1 CBE 341 Lecture 1 (9/13) – Concept of a fluid, fluid accelerating as a rigid body, buoyancy Definitions and concepts Elastic solid When an elastic solid is subjected to a small shear force, it will deform slightly and assume a new static configuration. The shear strain is defined as tan (g) ~ g for small values of g. If A is the area over which the normal and shear forces (Fn and Ft, respectively) are acting, then Normal stress = Fn/A Shear stress, t = Ft/A The shear modulus is defined as Shear modulus = Shear stress/shear strain = t/g. For a Hookean solid, the shear modulus is constant, independent of the magnitude of the shear strain (or equivalently, shear stress). FnFnFtFt2 Viscous fluid When a fluid element is subjected to the same set of forces as shown above, the fluid element will also deform; however, it will not assume a static configuration at any particular value of strain, g. Instead, the element will continue to deform indefinitely as long as the shear stress is acting on the fluid. Hence, it is not the deformation g, but the rate of deformation ddtggæö=ç÷èø, that is a measure of how a fluid responds to applied shear stress. g is called the shear rate or the strain rate or the rate of deformation. A typical velocity profile in the fluid element is sketched below. The viscosity, µ is defined as µ = Shear stress/shear rate = t/g. For Newtonian fluids, the viscosity is a constant independent of the shear stress (or equivalently, the shear rate). In this course, we will be mostly concerned with Newtonian fluids, but will touch upon non-Newtonian fluids briefly. All gases and common liquids such as water, liquid hydrocarbons, alcohols, glycerol and honey fall in the category of Newtonian fluids. In chemical industry, we also encounter a large variety of non-Newtonian fluids. Paints, colloidal suspensions and polymer melts are common examples. It suffices to say at the present time that for some fluids, referred to as shear thinning fluids, the viscosity decreases with increasing shear rate, while for others it increases with shear rate – shear thickening fluids.3 Units and dimensions M, L and t denote units of mass, length and time. - Shear or normal stress = force/area = F/L2 [units = M/(L.t2)] - Shear (strain) rate [units = 1/t] - Viscosity (common symbols, µ or h) = F.t/L2 [units = M/(L.t)]. Note that viscosity has the same units as Vr, where r is the density, V is velocity and denotes a length. - Kinematic viscosity, /n µ r= [units = L2/t]. Note that kinematic viscosity has the same units of V. - SI Units of viscosity: 1 N.s/m2 = 1 Pa.s = 1 pascal.second - CGS units of viscosity: 1 dyne.s/cm2 = 1 Poise - 1 Pa.s = 10 Poise Typical values to remember Viscosity of water ~ 1 centipoise (CP) = 10-2 Poise = 1 mPa.s [mPa = millipascal.] Viscosity of air = 10-2 CP = 10-5 Pa.s Kinematic viscosity of gases: In low density gases, where “viscous transport” of momentum occurs through the Brownian motion of the molecules, the kinematic viscosity scales as ~vln where v is the random velocity of the molecules and l is the mean free path between collisions. As molecules can execute random motion in three directions, only a third of the motion will be in any one direction, so a somewhat better estimate is 1~v3ln. A more accurate prefactor has been derived for low density gases, but a discussion of this derivation is not needed for this course. The random (also known as the Brownian) velocity, v scales as 1/ 2T where T is the absolute temperature, while l scales as (T/P) where P is the pressure. Thus, the4 kinematic viscosity of low density gases scale roughly as 3/ 2/.TP Note that the density scales as P/T, so that the viscosity is independent of pressure and it varies as 1/ 2T only. Experiments by Maxwell confirmed that the viscosity of gases is indeed essentially independent of pressure over a wide range of pressure – except when the pressure is so large that non-ideal behavior sets in or when the pressure is so low that the mean free path becomes comparable to the dimensions of the container in which flow occurs. The temperature dependence of viscosity of gases and liquids are distinctly different; while it increases with temperature for gases, it decreases with increasing temperature for liquids (see below).5 Control volume, Control surface, Traction In the study of mechanics, we will frequently consider a volume of fluid or solid or fluid-solid mixture, V, enclosed by a closed surface and apply balance laws. This volume V may be entirely arbitrary (which is the case when we derive general balance equations) or naturally suggested by the specific problem that we are trying to solve. Such a volume is commonly referred to as Control Volume and denoted by V or C.V. The closed surface bounding this volume of material is commonly referred to as Control Surface and denoted by S or C.S. If no mass transfer occurs across the control surface, the material bounded by the control volume is commonly referred to as a closed system in thermodynamics courses. Heat transfer is permitted in closed systems and the system was allowed to do work on the surrounding, the First law of Thermodynamics applied to closed systems is simply a relationship between the change in the internal energy of the system, amount of heat transferred and the amount of work done. In open systems, mass exchange between the system and the surrounding is permitted. The earlier courses extended the first law of thermodynamics to open systems as well. Please review your notes from CBE 245 and CBE 246 to refresh your memory of the first law. In the context of fluid mechanics, we will consider both closed and open systems. We will be concerned with forces acting on the material inside the control volume, some of which (such as the force due to gravity) act directly on the material inside – henceforth referred to as body force; force is also imparted on the material inside the control volume through the control surface – henceforth referred to as the surface force.6 Traction refers to force per unit area acting on the surface. Consider a differential surface element, dS, as shown on the figure below. The vector n in this figure denotes unit outward normal to the control surface at this surface element