The course aims to provide Mechanical Engineering students in their sophomore year with the fundamentals of Dynamics. The emphasis is on understanding the physical principles governing motion of rigid bodies and applying them to solve engineering problems. This course has a strong component on modeling and simulation.

Prerequisite: Grade of C- or better in MEEG112 or equivalent.

R. Valéry Roy, A First Course in Engineering Dynamics, (2nd Edition), Hyperbolic Press, 2016.

- Optimal projectile trajectory to reach top of “spherical” building.
- Computation of pursuit trajectory.
- Simulation of the drop of a toast from edge of table.
- Simulation of the toppling of a line dominos.

This course treats the study of the motions of material systems and the mechanical actions responsible for these motions. Material systems are assumed to be rigid bodies. The course starts with the topic of Kinematics, that is, the study of the motion of particles and rigid bodies without reference to the physical causes (forces) of the motion. Kinematics is the foundation of this course. The ultimate goal is the formulation of equations of motion and the extraction of information from these equations. This may include the search for first integrals of motion, the study of stability of a particular motion, the determination of external forces/moments to guarantee a particular trajectory, etc.

R. Valéry Roy, Advanced Engineering Dynamics, Second Edition, Hyperbolic Press, (2018).

- Parametrization of rigid body 2. Particle kinematics 3. Rigid body kinematics 4. Mechanisms 5. Mass distribution 6. Mechanical actions 7. Newton-Euler formalism 8. Kinetic energy theorem 9. Gyroscopic phenomena.

This course exposes graduate students to fundamental mathematical concepts with applications to enginnering problem. Specifically, four topics are covered: (i) Complex Analysis, (ii) Green’s function, (iii) Variational Calculus and (iv) Asymptotic Analysis. Emphasis is placed on problem-solving and application to mechanical engineering problems.

**I. Complex Analysis:** 1. Complex numbers, complex variables. 2. Analytic functions.
3. Multivalued functions, branch points, branch cuts. 4.
Integration in the complex plane. 5. Cauchy’s theorem. 6. Derivatives of analytic functions,
Taylor and Laurent series, singularities. 7. Conformal Mapping.

**II. Green’s Functions:** 1. Generalized functions and Dirac’s delta function.
2. Green’s Functions for one-dimensional boundary value problems. 3. Green’s function for the Poisson equation.
4. Green’s function for the Helmholtz operator and the diffusion operator.

**III. Calculus of Variations:** 1. Examples of variational problems. 2. The Euler-Lagrange equation.
3. Constraints and Lagrange multipliers. 4. The Second Variation. 5. Multi-dimensional Variational Problems.
6. Examples in mechanics: Hamilton’s Principle, Optimal control.

**IV. Asymptotic Analysis:** 1. Dimensional analysis and basic definitions. 2.
Asymptotic solution of algebraic equations. 3. Asymptotic expansion of integrals.
4. Regular Perturbation of ODEs and PDEs. 5. Singular Perturbation of ODEs:
Asymptotic matching, Boundary layer theory, WKB approximation. 6. Multiple scales analysis
of temporal systems. 7. The method of homogenization: application to the diffusion
equation.

Many natural and technological processes involve phenomena which take place over widely differing scales of time and space. The advent of novel micro and nanofabrication techniques is challenging the scientific community to design engineering structures with submicrometer feature size in many fields of applications. However, the development of these new technologies must be based on comprehensive theoretical modeling and computational predictions.

Multiscale systems abound in many modern technologies: composite materials, photonic and phononic materials, metamaterials, energy conversion systems, and more.

This course will specifically focus on the modeling and simulation of transport processes in multiscale systems,
by using a combination of mathematical modeling, continuum theories, and advanced computational techniques.
The role of multiscale methods is to provide the link between the microscopic features of a structure
(e.g. porous material) and its macroscopic or bulk properties.
They provide a fundamental understanding of how microstructure affects a system’s performance.
Although the microscopic details of the
system have a significant effect on its behavior, we are often not interested in the knowledge of its
behavior on the smallest scale. In fact, such solutions are not computationally feasible.
A classical method for multiscale periodic and random media is the method of *Homogenization* which will
be described at length for a variety of problems. When homogenization fails to yield a model in closed form,
or when the numerical solution of the homogenized model is untractable,
then one must resort to numerical homogenization techniques based on finite element
schemes.

**Part I: Introduction**

- Examples of multiscale problems in science and engineering.
- The challenge of multiscale problems & goals of a multiscale theory.
- Modeling strategies.
- An overview of classical analytical & numerical methods.
- 1D Example: Homogenization by Multiple Scale Expansion.

