Fluid Mechanics: Properties, Principles, and Biological Applications

States of Matter: Solids, Liquids, and Gases

Classification and Properties

The three primary states of matter—solid, liquid, and gas—are distinguished by the arrangement and binding of their molecules. Understanding these differences is fundamental to fluid mechanics and thermodynamics.

• Solids: Molecules are tightly bound, resulting in a definite shape and volume.

• Liquids: Molecules are less tightly bound than in solids, allowing liquids to flow and take the shape of their container, but they retain a definite volume.

• Gases: Molecules are not bound together, so gases have neither definite shape nor volume and expand to fill their container.

• Fluids: Both liquids and gases are classified as fluids due to their ability to flow.

Example: Water is a liquid at room temperature, ice is its solid form, and steam is its gaseous form.

Force and Pressure in Fluids

Transmission of Forces

Solids and fluids transmit forces differently due to their molecular structure and ability to flow.

• Solids: Forces applied to one section are transmitted unchanged in direction to other parts.

• Fluids: Forces are transmitted uniformly in all directions due to the ability to flow.

• Pressure in Fluids: At any point in a fluid at rest, pressure is the same in all directions and acts perpendicular to any surface in contact with the fluid.

Application: Hydraulic systems use fluid pressure to transmit force.

Pressure Variation with Depth

Fluid pressure increases with depth due to the weight of the fluid above.

• Pressure Difference Formula:

$P_2 - P_1 = \rho g h$

• $\rho$ = density of the fluid

• $g$ = acceleration due to gravity

• $h$ = vertical height difference

Units: Pressure is measured in Pascals (Pa) in SI units, and in millimeters of mercury (Torr) in other contexts. 1 Torr = pressure exerted by a 1 mm high column of mercury.

Pascal's Principle

Transmission of Pressure in Fluids

Pascal's Principle states that when pressure is applied to a confined fluid, the increase in pressure is transmitted equally to all parts of the fluid and to the walls of its container.

• Pressure Formula:

$P = \frac{F_1}{A_1}$

• $F_1$ = applied force

• $A_1$ = area over which force is applied

• Force Transmission:

$F_2 = P A_2 = \left(\frac{A_2}{A_1}\right) F_1$

• Pressure increase at one point is transmitted to all other points in the fluid.

Example: Hydraulic lifts use Pascal's principle to multiply force.

Hydrostatic Skeletons

Biological Application of Pascal's Principle

Some soft-bodied animals, such as worms, use a hydrostatic skeleton—a closed, elastic cylinder filled with fluid—to produce movement. Muscle contractions change the shape of the body by redistributing the fluid.

• Circular muscle contraction: Makes the animal thinner and longer.

• Longitudinal muscle contraction: Makes the animal shorter and fatter.

• Directional movement: Contraction on one side causes bending and changes direction.

Example: Earthworms move by sequential contraction of muscles, utilizing the principles of fluid pressure.

Bernoulli's Equation

Energy Conservation in Fluid Flow

Bernoulli's equation describes the relationship between pressure, velocity, and elevation in the flow of an incompressible fluid. It is derived from the conservation of energy principle.

• Bernoulli's Equation:

$P + \rho g h + \frac{1}{2} \rho v^2 = \text{constant}$

• $P$ = pressure in the fluid

• $\rho$ = density

• $g$ = acceleration due to gravity

• $h$ = height above reference point

• $v$ = velocity of the fluid

• Term meanings:

• Pressure energy per unit volume

• Gravitational potential energy per unit volume

• Kinetic energy per unit volume

Application: Explains why pressure decreases as fluid velocity increases in a constriction (e.g., narrowed arteries).

Viscosity and Poiseuille’s Law

Viscous Friction in Fluids

Real fluids experience internal friction, known as viscosity, which opposes relative motion between molecules and affects flow rates.

• Viscous friction: Proportional to fluid velocity and the coefficient of viscosity ($\eta$).

• Laminar flow: Fluid velocity is highest at the center of a pipe and zero at the walls.

Poiseuille’s Law for Laminar Flow

Poiseuille’s Law quantifies the rate of laminar flow through a cylindrical tube, accounting for viscosity.

• Poiseuille’s Law:

$Q = \frac{\pi R^4 (P_1 - P_2)}{8 \eta L}$

• $Q$ = volumetric flow rate

• $R$ = radius of the tube

• $L$ = length of the tube

• $P_1 - P_2$ = pressure difference between ends

• $\eta$ = coefficient of viscosity

Temperature dependence: Viscosity generally increases as temperature decreases.

Fluid Mechanics in Biology: Circulation of the Blood

Blood Flow and Pressure

The human circulatory system is often compared to a plumbing system, but blood is a complex fluid containing cells, and blood vessels are elastic, not rigid.

• Heart function: Electrical pulses trigger contraction of atria and ventricles, producing pulses of blood flow.

• Blood pressure: Systolic pressure (peak) ≈ 120 Torr; Diastolic pressure (minimum) ≈ 80 Torr; Average ≈ 100 Torr.

Control of Blood Flow

Blood flow to specific body regions is regulated by arterioles, which can constrict or dilate in response to nerve impulses and hormones.

• Arterioles: Small vessels (~0.1 mm diameter) with smooth muscle fibers.

• Constriction: Reduces blood flow to a region and diverts it elsewhere.

• Poiseuille’s Law: A 20% decrease in radius halves the flow rate (if pressure drop is constant).

Arteriosclerosis and Blood Flow

Arteriosclerosis is a disease where arterial walls thicken and narrow due to plaque deposits, severely affecting blood flow.

• Severity classification:

  • 50% narrowing: moderate

  • 60-70%: severe

  • 80%+: critical

• Bernoulli’s effect: Narrowing increases velocity and kinetic energy, but decreases pressure in the constricted region.

• Example: If artery radius is reduced by a factor of 3, area decreases by 9, velocity increases by 9, and pressure drops by 80 Torr.

• Clinical consequence: If pressure falls too low, external pressure may close the artery, blocking blood flow—potentially causing heart failure if the coronary artery is affected.

Condition	Arterial Narrowing	Severity
Moderate	50%	Moderate
Severe	60-70%	Severe
Critical	80%+	Critical

Additional info: The notes integrate fundamental fluid mechanics with biological applications, illustrating the importance of physics in understanding physiological processes.