## Chapter 2

Pressure Distribution in a Fluid

## 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.

$$\frac{d}{dt}\iiint_{\Omega}\rho d{\mathscr{V}}+\oint_{\partial \Omega}\rho {\mathbf{v}}\cdot {\mathbf{n}}dS=0$$

#### 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.

$$\frac{d}{dt}\iiint_{\Omega}\rho{\mathbf{v}} d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho ({\mathbf{v}}\cdot {\mathbf{n}}){\mathbf{v}}+p{\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}} d{\mathscr{V}}$$

#### 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.

$$\frac{d}{dt}\iiint_{\Omega}\rho e_o d{\mathscr{V}}+\oint_{\partial \Omega}\left[\rho h_o {\mathbf{v}}\cdot {\mathbf{n}}\right]dS=\iiint_{\Omega}\rho{\mathbf{f}}\cdot{\mathbf{v}} d{\mathscr{V}}$$

### Study Guide

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

1. If you would hold your thumb tightly over the upper end of straw filled with water and held vertically, would the water not pour out - explain why. Is there a limit for how heigh a water pilar can be without pouring out? If so, how heigh can it be?
2. Show that the pressure difference in a fluid can be calculated as $$\Delta p=-\rho g\Delta z$$ starting from Newton's second law ($$F=ma$$), the fluid density can be assumed to be constant.
3. Show that the force balancing gravity in a fluid at rest is caused by the pressure gradient.
4. Show that the normal of a constant-pressure surface must be aligned with the gravity vector in a fluid at rest.
5. How does the hydrodynamic pressure distribution differ in liquids and gases?
6. How can we make use of Pascal's law when analysing manometer tubes?
7. Describe the implication of buoyancy for immersed bodies and floating bodies, respectively.
8. How is the buoyancy of a body immersed in a fluid calculated? Show how this relation is obtained.

 Document Archive MTF053_C02.pdf Lecture notes chapter 2 MTF053_Formulas-Tables-and-Graphs.pdf A collection of formulas, tables, and graphs