What is Fluid Mechanics? A fluid is either a liquid or a gas. A fluid is a substance which deforms continuously under the application of a shear stress. A stress is defined as a force per unit area Next, what is mechanics? Mechanics is essentially the application of the laws of force and motion. Fluid statics or hydrostatics is the study of fluids at rest. The main equation required for this is Newton's second law for non-accelerating bodies, i.e. ∑ F = 0 . Fluid dynamics is the study of fluids in motion. The main equation required for this is Newton's second law for accelerating bodies, i.e. ∑ F = ma . Mass Density, ρ, is defined as the mass of substance per unit volume. ρ = mass/volume Specific Weight γ, is defined as the weight per unit volume γ = ρ g = weight/volume Relative Density or Specific Gravity (S.g or Gs) S.g (Gs) = ρsubstance / ρwater at 4oC Dynamic Viscosity, µ µ = τ / (du/dy) = (Force /Area)/ (Velocity/Distance) Kinematic Viscosity, v v = µ/ ρ Primary Dimensions In fluid mechanics there are only four primary dimensions from which all other dimensions can be derived: mass, length, time, and temperature. All other variables in fluid mechanics can be expressed in terms of {M}, {L}, {T} Quantity SI Unit Dimension Area m2 m2 L2 Volume m3 m3 L3 Velocity m/s ms-1 LT-1 Acceleration m/s2 ms-2 LT-2 -Angular velocity s-1 s-1 T-1 -Rotational speed N Force kg m/s2 kg ms-2 M LT-2 Joule J energy (or work) N m, kg m2/s2 kg m2s-2 ML2T-2 Watt W Power N m/s Nms-1 kg m2/s3 kg m2s-3 ML2T-3 Pascal P, pressure (or stress) N/m2, Nm-2 kg/m/s2 kg m-1s-2 ML-1T-2 Density kg/m3 kg m-3 ML-3 N/m3 Specific weight kg/m2/s2 kg m-2s-2 ML-2T-2 a ratio Relative density . no dimension no units N s/m2 N sm-2 Dynamic viscosity kg/m s kg m-1s-1 M L-1T-1 Kinematic viscosity m2 s-1 m2 s-1 L2 T-1 N/m Nm-1 surface tension kg /s2 kg s-2 MT-2 Table (1.1) Dimensions and Systems Dimension Length Time Mass Force (L) (T) (M) (F) System Slug Pound 1-American system Foot (ft) Sec 1 Slug = 32.2 Ib (Ib.wt) mass Pound mass Poundal 2-English system Foot (ft) Sec ( Ib mass) (pdl) 3-French system gm.mass Dyne Cm Sec (Metric system ) metric slug (gm.wt) 4-International Newton m Sec Kg.mass system (S.I system) (N) Newton's Second Law The fundamental relation between the mass and the weight is defined by Newton's Second Law and can be expressed as F = m a (1) where F = force (N) m = mass (kg) a = acceleration (m/s2) Mass Mass is a measure of the amount of material in an object, being directly related to the number and type of atoms present in the object. Mass does not change with a body's position, movement or alteration of its shape, unless material is added or removed. Weight Weight is the gravitational force acting on a body mass. Transforming Newton's Second Law related to the weight as a force due to gravity can be expressed as W = m g (2) where W = weight (N) m = mass (kg) g = acceleration of gravity (m/s2) The handling of mass and weight depends on the systems of units that is used. The most common systems of units are the International System - SI British Gravitational System - BG English Engineering System - EE 1 kg = 2.20462 lbf 1000 kg = 1 tonne (metric) = 0.9842 tons (imperial) 1 ton (US) = 2000 lb = 907.185 kg The International System - SI In the SI system the mass unit is the kg and since the weight is a force - the weight unit is the Newton (N). Equation (2) for a body with 1 kg mass can be expressed as: w = (1 kg) (9.807 m/s2) = 9.807 (N) (2b) where 9.807 m/s2 = standard gravity close to earth in the SI system As a result: a 9.807 N force acting on a body with 1 kg mass will give the body an acceleration of 9.807 m/s2 a body with mass of 1 kg weighs 9.807 N 1N= 1 kg m/s2 1N = 105 dyne 1 dyne = 1 gm cm/s2 The British Gravitational System – BG The British Gravitational System (Imperial System) of units is used by engineers in the English-speaking world with the same relation to the foot - pound - second system The three base units in the Imperial system are the foot, the second, and the pound-force. In the BG system the mass unit is the