CE 204 FLUID MECHANICS Onur AKAY Assistant Professor Okan University Department of Civil Engineering Akfırat Campus 34959 Tuzla-Istanbul/TURKEY Phone: +90-216-677-1630 ext.1974 Fax: +90-216-677-1486 E-mail: onur.akay@okan.edu.tr Onur Akay, Ph.D. CE 204 Fluid Mechanics 1
Mass Flow Rate(Review): The mass flow rate,, is the mass of fluid passing through a cross-sectional area per unit time [M/T]. (velocity vector is aligned with the area vector.) *The equations for discharge and mass flow rate are summarized in Table F.2. Onur Akay, Ph.D. CE 204 Fluid Mechanics 2
Reynolds Transport Theorem (Review): Onur Akay, Ph.D. CE 204 Fluid Mechanics 3
General Form of the Continuity Equation(Review): -The continuity equation derives from the conservation of mass (dm sys / dt= 0): Onur Akay, Ph.D. CE 204 Fluid Mechanics 4
Momentum Equation: Derivation Newton s second law: Force is equal to the time derivative of momentum. For a fluid system composed of a group of particles: Onur Akay, Ph.D. CE 204 Fluid Mechanics 5
Momentum Equation: Derivation How can we write this equation in Eulerian form? Recall the Reynolds transport theorem: B sys = Mom sys b= v Momentum Equation: Remember that the momentum equation is a vector equation! Onur Akay, Ph.D. CE 204 Fluid Mechanics 6
Momentum Equation: Derivation -If the flow crossing the CS occurs through a series of inlet and outlet ports, - and the velocity v is uniformly distributed across each port: The three components of the Momentum Equation for the Cartesian coordinate system: Onur Akay, Ph.D. CE 204 Fluid Mechanics 7
Force Diagram: In most cases we will use force diagrams (FD)to determine forces acting on the matter in the CV. Pipe schematic CV inside the pipe CV surrounding the pipe Onur Akay, Ph.D. CE 204 Fluid Mechanics 8
Force Diagram: Body force: Acts on mass elements within the body (gravitational forces). CV inside the pipe Surface force: Acts at the control surface (pressure and shear forces). CV surrounding the pipe Onur Akay, Ph.D. CE 204 Fluid Mechanics 9
Momentum Diagram: Momentum diagram is created by sketching a CV and then drawing a vector to represent momentum flow at each section. Onur Akay, Ph.D. CE 204 Fluid Mechanics 10
Application of the Momentum Equation: Fluid Jets: A fluid jet is created by a high-speed stream of fluid leaving a nozzle. p B = p C =p s (for subsonic jets) Onur Akay, Ph.D. CE 204 Fluid Mechanics 11
Application of the Momentum Equation: Nozzles: Flow devices used to accelerate a fluid stream by reducing the cross-sectional area of the flow. Onur Akay, Ph.D. CE 204 Fluid Mechanics 12
Chapter 6 Momentum Equation Application of the Momentum Equation: Vanes: Used to turn a fluid jet. Onur Akay, Ph.D. CE 204 Fluid Mechanics 13
Application of the Momentum Equation: Pipe Bends: Calculating the force on pipe bends is important in engineering applications using large pipes to design the support system. Onur Akay, Ph.D. CE 204 Fluid Mechanics 14
Navier-Stokes Equation: Differential equation for momentum at a point in the flow. Claude-Louis Navier Born: 10 February 1785, Dijon Died: 21 August 1836, Paris Sir George Gabriel Stokes Born: 13 August 1819, Ireland Died: 1 February 1903, England Onur Akay, Ph.D. CE 204 Fluid Mechanics 15
Navier-Stokes Equation: Onur Akay, Ph.D. CE 204 Fluid Mechanics 16