**Part II: Multiscale Conduction Problems**

- Historical Background
- General Notions on Elliptic Equations: physical relevance, Variational Formulation, Finite Element Approximations, Finite Element Spaces (2D), Application to Poisson Equation, Illustration with FreeFem.
- Homogenization and multiscale asymptotic expansions applied to the heat conduction equation.
- Properties of the homogenized conductivity tensor.
- Closed-form/approximate/numerical solutions of the homogenized conductivity tensor.
- Optimal bounds. 5. Convergence properties & error estimates.
- Correctors.
- Generalization to other problems: perforated microstructures, Robin boundary conditions, elasticity, evolution equation.
- Extension to quasi-periodic and random materials.
- Multiscale Finite Element Methods for Elliptic Problems.

**Part III: Fluid Flow in Porous Media**

- Mathematical notions on Stokes equations. Illustration with FreeFem.
- From Stokes equations to Darcy’s law: derivation by 2-scale asymptotic equations for periodic medium.
- Properties of the permeability tensor.
- Numerical determination of the permeability tensor for fiber geometries and spherical packings.
- Extensions: inertial, compressibility and nonlinear effects. Locally periodic and random porous media.
- Effective boundary conditions and existence of boundary layers.

**Part IV: Solute Transport and Reaction in Porous Medium**

- What is Taylor dispersion?
- Application: transport of DNA particles in engineered porous media.
- Passive solute transport through saturated porous media.
- Numerical determination of the dispersion tensor.
- Reactive Transport in Porous Medium.

Why do small drops stay attached to a windshield? While the formation of foam is a necessity in shampoos, how can it be avoided in dishwashers? How can leaves be wetted optimally during the application of insecticides while preventing the formation of drops on the soil? The answers to these questions lie in the study of interfacial phenomena, the subject of this course. Many technological and natural processes involve phenomena dominated by interfacial mechanics, such as in coating, printing, pharmaceutical, microfluidics and oil technologies, but also in the biology of the lungs or the cell. In addition to capillary, viscous and gravitational forces, interfacial phenomena involve the interplay of other complex physico-chemical processes such as intermolecular forces, surface active materials, and evaporation.

**I. Capillarity:** 1. Surface tension: physical origin, energy and force
interpretation, physical measurement. 2. Static capillary interfaces:
Laplace equation, pression discontinuity, minimal area surfaces.
3. Three-phase contact and wetting: classification of wetting behavior,
the spreading parameter, Young equation and equilibrium contact
angle, Zisman rule, liquid substrate.

**II. Capillarity and Gravity:** 1. Capillary length. 2. Shape of drops in
partial wetting. 3. Meniscus: characteristic shape ans scales, meniscus
on thin fiber. 4. Capillary rise in tubes. 5. Liquid lenses.
6. Stability of capillary surfaces.

**III. Contact Angle Hysteresis:** 1. Advancing and receding angles,
hysteresis, contact line pinning. 2. Elasticity of contact line.

**IV. Wetting and Intermolecular Forces:** 1. Energetics of submicroscopic thin
film. 2. Disjoining pressure. 3. Classification of wetting. 4. Long-range
and short range molecular forces. 5. Physical manisfestation of
molecular forces.

**V. Interfacial Hydrodynamics of Thin Films:** 1. Scales and the lubrication
approximation. 2. Hydrodynamics of thin films: gravity drainage on an
inclined plane, leveling of liquid interface,
Rayleigh instability, Plateau-Rayleigh instability. 3. Forced wetting:
Laudau-Levich and Bretherton problems, forced wetting on thin fibers.
4. Impregnation: Washburn law, corrections, inertial regime.
5. Capillary waves.

**VI. Moving Contact Lines:** 1. Experimental observations, dynamic contact
angles, dynamic wetting transition. 2. Moving contact line singularity.
3. Contact-angle/Contact-line-velocity relations: hydrodynamic models,
other models. 4. Kinetics of total wetting.

**VII. Kinetics of Dewetting:** 1. Experimental observations, critical thickness for dewetting. 2. Viscous regime. 3. Inertial regime. 4. Visco-elastic regime.

**VIII. Surfactants:** 1. Characteristic features of surfactant. 2. Structural features and behavior of Surfactants. 3. Micelle formation. 4. Adsoption of surfactant at interfaces. 5. Reduction of surface and interfacial tension by surfactants. 6. Wetting and its modification by surfactants. 7. Surfactant and soap films. 8. Application of surfactants.

**IX. Other Topics:** 1. Wetting on textured substrates: super-hydrophobicity. 2. Wetting and porous media.
3. Entrainment phenomena: Marangoni effects, electrocapillarity, electro-osmosis.