slug and is defined from the Newton's Second Law (1). The unit of mass, the slug, is derived from the pound-force by defining it as the mass that will accelerate at 1 foot per second when a 1 pound-force acts upon it: 1 lb = (1 slug)(1 ft/s2) 1slug = 1 Ib / 1ft/s2 In other words, 1 lb (pound) force acting on 1 slug mass will give the mass an acceleration of 1 ft/s2. The weight of the mass from equation (2) in BG units can be expressed as: w (lb) = m (slugs) g (ft/s2) With a standard gravity - g = 32.17405 ft/s2 - the mass of 1 slug weights 32.17405 lbf (pound-force). The English Engineering System - EE In the English Engineering system of units the primary dimensions are force, mass, length, time and temperature. The units for force and mass are defined independently the basic unit of mass is pound-mass (lbm) the unit of force is the pound (lb) alternatively pound-force (lbf). In the EE system 1 lb of force will give a mass of 1 lbm a standard acceleration of 32.17405 ft/s2. Since the EE system operates with these units of force and mass, the Newton's Second Law can be modified to F = m a (3) or transformed to weight w = m g (4) 2 1 lbf = (1 lbm)(32.174 ft/s ) or 2 Since 1 lbf gives a mass of 1 lbm an acceleration of 32.17405 ft/s and a mass of 1 slug an acceleration of 1 ft/s2, then 1 slug = 32.17405 lbm Density Density is defined as an objects mass per unit volume. The density can be expressed as ρ = m / V = 1 / vg (5) where ρ = density (kg/m3) m = mass (kg) V = volume (m3) 3 vg = specific volume (m /kg) The SI units for density are kg/m3. The imperial (U.S.) units are lb/ft3 (slugs/ft3). While people often use pounds per cubic foot as a measure of density in the U.S., pounds are really a measure of force, not mass. Slugs are the correct measure of mass. You can multiply slugs by 32.2 for a rough value in pounds. Relative Density (Specific Gravity) Relative density of a substance is the ratio of the substance density to the density of water at 4oC. Substance Relative density Alcohol 0.82 Mercury 13.95 Paraffin 0.8 Petrol 0.72 Water (4oC) 1 Sea water 1.02 Aluminum 2.72 Brass 8,48 Specific Weight Specific Weight is defined as weight per unit volume. Weight is a force. Specific Weight can be expressed as γ = ρ g (6) where γ = specific weight (N/m3) ρ = density (kg/m3) g = acceleration of gravity (m/s2) The SI-units of specific weight are N/m3. The imperial units are lb/ft3. The local acceleration g is under normal conditions 9.807 m/s2 in SI-units and 32.174 ft/s2 in imperial units. Example - Specific Weight Water Specific weight for water at 39 oF (4 oC) is 62.4 lb/ft3 (9.81 kN/m3) in imperial units. Specific weight in SI units can be calculated like γ = (1000 kg/m3) (9.81 m/s2) = 9810 N/m3 = 9.81 kN/m3 Specific Weight Some Materials Specific Weight - γ Product Imperial Units SI Units (lb/ft3) (kN/m3) Ethyl Alcohol 49.3 7.74 Gasoline 42.5 6.67 Glycerin 78.6 12.4 Mercury 847 133.7 SAE 20 Oil 57 8.95 Seawater 64 10.1 Water 62.4 9.81 Specific Gravity (Relative Density) The Specific Gravity - SG - is a dimensionless unit defined as the ratio of density of the material to the density of water at a specified temperature. Specific Gravity can be expressed as SG = ρ / ρH2O (7) where SG = specific gravity ρ = density of fluid or substance (kg/m3) 3 ρH2O = density of water (kg/m ) It is common to use the density of water at 4 oC (39oF) as reference - at this point the density of water is at the highest - 1000 kg/m3 or 62.4 lb/ft3. Example - Specific Gravity If the density of iron is 7850 kg/m3, 7.85 grams per cubic centimeter (cm3), 7.85 kilograms per liter, or 7.85 metric tons per cubic meter - the specific gravity of iron is: SG = (7850 kg/m3) / (1000 kg/m3) = 7.85 water density is 1000 kg/m3 Dynamic (absolute) Viscosity Absolute viscosity or the coefficient of absolute viscosity is a measure of the internal resistance. Dynamic (absolute) viscosity is the tangential force per unit area required to move one horizontal plane with respect to the other at unit velocity when maintained a unit distance apart by the fluid.