Fluid Statics and Dynamics

• Fluid statics, dynamics, and determinants of flow are important topics.
• Topics include: mass density, ideal fluids, viscous fluids, pressure, resistance, flow concepts, laminar and turbulent flow.

Mass Density

• Mass density of a substance is the mass of the substance divided by its volume.
• Mass density is expressed as ρ = m/V.
• The SI unit of mass density is kg/m³.

Mass Density Values for Common Substances

• Aluminum has a mass density of 2700 kg/m³.
• Brass has a mass density of 8470 kg/m³.
• Concrete has a mass density of 2200 kg/m³.
• Copper has a mass density of 8890 kg/m³.
• Diamond has a mass density of 3520 kg/m³.
• Gold has a mass density of 19,300 kg/m³.
• Ice has a mass density of 917 kg/m³.
• Iron (steel) has a mass density of 7860 kg/m³.
• Lead has a mass density of 11,300 kg/m³.
• Quartz has a mass density of 2660 kg/m³.
• Silver has a mass density of 10,500 kg/m³.
• Wood (yellow pine) has a mass density of 550 kg/m³.
• Blood (whole, at 37 degrees Celsius) has a mass density of 1060 kg/m³.
• Ethyl alcohol has a mass density of 806 kg/m³.
• Mercury has a mass density of 13,600 kg/m³.
• Oil (hydraulic) has a mass density of 800 kg/m³.
• Water (at 4 degrees Celsius) has a mass density of 1.000 × 10³ kg/m³.
• Air has a mass density of 1.29 kg/m³.
• Carbon dioxide has a mass density of 1.98 kg/m³.
• Helium has a mass density of 0.179 kg/m³.
• Hydrogen has a mass density of 0.0899 kg/m³.
• Nitrogen has a mass density of 1.25 kg/m³.
• Oxygen has a mass density of 1.43 kg/m³.
• Unless otherwise noted, densities are given at 0 °C and 1 atm pressure.

Mass Density - Blood as Fraction of Body Weight (Example)

• A man with a weight of 690 N has approximately 5.2 × 10⁻³ m³ of blood.
• To find the blood's weight, multiply the volume of the blood by its density: m = Vρ = (5.2 × 10⁻³ m³)(1060 kg/m³) = 5.5 kg.
• The weight of the blood is then calculated as: W = mg = (5.5 kg)(9.80 m/s²) = 54 N.
• Expressing the blood's weight as a percentage of the body weight: precentage = (54 N / 690 N) × 100% = 7.8%.

Concepts of Pressure, Resistance, and Flow

• Pressure is the force applied perpendicular to the surface of an object per unit area.
• Pressure is expressed as P = F/A.
• The unit for pressure is 1 Pascal, which equals 1 N/m².
• Pressure is a scalar quantity.
• Gas pressure refers to the force exerted by gas particles colliding with the walls of its container.

Pressure and Depth in a Static Fluid

• Pressure equations in static fluids are presented, though specific equations are not provided in this summary.

Pressure Gauges

• Mercury barometers used to measure atmospheric pressure.
• The atmospheric pressure can be calculated using the formula Patm = ρgh, where ρ is the density of mercury, and h is the height of the mercury column.
• Standard atmospheric pressure is 1 atm = 760 mmHg.
• Mercury Manometers are used to measure pressure.

Pascal's Principle

• Pascal's principle states that a pressure change in one part of a fluid in a closed container is transmitted without loss to every portion of the fluid and to the walls of the container.
• The principle is mathematically expressed as F1/A1 = F2/A2.

Applications of Pascal's Principle

• Pascal's principle explains how sound is amplified in the middle ear.
• The effective area of the stapes footplate is about 20 times smaller than that of the malleus.
• Since the mechanical force remains constant through the bones, the pressure at the oval window is greatly amplified due to the smaller area.

Archimedes' Principle

• Archimedes' principle states that a fluid applies a buoyant force to an object that is partially or completely immersed in it. The magnitude of the buoyant force equals the weight of the fluid that the object displaces.
• The buoyant force (FB) is equal to the weight of the displaced fluid (Wfluid).
• If an object is floating, the magnitude of the buoyant force is equal to the magnitude of its weight.

