DALMINE
Overview
Date/time interval
Syllabus
Course Objectives
Engineering problem-solving in the field of fluid mechanics is addressed by means of mathematical modeling, following the continuum mechanics approach.
Fluid properties (density, viscosity, pressure) are interpreted in view of molecular-level interactions modeled by means of suitable constitutive laws (ideal vs real, inviscid vs viscous, compressible vs incompressible fluid). Students are expected to familiarize with the physical principles that provide the flow governing equations (conservation of mass, balance of linear momentum and angular momentum, balance of mechanical energy and internal energy).
In order to derive the flow governing equations, students
- master index and tensor notations and are able to apply differential operators depending on the coordinate system,
- describe the motion of a fluid particle from both Lagrangian and Eulerian perspectives using kinematic properties,
- analyze volume and area variations along the trajectory and the state of deformation.
In order to solve basic engineering problems in fluid mechanics (fluid statics and forces on submerged surfaces/gates; ideal and real fluid flows in ducts and complex systems involving nozzles, valves, pumps and turbines; flow meters; external flows over airfoils and aerodynamics; De Laval nozzles with normal shock waves) students apply the flow governing equations relying on the control volume method and differential analysis.
By the end of the course, students are expected to autonomously select the most correct and appropriate simplifying assumptions to model real-world problems in fluid mechanics (e.g., steady, incompressible, one-dimensional flow) and assess the numerical and analytical results, evaluating their validity and physical consistency.
Furthermore, students are supposed to present the principles of fluid mechanics using correct mathematical formalism and appropriate technical terminology.
Students have developed the necessary methodological foundation to undertake the independent study of specialized topics (e.g., Computational Fluid Dynamics, Fluid Machinery), and will be able to consult and understand advanced textbooks, technical manuals, and scientific papers in the field.
Course Prerequisites
Students have knowledge of differential and integral calculus for multivariate functions. Students have background knowledge in physics, algebra and vector calculus.
In order to attempt the written test, students should have already cleared the exams of Fisica I, Geometria e Algebra Lineare, and Analisi matematica I. For students enrolled starting from the 2025/2026 academic year, inclusive, it is also necessary to have passed the course Analisi Matematica II.
Teaching Methods
Teaching will take place through lectures. All the lecturing material will be made available on the moodle platform. Practical training sessions will be organized sideways to theoretical sessions, aiming at solving engineering problems in the field of fluid mechanics. The simplifying assumptions introduced in the mathematical model will be discussed with the goal to critically analyze the reliability of the results.
Assessment Methods
The teacher reserves the right to require a supplementary oral interview, even after the written exam has been cleared, to properly assess the acquisition of the knowledge and skills outlined in the course learning objectives, and ensure a fair evaluation of the exam grade.
Contents
Introduction: purpose of fluid mechanics
Vector and tensor algebra
Change of basis for vectors and tensors
Tensor analysis, directional derivative and differential operators: gradient, divergence, curl
Fluid statics
- Pressure and pressure gradient
- Hydrostatic forces and their moments on plane and curved surfaces
- Archimedes force
Kinematics, analysis of deformation and motion
- Law of motion
- Lagrangian and Eulerian description
- Velocity acceleration and material derivative
- Trajectories and streamlines
- Homogeneous and rigid body deformation
- Strain tensor and spin tensor
- Deformation of volume and area
- Rate of change of volume
Continuum mechanics: fundamental physical principles in differential form and integral form over fixed, moving and deformable control volumes
- Conservation of mass
- Reynolds transport theorem
- Surface and body forces, Cauchy stress
- Balance law for linear momentum
- Balance law for angular momentum
- Kinetic energy and power balance
Objectivity of tensor field
- Objective physical properties and objective kinematic properties
- Inertial and non-inertial reference frames
Thermodynamics
- First and second law of thermodynamics
- Balance of internal energy
Constitutive laws
- Mechanic and thermodynamic constitutive restrictions
- Elastic fluids: Cauchy-Lagrange theorem, irrotational motion, Bernoulli theorem
- Compressible and incompressible viscous fluids: Newtonian fluids, vorticity transport equation, pressure Poisson equation
- Compressible non-isothermal viscous fluids: balance of internal energy and entropy
Dimensional analysis and similarity
- Nondimensionalization of the Navier-Stokes equations
- The Buckingam π theorem
- Ipsen method of repeated variables
Internal incompressible flows
- Fully developed flow in pipes
- Integral and differential analysis of Poiseuille flow in laminar and turbulent flow regime
- Modeling of viscous dissipation: distributed and localized pressure drops
- Piping systems and evaluation of pressure drops
External incompressible flows
- Differential and integral analysis of the flat plate boundary layer
- Displacement thickness and momentum thickness
- Boundary layer equations in laminar and turbulent flow regime
- Adverse pressure gradient, flow separation
Internal adiabatic compressible flows with area variation
- Axisymmetric nozzle and diffuser: consequences of area variation on the mass flow per unit area
- Stagnation properties and critical conditions
- Isentropic flow in nozzle and nozzle-diffuser
- Normal shocks
Helmholtz-Hodge decomposition, potential and stream function
Online Resources
More information
Books
Theory: Gurtin, An introduction to continuum mechanics;
Practical problems: White, Fluid-mechanics; Fox-McDonald's, Introduction to fluid mechanics.
Comprehensive lecture notes covering all the course topics will be provided.