Chapter 1. Introduction 1.1 Some Characteristics of Fluids We may have a general idea that a solid is hard and a fluid is soft. This is not satisfactory from scientific or engineering point of view. In reality, the solid has closely-spaced molecules with large inter-molecular cohesive forces that allow the solid to maintain its shape. However, for a liquid, the molecules are spaced further apart. (Q) What is the difference between liquids and fluids? A fluid is defined as a substance that deforms continuously when acted on by a shear stress of any magnitude. (Q) What is the shear stress and what is the normal stress? Rheology concerns the level of deformation of fluids when the shear stress is applied. Some materials such as slurries, tar, putty, toothpaste, and etc are like fluids, but behave as a solid if the applied shear stress is small. When we say that the velocity at a certain point in a fluid is so much, we are indicating that the average velocity of the molecules in a small volume surrounding the point. The number of molecules in a cubic mm is on the order of 10 21 for liquids. 1.2 Dimensions, Dimensional Homogeneity, and Units The qualitative description is conveniently given in terms of certain primary quantities such as length (L), time (T), mass (M), and temperature (θ ). The primary quantities are referred to as basic dimensions. These can be used to provide qualitative description of secondary quantities such as 1
area (L 2 ), velocity (L/T), and density (M/L 3 ). All theoretically derived equations are dimensionally homogeneous. That is, the dimensions at both sides coincide with each other. For example, V = V0 + at where all terms have a dimension of LT -1. 1.2.1 SI Unit The unit of work in SI unit is joule (J), which is given by 1 J = 1 N m and the unit of power is the watt (W) defined as a joule per second, i.e., 1 W = 1 J/s = 1 N m/s 1.3 Analysis of Fluid Behavior Like other mechanics, fluid mechanics uses such laws as Newton s law, conservation of mass, and first and second laws of thermodynamics. There are strong similarities between fluid mechanics and solid mechanics for rigid-body and deformable-body. Fluid statics is for fluid at rest, and fluid dynamics is for moving fluid. 1.4 Measure of Fluid Mass and Weight 1.4.1 Density The density ρ is the mass per unit volume. 2
1.4.2 Specific Weight The specific weight γ is fluid s weight per unit volume. That is, γ = ρg (1.5) 1.4.3 Specific Gravity The specific gravity (SG) is the ratio of density of the fluid to the density of water, i.e., SG = ρ (1.6) ρ w where ρ w is the water density at 4 C (= 1,000 kg/m 3 ). The specific gravity of mercury is 13.5. This means ρ Hg = 13.6 kg/m 3. 1.5 Ideal Gas Law The ideal gas law is given by p ρ = (1.7) RT where p is the absolute pressure, T the absolute temperature, and R is a gas constant. 1.6 Viscosity Consider a hypothetical experiment in which a fluid is placed between two plates. The bottom plate is fixed, and the upper plate is free to move. The experimental observation reveals that the fluid sticks to the bottom, which is referred to as no-slip condition. When the force P is applied to the upper plate, it will move continuously at a speed of U. The velocity distribution between two plates is given by u = u( y) = Uy / b. This indicates that the 3
velocity gradient is du / dy = U / b. Figure 1.3 Figure1.4 Behavior of a fluid between two plates In a small time δ t, the vertical line AB rotates through δβ, which is given by δa Uδt tanδβ δβ = = b b because δa= Uδt. Since δβ is a function of not only the force P but also time. So we define the rate of shearing strain, the rate at which δβ is changing, such as δβ γ = lim δ t which is equal to U du γ = = b dy Experiments reveal that the shear stress τ ( = P/ A) shearing strain, i.e., ) is directly proportional to the rate of 4
τ γ or du τ dy which can be given in the form of du τ = µ dy (1.8) where the constant of proportionality ( µ ) is the viscosity (= dynamic viscosity, absolute viscosity). Fluids for which the shear stress is not linearly related to the rate of shearing strain are designated as non-newtonian fluids. They are grouped into three types: For shear thinning fluids, the viscosity decreases with increasing shear rate. Examples are colloidal suspensions and polymer solutions. For shear thickening fluids, the viscosity increases with increasing shear rate. Examples are water-corn starch mixture and water-sand mixture (quick sand). The difficulty of removing an object from quicksand increases dramatically as the speed of removal increases. The third type is that of a Bingham plastic. This is neither a fluid nor a solid. The material can withstand the yield stress without deformation, but once the yield stress is exceeded it flows like a fluid. Examples include toothpaste and mayonnaise. 5
1.7 Compressibility of Fluids A property that is commonly used to characterize compressibility is the bulk modulus defined by E v = dp dv / V (1.11) where dp is the differential change in pressure needed to create a differential change in volume dv. 1.9 Surface Tension At the interface between a liquid and a gas, or two immiscible liquids, forces develop in the liquid surface that cause the surface behave as if it were a skin or membrane stretched over the fluid mass. Various types of surface phenomena are due to the unbalanced cohesive forces acting on the liquid molecules at the fluid surface. Molecules in the interior of the fluid mass are surrounded by fluid molecules that are attracted to each other equally. However, molecules along the surface are subjected to a net force toward the interior. The consequence of this unbalanced force along the surface is to create the hypothetical skin or membrane. The pressure inside a drop can be calculated by the free-body diagram below. If the surface tension is σ [F/L] around the edge, then we have 2πRσ = pπr 2 or 6
2σ p = R Figure 1.8 Forces acting on one-half of a liquid drop The rise of a liquid in a cube (capillary rise) is governed by the surface tension. That is, 2 γπ Rh= 2π Rσ cosθ where θ is the angle of contact. Therefore, the height is given by 2σ cosθ h = γ R Figure 1.9 Effect of capillary action in small tubes 7