AMME2261: Fluid Mechanics 1 Course Notes


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1 Module 1 Introduction and Fluid Properties Introduction Matter can be one of two states: solid or fluid. A fluid is a substance that deforms continuously under the application of a shear stress, no matter how infinitesimal it is. Therefore, any fluid at rest has zero shear stress. Fluids are divided into liquids and gases. Liquids have closely packed molecules with strong cohesive forces and retain their volume in a gravitational field. Methods of Analysis Analytical fluid mechanics involves obtaining general analytical solutions for basic laws of fluid mechanics, such as the conservation of mass and thermodynamic laws. Numerical fluid mechanics involves obtaining numerical solutions for the governing equations for a set of specific initial and boundary conditions. There is also a field of experimental fluid mechanics. A system is defined as a quantity of matter or a region in space chosen for study. The mass or region outside the system is called the surroundings, and the real or imaginary surface that separates a system from its surroundings is called the boundary. Boundaries may be fixed or movable and a system may be comprised of several of each type. A closed system or control mass, such as a pistoncylinder, has a fixed amount of mass and no mass flow across its boundary. A closed system is assumed if not otherwise specified. Energy, in the form of heat or work, can cross the boundary. The volume of a closed system may not be fixed. In special cases, when even energy is not allowed to cross the boundary, the system is called an isolated system. An open system or control volume encloses a device which has mass flow across its boundary, for example a turbine. Both mass and energy (heat or work) can cross the system boundary, which is called a control surface and can be real or imaginary. In general, any arbitrary region in space can be selected as a control volume. Reference Frames A Lagrangian reference frame is a moving reference frame considering individual components governed by Newton s second law. ΣF = ma = m dv dt = m d) r dt ) For the Eulerian or fixed reference frame, a fixed point in space is monitored and flow past it is observed to obtain field solutions v. Dimensions, Units and Consistency v = v(r, t) Fundamental dimensions are length, mass, time and temperature, while others such as force are expressed in terms of the primary dimensions and are called secondary or derived dimensions.
2 Density and Continuum Fluids are aggregations of molecules, in which the distance between molecules is very large compared to the molecular diameter. Molecules are not fixed in a lattice, so there is microscopic uncertainty. Also, macroscopic uncertainty occurs when the observation volume is very large and aggregate variations occur. However, this uncertainty diminishes when the observation volume is large compared to the molecular spacing. ρ = m V v = V m = 1 ρ SG = ρ ρ H2 O γ = ρg In large engineering applications, a fluid is called a continuum, as the variation of properties is sufficiently smooth that differential calculus can be used in the analysis. However, at very low pressures, the molecular spacing can be comparable to the size of the system. Then, the continuum approximation must be replaced by the molecular theory of rarefied gas flows. Viscosity Viscosity is a measure of a fluid s resistance to a shear stress. It determines the fluid strain rate that is generated by an applied shear stress F 9 /A <. A Newtonian fluid, such as water, oil or air, is a fluid in which there is a linear relationship between the applied shear and the resulting strain rate. δθ is the shear angle and the upper surface travels at velocity δu greater than the lower surface. Dynamic viscosity μ has units ML 1 t 1 and kinematic viscosity v has units L 2 t 1. τ = μ dθ du = μ dt dy v = μ ρ 2
3 NonNewtonian fluids do not have an exact linear relationship between shear stress and deformation rate. Fluids for which η decreases with increasing deformation ate are called pseudoplastic or shear thinning fluids, such as toothpaste or paint. Fluids for which η increases with increasing deformation rate are called dilatant or shear thickening fluids, such as sandy solutions. For these kinds of fluids, apparent viscosity η is used. τ = η du dy Surface Tension At an interface between two liquids, a liquid and a gas or a liquid and a solid, the liquid surface acts like a stretched elastic membrane in tension. This is caused by attractive intermolecular forces. For a droplet, these forces are not symmetrical, thus a net force pulls the surface towards the interior of a droplet. A liquid is wetting a surface when the contact angle between the surface and the tangent at the droplet edge is below 90. A nonwetted surface has the contact angle at above 90. F E = σπd 4σ cos θ Δh = ρgd 3
4 Module 2 Fluid Statics Pressure Pressure is the normal force exerted by a fluid per unit area, its units being the Pascal. Absolute pressure, the actual pressure at a given position, is measured relative to absolute zero pressure (or absolute vacuum ). Gauge pressure, the difference between the absolute pressure and the local atmospheric pressure, and vacuum pressure, which is pressure below atmospheric pressure, are measured relative to atmospheric pressure. Atmospheric pressure is measured by a barometer. We assume we are working with absolute pressure unless the problem explicitly states gauge pressure is used. +P gage is used when P abs > P atm and P gage is used for a vacuum gauge. P = F A While a static fluid at rest does not have shear stresses, there are normal stresses due to the weight, which cause pressure. Pascal s law states that in a fluid at rest, the pressure at a point is the same in all directions. P gauge = P abs P atm P vac = P atm P abs P abs = P atm ± P gauge Liquids are incompressible and therefore liquid density does not vary with depth. For bodies of liquid, it is convenient to take the reference point as the free surface exposed to air and measure positive distance down. P = P gauge = ρgh Gases are compressible and therefore gas density varies with depth due to changes in pressure and temperature. Hydraulic Pressure and Manometers Blaise Pascal ( ) knew that the force applied by a fluid is proportional to the surface area involved. He realised that two hydraulic cylinders of different areas could be connected, and the larger one could be used to exert a proportionally greater force than that applied to the smaller one (Pascal s Machine). Pascal s law states that the pressure applied to a confined fluid increases the pressure throughout by the same amount. The area ratio A 2/A 1 is called the ideal mechanical advantage of the hydraulic lift. P Z = P ) F Z A Z = F ) A ) F ) F Z = A ) A Z 4
5 Hydrostatic Forces on Plane Surfaces To determine the resultant force acting on a submerged surface, we must specify its magnitude, direction and line of action through which it acts at the centre of pressure with coordinates (x, y ). Here, the surface lies in the xyplane and the origin is at the intersection of the free surface and a line extending from the submerged surface. The hydrostatic force acts normal to the surface. y \ is the y coordinate of the centroid of area and P \ is the absolute pressure at the centroid. F ] = P^A + ρg sin θy \ A = (P^ + ρg sin θy \ )A = P \ A If there is air on one side of the plate, the atmospheric pressure component of F ] cancels out. F ] = ρg sin θy \ A = P \ gauge A Even though the magnitude of F ] can be calculated from the pressure at the centroid of area, this is not the point through which F ] acts (i.e. y \ y ; centroid of area centre of pressure). y d = y \ + ρg sin θi ff F ] x d = x \ + ρg sin θi hh F ] Where there is air at atmospheric pressure on the underside surface, the atmospheric pressure P^ cancels, which leads to the following equations. y d = y \ + I ff Ay \ x d = x \ + I hh Ax \ 5
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