# FLUID MECHANICS

**Academic Year 2023/2024**- Teacher:

**Pietro SCANDURA**

## Expected Learning Outcomes

The main objective of the course is to provide the basic knowledge of Fluid Mechanics. After a preliminary part in which the physical characteristics of fluids are described, with particular emphasis on those that distinguish them from solids, the course introduces the fundamental topics of Fluid Mechanics, accompanied by the necessary theoretical framework. The course also includes lessons to be dedicated to carrying out exercises in the classroom, relating to the application of the principles of Fluid Mechanics to the solution of engineering problems.

## Course Structure

The teaching will be carried out through lectures and classroom exercises.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

**Introduction to the course **

Definition of fluid substance. The continuum hypothesis. Dimensions and measurement units. Mass forces and surface forces. Stress tensor and its properties. Fluid properties: compressibility, thermal elasticity, surface tension, viscosity. Non-Newtonian fluids. Gas absorption.

**Fluid statics**

Stress in fluids at rest. Equations of fluid statics in differential form. Equation of fluid statics in global form. Statics of incompressible fluids under the action of gravity force. Stevin's law. Pressure measurements. Hydrostatic thrusts on plane surfaces. Hydrostatic thrusts on curved surfaces. Hydrostatic thrusts on immersed body. Fluids of low specific weight.

**Fluid kinematics **

Generalities on fluid kinematics. Material volume and control volume. Lagrangian and Eulerian approaches. Local derivative and material derivative. Velocity and acceleration. The Reynolds transport theorem. Trajectories. Streamlines. Smoke lines. Flux tube. Steady flow. Unsteady flow. Uniform flow. Two-dimensional flow. Continuity equation in differential form. Continuity equation in global form. Continuity equation for currents. Rotation and deformation of fluid elements. Irrotational flows.

**Fluid dynamics**

Momentum equation in differential form. Euler equation. Boundary conditions. Momentum equation in global form. Example of application.

Euler equation in the intrinsic coordinate system. Pressure distribution in gradually varied currents. Bernoulli's theorem. Geometric representation of Bernoulli’s theorem. Energy implications of Bernoulli's theorem. Mechanical energy equation. Outflow phenomena. Extension of Bernoulli theorem to a current. Exchange of energy between a current and a machine. Venturimeter. Pitot tube. Extension of Bernoulli's theorem to real fluids. Continuous head losses. Localized head losses.

Constitutive relationship of viscous fluids. The Navier-Stokes equations. Momentum equation of viscous fluids in global form.

**Confined flows**

Reynolds experiment. Flow regimes. Laminar regime. Turbulent regime. The Reynolds number. Tangential stresses in uniform flow. Laminar flow in a pipe. Poiseuille formula. Laminar flow between parallel plates. Turbulent flow. The mean flow equation. Viscous and turbulent shear stresses. Application of the Buckingham theorem to the determination of the resistance law formula. Darcy-Weisbach formula. Friction factor. Flow in smooth pipes. Friction factor in smooth pipes. Mean velocity distribution of a turbulent flow in a smooth pipe. Flows in rough pipes. Nikuradse's experiments. Friction factor in rough pipes. Velocity distribution in turbulent flow in a rough pipe. Commercial pipes. Colebrook and White formula. Moody chart. Empirical formulas for the calculation of the head losses. Localized head losses. Head loss due to abrupt enlargement, sharp-edged inlet and outlet in a reservoir. Flows subject to negative pressure.

## Textbook Information

1) *Y. A. Cengel, J.M. Cimbala "Meccanica dei Fluidi " Quarta Edizione, McGraw-Hill.*

2) G. Alfonsi, E. Orsi "*Problemi di Idraulica e Meccanica dei Fluidi*" CEA Milano, 1984.

3) G. Pezzinga "*Esercizi di Meccanica dei Fluidi*" Aracne editrice, 2008.

4) Notes provided by the teacher available during the course on the Studium Platform.

