Introduction to Fluid Mechanics

• Fluids are liquids and gases.

Mechanics of Fluids

• Statics studies fluids at rest.
• Kinematics studies fluids in motion, focusing on velocities.
• Dynamics examines fluids in motion, considering velocities, accelerations, forces, energies, impulse, and momentum.
• Fluid mechanics uses solid mechanics principles but is more complex due to the lack of separate elements.

Development of Fluid Mechanics

• Classical hydrodynamics is a branch of mathematics dealing with imaginary ideal fluids, which limits its practical application.
• Historically, engineers used experiments to develop empirical formulas for practical problems, known as hydraulics.
• As aeronautics, chemical engineering, and petroleum industries evolved, a broader approach was needed, combining classical hydrodynamics with real fluids (hydraulics).
• Fluid mechanics combines hydrodynamics principles with experimental hydraulics techniques.

Solid vs. Fluid

• Solids have closer molecules and stronger attractive forces compared to fluids.
• Solids maintain a fixed shape, while fluids have a variable shape in static conditions.
• Tangential stresses are always present in solids, but in fluids, they depend on the magnitude of velocity.

Gas vs. Liquid

• Gases have molecules that are far apart and are highly compressible, whereas liquids have closer molecules and are almost incompressible.
• Gases expand indefinitely without external pressure, but liquids do not due to cohesive forces, except for vapor pressure.
• Gas volume depends on temperature and pressure, while liquid volume mainly depends on temperature and is independent of pressure under normal conditions.

Gas vs. Vapor

• All vapors can be gases but not all gases can be vapor.

Density

• Density (ρ) is a fluid's mass per unit volume; Density = mass/volume.
• Mass is the amount of matter contained, measured in kilograms (kg) or pounds (lb).
• 1 kg = 2.204 lb
• Formula: ρ = m/V, measured in kg/.
• Density (ρ) is absolute, depending on mass and independent of location.
• Density is used to determine the mass of a fluid: mass of fluid = density x volume occupied.
• Volume calculation: Volume = length x width x height.

Specific Weight

• Specific Weight is the weight of fluid per unit volume; Specific weight = weight/volume.
• It is the force exerted by gravity on a unit volume of fluid.
• Weight is the measurement of gravitational pull: Weight = m*g.
• Formula: = W/V,
• Specific weight changes with location due to variations in g (gravity), influenced by latitude and elevation above mean sea level, hence it isn't absolute.
• Specific weight determines the weight of fluid: Weight of fluid = specific weight x volume and Volume = length x width x height
• Gravity on the moon is 1.625 m/s².
• Gravity at the equator is 9.780 m/s².
• Gravity at the poles is 9.832 m/s².

Density and Specific Weight Relationship

• Specific Weight = weight/volume = (mass x Acc. due gravity)/Volume = mass/volume x Acc. due to gravity
• Derived formulas: = ρg or ρ = /g

Specific Volume

• Specific volume (v) is the volume occupied by a unit mass of fluid and is commonly used for gases.
• Defined as specific volume = volume/mass.
• Formula: v = V/m.
• Specific volume and density have a reciprocal relationship: v = 1/ρ.

Specific Gravity

• Specific gravity (s) of a liquid is the ratio of its density to pure water density at a standard temperature - Physicists use 4, while engineers often use 15 degrees.
• Formula: Specific gravity of liquid = liquid density/water density.
• Specific gravity (s) of a gas is the ratio of its density to either hydrogen or air density at a specified temperature and pressure.
• Formula: Specific gravity of gas = gas density / air or hydrogen density.

Summary of Formulas

• = ρg or ρ = /g
• v = 1/ ρ

Typical Conditions on Earth

• Density of water, p = 1000 kg/= 1 g/ = 1 Mg/
• Specific weight of water, = 9810 = 9.81
• Specific gravity of water, s = 1
• Specific gravity of mercury, s = 13.55 or 13.6

Compressibility of Fluids

• Compressible fluids change density, like gases.
• Incompressible fluids maintain constant density, like liquids.
• No fluid is truly incompressible; the term applies when density change with pressure is negligible.
• Liquids are generally considered incompressible, but sound waves and water hammer show their compressibility.
• Gases are compressible but can act as incompressible if pressure changes are small, such as in ventilation systems.

