## Chapter 2

Integral Forms of the Conservation Equations for Inviscid Flows

## Integral Forms of the Conservation Equations for Inviscid Flows

### Overview

The second chapter is devoted to the derivation of the integral formulation of the governing equations for inviscid compressible flow.

### Sections

#### 2.2 Approach

The equation derivation approach is described.

#### 2.3 Continuity Equation

*Mass can be neither created nor destroyed*

Conservation of mass is applied on a control volume which leads to the continuity equation on integral formulation.

#### 2.4 Momentum Equation

*The time rate of change of momentum of a body equals the net force exerted on it*

Newtons's second law is applied on a control volume leading to the integral formulation of the momentum equation.

#### 2.6 Energy Equation

*Energy can be neither created nor destroyed; it can only change in form.*

The first law of thermodynamics is applied to a control volume leading to the integral formulation of the energy equation.

### Study Guide

The questions below are intended as a "study guide" and may be helpful when reading the text book.

- What is the physical interpretation of each of the terms in the continuity equation on integral form $$\frac{d}{dt}\iiint_{\Omega}\rho d{\mathscr{V}}+\oint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=0$$
- What is the physical interpretation of each of the terms in the momentum equation on integral form $$\frac{d}{dt}\iiint_{\Omega}\rho{\mathbf{v}} d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}+p{\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}} d{\mathscr{V}}$$
- What is the physical interpretation of each of the terms in the energy equation on integral form $$\frac{d}{dt}\iiint_{\Omega}\rho e_o d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho e_o ({\mathbf{v}}\cdot{\mathbf{n}}) +p {\mathbf{v}}\cdot {\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}}\cdot{\mathbf{v}} d{\mathscr{V}}$$
- How can the control volume formulations of the governing flow equations be used?
- Check through the three conservation theorems (the integral forms for the conservation of mass, momentum and energy) and make sure you understand how to apply them for a specific case. As a simple example, look at the derivation on the slides of Lecture 2. There are also some examples on how to use these relations in the book.
- In Formulas, Tables & Graphs, check the conservation laws. Make sure you are familiar with the notation.