes of Fluid Transport: Diffusive vs. Convective

Fluid properties, such as mass, momentum, and heat, can be transported through two primary mechanisms:

Convective Transport

Convective transport involves the bulk, macroscopic motion of a fluid from one location to another. This mode of transport requires the physical displacement of fluid parcels, carrying their properties with them (e.g., a heated fluid rising).

Diffusive Transport

Diffusive transport relies on the random molecular motion within a fluid. It leads to the gradual mixing and homogenization of properties, even in the absence of bulk fluid flow (e.g., sugar dissolving in water, or milk slowly spreading through coffee).