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Definition of fluid substance. The hypothesis of continuity. Dimensions and units of measurement | 1, 4 |

2 | Mass forces and surface forces. Stress tensor and its properties. | 1, 4 |

3 | Compressibility, thermal expansion, surface tension, viscosity. | 1,4 |

4 | Non-Newtonian fluids. Absorption of gases. | 1, 4 |

5 | Stresses in fluids at rest. Indefinite equation of the statics of fluids. Global equation of static equilibrium. Statics of heavy and incompressible fluids. Measurement of pressures. Thrusts on flat surfaces. Thrusts on curved surfaces. | 1, 4 |

6 | Fluid kinematics. Eulerian and Lagrangian approaches. Velocity and acceleration. Trajectories. Streamlines. Smoke lines. Flux tubes. Steady motion. Unsteady motion. Uniform motion. | 1, 4 |

7 | Two-dimensional flows. Laminar flow. Turbulent flow. Differential equation of continuity. Global equation of continuity for fixed control volumes in space. Continuity equation applied to currents. | 1, 4 |

8 | Deformation of fluid elements. Rotation and deformation speed. | 1, 4 |

9 | Differential equation of motion. Euler equation. Boundary conditions. Global equation of dynamic equilibrium. | 1, 4 |

10 | Equations of motion in the triad intrinsic to a trajectory. Distribution of pressures. Bernoulli's theorem. Geometric representation of Bernoulli's theorem. Energetic significance of Bernoulli's theorem. Outflow processes. Venturi meter. Pitot tube. | 1, 4 |

11 | Extension of Bernoulli's theorem to real fluids. Extension of Bernoulli's theorem to currents. Exchange of energy between a current and a machine. | 1, 4 |

12 | Constitutive bond of viscous fluids. The Navier-Stokes equations. Global equation of the dynamic equilibrium of viscous fluids. | 1, 4 |

13 | Drag action of a current. Tangential stresses. Laminar flow in a circular section duct. Poiseuille formula. Flow between flat and parallel plates. Turbulent flow. The equation of mean motion. | 1, 4 |

14 | Viscous and turbulent tangential stresses. Application of Buckingham's theorem to determine the shape of the resistance law. Darcy-Weisbach formula. Resistance index. | 1, 4 |

15 | Flow in smooth tubes. Resistance index in smooth tubes. Notes on the velocity distribution in turbulent flow in a smooth tube. Flow in rough pipes. Resistance index in rough pipes. Moody's abacus. | 1, 4 |

16 | Velocity distribution in turbulent flow in a rough tube. Practical formulas for the calculation of resistance. Localized pressure drops. Energy dissipation due to abrupt widening, sharp edge inlet and outlet in a tank. | 1, 4 |

17 | Flows in pipes at pressures lower than atmospheric. | 1, 4 |

## Learning Assessment

### Learning Assessment Procedures

The examinations consist of a written and an oral test. The written test consists of the numerical solution of fluid mechanics problems and the answer to a theoretical question. There are 10 questions in total. The student will be admitted to the next oral test if he/she scores 15/30 or higher in the written test. The student who achieves a score of 18/30 or higher in the written test has the right to acquire teaching credits without taking the oral test.

There are three written tests during the course. Students who, on the day of the test, have attended at least 70% of the lessons of the course already held will be admitted to the test.

In each test the student shall report a mark expressed out of thirty. If the mark given in each test is greater than or equal to 15/30 and the average of the marks given in the three tests is greater than or equal to 18/30, the student is entitled to acquire teaching credits with a mark equal to the average of the marks given in the three tests.

The results of the ongoing tests are valid only for the exam session that immediately follows the end of the course.

When the average of the marks obtained in the ongoing tests is greater than or equal to 15/30, during the exam session following the end of the course, the student has the right to take an oral test in addition to the three written tests.

In one of the examination sessions immediately after the end of the course, the student has the right to repeat one of the three written tests. The mark obtained in this test will be added to the marks obtained in the other two tests and, if the average is greater than or equal to 18/30, the student will pass the exam without having to take the oral test, which remains optional.

Verification of learning can also be carried out electronically, should the conditions require it.

They can also contact the CInAP (Centre for Active and Participatory Integration - Services for Disabilities and/or SLDs) reference teacher in their Department.