Course: 2020/2021

Biomechanics of continuum media II (fluids)

(15544)

Students are expected to have completed

Calculus I and II
Linear algebra
Differential equations
Biomechanics of continuum media I (solid mechanics)

Competences and skills that will be acquired and learning results. Further information on this link

- The students must become familiar with the basic concepts of Fluid Mechanics: conservation laws, dimensional analysis, simplification of the general equations, etc.
- The students must become fluent in the usage of the mathematical tools commonly used in fluid mechanics: partial differential equations, usage of different coordinate systems, surface and volume integrals, complex variable, etc.

Description of contents: programme

1.- Introduction to fluid mechanics
1.1. Solids, liquids and gases
1.2. The continuum hypothesis
1.3. Density, velocity and internal energy
1.4. Local thermodynamic equilibrium. Equations of state.
2.- Kinematics of the fluid flow
2.1. Eulerian and Lagrangian descriptions
2.2. Uniform flow. Steady flow. Stagnation points.
2.3. Trajectories. Paths. Streamlines.
2.4. Substantial derivative. Acceleration.
2.5. Circulation and vorticity. Irrotational flow. Velocity potential.
2.6. Stream function
2.7. Strain-rate tensor
2.8. Convective flux. Reynolds transport theorem.
3.- Conservation laws in fluid mechanics
3.1. Continuity equation in integral form
3.2. Volume and surface forces
3.3. Stress tensor. Navier-Poisson law
3.4. Forces and moments on submerged bodies.
3.5. Momentum equation in integral form. Angular momentum equation.
3.6. Heat conduction vector. Energy equation in integral form.
4.- The Navier-Stokes equations
4.1. Navier-Stokes equations.
4.2. Initial and boundary conditions.
4.3. Bernoulli¿s equation
5.- Dimensional analysis
5.1. Dimensional analysis. The Pi theorem.
5.2. Applications
5.3. Nondimensionalization of the Navier-Stokes equations
5.4. Dimensionless numbers in fluid mechanics
6.- Viscous flows with applications to biomedical problems: circulatory flow, flow in airways, flow at the cell's scale
6.1. Unidirectional flows
6.2. The Stoke's problem
6.3. Quasi-one-directional flow
6.4. Applications to flows of interest in biology

Learning activities and methodology

Lectures: the main concepts of fluid mechanics are derived rigorously using physical and mathematical tools.
Seminars: the concepts derived in the lectures are used to solve problems. Also, new concepts are introduced through examples.
Homework: two homework covering different areas of Fluid Mechanics are given to the students.
Lab sessions: the students will become familiar with the usage of numerical (computational) and experimental tools to investigate a canonical flow of biomedical interest.

Assessment System

- % end-of-term-examination 40
- % of continuous assessment (assigments, laboratory, practicals...) 60

Basic Bibliography

- G.I. Barenblatt. Scaling. Cambridge University Press. 2003
- G.K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press. 2000
- Landau L.D., Lifshitz E.M.. Fluid Mechanics. Pergamon Press. 1989
- Y.C. Fung. Biomechanics: Mechanical Properties of Living Tissues, Second Edition. Springer; 2nd edition. 1993
- Y.C. Fung. Biomechanics: Circulation. Springer; 2nd edition. 1996
- Y.C. Fung. Biomechanics: Motion, Flow, Stress, and Growth. Springer. 1998