Chapter 2 Integral Forms of the Conservation Equations for Inviscid Flow 2-1 Problem Solving 1. Physics  2. Approach methods  3. Tools (mathematics)  4. Solve the problem Approach methods : Classical mechanics , no relativity： 1.Mass is conserved 2.Force = ma 3.Energy is conserved

A. Finite Control Volume Approach (macroscopic , Integral Form ) FIG. 2-1 Finite control volume approach

Statistical averaging → Boltzmann equation from kinetic theory B. Infinitesimal Fluid Element Approach (Macroscopic , Differential Form) Eulerian coordinate Lagrangian coordinate FIG.2-2 Infinitesimal fluid element approach C. Molecular Approach (Microscopic) Statistical averaging   → Boltzmann equation from kinetic theory Ref. Hirschfelder , Curtiss and Bird “Molecular Theory of Gases and Liquids”

Physical principle : Mass can be neither created nor destroyed. 2-3 Continuity Equation Physical principle : Mass can be neither created nor destroyed. denotes the mass flow through dS u are local velocity and density at B FIG.2-3 Fixed control volume

The total mass inside the control volume The net mass flow into the control volume through the entire control surface S Note : (- )  inflow (+)  outflow The total mass inside the control volume The time rate of change of this mass inside the C.V. Continuity equation (Integral Form) Mass is conserved Applies to all flows , compressible or incompressible , viscous or inviscid

Physical principle : The time rate of change of momentum of a 2-4 Momentum Equation Physical principle : The time rate of change of momentum of a body equals the net force exerted on it . Body forces on V , eg. Gravitational and E.M forces Total body force = Forces on the control volume Body force / mass 2. Surface forces on S , eg. P and τ Total surface force due to pressure = We consider inviscid flows only The total force acting on the C.V. is

= the net rate of flow of momentum The total time rate of change of momentum of the fluid as it flows through a fixed control volume = the net rate of flow of momentum across the surface S + the change in momentum in V due to unsteady fluctuations in the local flow (For fixed C.V.) Therefore : Momentum Equation

change in form. ------- the 1st law of thermodynamics (δQ+δW=dE) 2-5 Energy Equation Physical principle : Energy can be neither created nor destroyed ; It can only change in form. ------- the 1st law of thermodynamics (δQ+δW=dE) B1= rate of heat added to the fluid inside the C.V. from the surroundings B2= rate of work done on the fluid inside the C.V. B3= rate of change of the energy of the fluid as it flows through the C.V. B1 + B2 = B3 B1 : volumetric heating of the fluid in V due to a. Radiation b. Thermal conduction & diffusion (viscous effect) : the rate of heat added to C.V./mass

B2 = the rate of doing work on a moving body = = rate of work done on the fluid inside V due to pressure forces on S + rate of work done on the fluid inside V due to body forces = B3 = net rate of flow of energy across the C.V. + time rate of change of energy inside V due to transient variations of the flow field variables =

Therefore , More general form Energy Equation (Integral Form , Inviscid) More general form