Compressibility of Liquids

• Compressibility is the change in volume relative to pressure change (dv/dp), inversely proportional to the bulk modulus of elasticity.
• Water is 80 times more compressible than steel; mercury is 8 times, and nitric acid is 6 times more compressible than water.

Specific Weight of Liquids

• Unless specified, use 9.81 for water.

Equations of State for Gases

• Perfect gases don't exist, but air and other gases can be considered perfect if they are far from liquid phase.
• Equation of state: (1.4)
• Absolute pressure, density, specific volume (1/)
• R is the gas constant, depending on the gas.
• T is absolute temperature in degrees Rankine or Kelvin.
• Air's R value is 287 N.m/(kg.K), and other equations can be derived.

Gas Laws

• General formula from general gas equation P1 V1= RT1
• Avogadro's law states that equal volumes of all gases, at the same temperature and pressure, contain the same number of molecules.

Compressibility of Gases

• Pressure of 100 kPa, the isothermal modulus of elasticity for a gas is 100 kPa, and for air in an isentropic process it is 1.4 x 100 kPa.

Ideal Fluids

• Ideal fluids have no friction and zero viscosity (inviscid).
• Internal forces are normal to sections within the fluid, even during motion.
• Purely pressure forces.
• Many fluids approach frictionless flow at a distance from solid boundaries allowing analysis as an ideal fluid.

Real Fluids

• Real fluids develop tangential or shearing forces when there is motion relative to a body and friction exists.
• Viscosity is a fluid property caused by friction forces.

Ideal vs Real Fluids

• Ideal fluids are Incompressible, Ir-rotational, Non-Viscous and have Internal forces at any section that are always normal to cross section even during motion
• Real fluids are Compressible, Rotational, Viscous and tangential or shearing forces always develop whenever there is motion relative to body.

Viscosity

• A fluid's viscosity measures its resistance to shear or angular deformation; Motor oil is cohesive substance.
• Gasoline exhibits low viscosity.
• Forces in flowing fluid result from cohesion and momentum interchange between molecules.
• As temperature elevates, the viscosities of liquids decrease, and gases increase because cohesion diminishes in liquids.
• A rapidly moving gas molecule affects a slower layer.

Classic Case of Viscosity

• Particles adhere to walls at boundaries for viscous fluids, creating a no-slip condition; Fluid velocities must equal plate velocities where in contact.
• The velocity variation form between the upper and lower boundaries is the velocity profile.
• The velocity profile will be linear if the separation distance is not too great.

Absolute Viscosity

• Proportionality µ (mu), can express the shearing stressτ(tau) between any two thin sheets of fluid by
• Newton's equation of viscosity, is where Sir Isaac Newton first suggested it and defines the proportionality constant
• known as the coefficient of viscosity, the absolute viscosity, the dynamic viscosity,

Additional Info About Viscosity

• A Poies (P) is the unit viscosity.
• Kinematic viscosity is the ratio of absolute viscosity to density.
• Absolute viscosity is independent of pressure.
• Kinematic viscosity of gases varies strongly with pressure du to density changed

Kinematic vs. Dynamic Viscosity

• Dynamic viscosity offers information regarding the force necessary for achieving a particular flow rate.
• Kinematic viscosity, conversely, indicates the velocity attained upon the application of a specific force

Reynolds Number

• Reynolds number is the ratio of inertial force and viscosity; R
• V is the velocity of flow in fps, L is a characteristic length in ft equal to the hydraulic radius R of a conduit.

Types of Flow

• If R<500; flow is laminar
• If R>2000; flow is turbulent
• Between 500-2000, flow is transitional

Surface Tension

• Liquids have cohesion and adhesion.
• Cohesion enables a liquid to resist tensile stress and adhesion enables it to adhere to another body
• At a liquids surface, out of balance molecules exert a tension force in the surface known as surface tension.
• Surface tension is measured by comparing such forces by measuring the tension force per unit length of surface.
• Surface tensions decrease slightly with temperature and liquid surface is in contact with air.

Capillarity

• Capillarity: Fluids exert forces in fine tubes/porous media due to cohesion/adhesion.
• Cohesion and adhesion both exert forces on fluids
• Adhesion greater than Cohesion = liquid wets surface and rises at contact
• Cohesion greater than Adhesion = surface depressed at point of contact.