Fluid Mechanics Introduction Fluid mechanics study the fluid under all conditions of rest and motion. Its approach is analytical, mathematical, and empirical (experimental and observation). Fluid can be classified either as gases or liquids, the most notable difference between the two states are that liquids are more denser than gases and gases are highly compressible compared to liquids. Fluid mechanics consider one of the most fundamental subjects not only in civil engineering while in chemical, mechanical, and geotechnical, in addition to aeronautical and marine engineering. Fluid mechanics play a major role in the design and industry of aircrafts, aircraft carries, ships, boats, and submarines which are consider the last technology in advanced countries. Hydraulics is the science that study specifically water as a liquid. What is a fluid? A fluid is a material which is incapable of supporting shear force. What is a shear force? A Shear is a force that is applied tangentially to a material element. 1
Solid can support static deflections; fluids need a container. Units and Dimensions The basic dimensions used in fluid mechanics and hydraulics are: MLT : mass- length- time FLT : force- length- time Dimension English System Metric System International System Mass (M) Slug Gram Kilogram Force (F) Pound Dyne Newton Length (L) Foot Centimeter Meter Time (T) Second Second Second Newton s second low can relate between these dimensions. Force = Mass * Acceleration Acceleration due to gravity, g in SI units = 9.81 m/s 2 Acceleration due to gravity, g in BS units = 32.2 ft/s 2 2
Some useful conversion tables: Table 1.1 Primary Dimensions in SI and BG Systems: 3
Table 1.2 Secondary Dimensions in Fluid Mechanics: Properties of fluid 1. Mass density ρ It is the mass per unit volume (kg/m 3 ) Density of water in SI = 1000 kg/m 3, in BS= 1.94 slug/ft 3 2. Weight density γ It is the weight per unit volume (N/m 3 ) γ water = 9810 N/m 3 = 9.81 KN/m 3 From Newton s second low: 4
3. Relative density r.d. or specific gravity S.g. It is the ratio of fluid density to water density at 1 atmosphere and 4 C. S.g. oil= 0.85 Hydrometer A direct-reading instrument for indicating the density or specific gravity of liquids is called hydrometer. Float Section Stem 4. Specific Volume S.V. It is the volume per unit weight ( m 3 / N) 5. Compressibility and Elasticity of Fluid: Compressibility reflects the stress-strain properties of a material. Stress: internal response of a material to an external pressure Strain: measure of the linear or volumetric deformation of a stressed material. 5
The modulus of elasticity of fluid is define as (N/ m 2 ) 6. Viscosity of fluid: Viscosity of fluid is due to cohesion between fluid particles and also due to interchange of molecules between the layers of different velocities, its units in( Pa.s.). Viscosity measures a fluids ability to resist shear stress Hypothetical Experiment: 6
Real Experiment: One moving shaft inside another hollow shaft filled with oil. In the case the thickness of oil film is equal to: L = R i R s The relationship between the shear stress and viscosity of fluid is given by Newton s law of viscosity: Where τ = shear stress between fluid layers dy = thickness of fluid slip. dv = difference in velocity between adjacent slips. μ = dynamic viscosity of fluid (Pa.s.). 7
When τ in viscous fluid at a rest = 0 When y is too small yielding linear velocity profile and Shear stress at the solid surface. Fluids obeying Newton s law of viscosity for which μ has constant value (does not change with the deformation) are known as Newtonian fluids, for which shear stress τ is linearly related to velocity gradient is the absolute viscosity μ. and the slope of the line The ideal fluid, with no viscosity (μ = 0), falls on the horizontal axis, while the true elastic solid plots along the vertical axis. Fluids which do not obey Newton s law of viscosity are known as non- Newtonian fluids, There are certain non-newtonian fluids in which μ varies with the rate of deformation. These are relatively uncommon in engineering usage. Typical non-newtonian fluids include paints, printer s ink, gels and emulsions as shown in graph below: 8
7. Surface tension and Capillarity: In the figure below, molecule A is attracted in all directions equally by surrounding molecules (cohesion forces) thus the resultant force on A is zero. Molecule B located on the free surface of a liquid will experience resultant force downward, all the molecules on the surface experience downward force, thus the surface of a liquid acts as an elastic membrane under tension. Surface tension (σ) is the work (energy) per unit area required to bring the molecules to the surface N.m/ m 2 = N/ m. Thus surface tension will be formed when the liquids is in contact with gases. Water in contact with air has a surface tension of about 0.073 N/m at usual ambient temperatures. Molecular forces: Cohesion: is the inner force between liquid molecules. Adhesion: is the attraction force between liquid and a solid surface. Capillarity: is define as a rise or fall of a liquid surface in a small tube relative to adjacent general level of a liquid when the tube is held vertically in a liquid, it is expressed in terms of cm or mm of liquid. Capillary attraction is caused by surface tension and relative value of adhesion to cohesion of the liquid. For capillarity rise, at equilibrium the weight of a liquid of height h is balanced by the force of surface tension of a liquid. 9
Where: h capillary rise in the tube. σ surface tension. θ angle of contact between water, and tube. γ specific weight of water. r radius of the tube. For clean glass tube For capillarity depression, at equilibrium, the force of surface tension acting downward is equal to the hydrostatic force acting upward. Tension force = weight of fluid displaced For mercury θ = 130, σ = 0.51 N/m 11
8. Vapor Pressure When evaporation take place within an enclosed space, the partial pressure created by the vapor molecules in a space is called vapor pressure and depend on temperature and increase with it. When vapor pressure of a liquid equals the pressure above the liquid (atmospheric) boiling will begin. Ordinary evaporation is a surface phenomenon - some molecules have enough kinetic energy to escape. If the container is closed, equilibrium is reached where an equal number of molecules return to the surface. The pressure of this equilibrium is called the saturation vapor pressure. Temperature C Vapor Pressure (Kpa.) absolute 0 0.61 10 1.23 20 2.34 30 4.24 40 7.38 50 12.3 60 19.9 70 31.2 80 47.4 90 70.1 100 101.3 11