Fluids in Motion

• Steady flow occurs if the velocity of the fluid particles is constant at any point as time passes.
• Unsteady flow exists when the velocity of the fluid particles at a point changes with time.
• Turbulent flow is an extreme case of unsteady flow, where the velocity of the fluid particles at a point changes erratically in both magnitude and direction.
• Fluid flow can be compressible or incompressible; most liquids are nearly incompressible.
• Fluid flow can be viscous or non-viscous.
• An incompressible, non-viscous fluid is called an ideal fluid.
• Streamlines are used to represent trajectories of fluid particles during steady flow.

The Equation of Continuity

• The mass of fluid per second that flows through a tube is called the mass flow rate.
• In a steady flow, if a certain mass of fluid flows into a region of space within a given time interval, the same amount of mass must flow out of that region during that same time interval.
• The equation of continuity is ρ1A1v1 = ρ2A2v2, and the SI unit of mass flow rate is kg/s.
• For incompressible fluids, A1v1 = A2v2.
• Volume flow rate is Q = Av.
• In the human circulatory system, the blood flow slows down in the capillaries.

Bernoulli's Equation

• A fluid will accelerate towards lower pressure regions.
• According to the pressure-depth relationship, pressure is lower at higher elevations, assuming the pipe's area does not change.
• Bernoulli’s equation relates pressure, fluid speed, and elevation in steady flow of a non-viscous, incompressible fluid at two points: P1 + ½ρv1² + ρgy1 = P2 + ½ρv2² + ρgy2..
• In hydrostatic conditions the equation simplifies to P1 + ρgy1 = P2 + ρgy2.

Application of Bernoulli's Equation

• Bernoulli's equation has applications in understanding aneurysms.
• A blood vessel with an aneurysm has a weakened wall that swells.
• A greater cross-sectional area leads to a slower velocity due to the continuity equation.
• From Bernoulli's equation, a smaller velocity will lead to higher pressure in the aneurysm, which is extremely dangerous because the vessel wall is already very weak.

Viscous Fluids

• Viscous fluids are fluids that have viscosity (μ > 0) and their motion is known as viscous flow.
• All fluids in nature are viscous, including water.

Viscous Flow

• The tangential force (F) needed to move a fluid layer at a constant speed is given by F = (ηAv)/y, where η is the coefficient of viscosity.

Poiseuille's Law

• The volume flow rate (Q) is given by Q = (πR⁴(P₂ - P₁))/(8ηL); ΔP is the pressure difference across the tube of length L.

Resistance in a Fluid System

• Resistance in a fluid system occurs due to friction between fluids and surfaces or between layers of a fluid.
• Resistance is calculated as R = ΔP/Q.
• The unit is N/m² or m³/sec.
• Fluid resistance has two effects: it decreases flow rate and creates pressure drop.
• Factors that influence fluid flow resistance are: the length of the pipe, inside surface, type of fluid flow, and fluid viscosity.

Concepts of Pressure, Resistance, and Flow

• Laminar flow is when the fluid moves smoothly in layers.
• Turbulent flow is when adjacent layers cross each other and do not move along a well-defined path.
• Turbulence is important in heart valves.

Reynolds Number

• The dimensionless Reynolds number helps describe the nature of fluid flow, whether laminar or turbulent.
• Reynolds number is defined as R = (Lρv) / η.
• L represents the characteristic size of the object.
• ρ is the fluid destiny.
• η is the viscosity of the fluid.
• v is the fluid flow velocity.

Blood and its Properties: Composition and Viscosity

• Blood consists of plasma (55%), white blood cells and platelets (4%), and red blood cells (41%).
• Water is 90% of plasma.
• Viscosity of whole blood increases with hematocrit (percentage of red blood cells).
• Velocity is inversely related to the cross-sectional area of the blood vessels.
• Blood flow speed is fastest in the aorta and large arteries and slowest in the capillaries.
• Determinants of blood pressure include heart rate, heart contraction force, resistance of blood vessels, blood volume, and blood viscosity.