Notes on Fluid Dynamics
                                                                 Rodolfo Repetto
                                                Department of Civil, Chemical and Environmental Engineering
                                                                 University of Genoa, Italy
                                                                  rodolfo.repetto@unige.it
                                                             phone number: +39 010 3532471
                                                            http://www.dicca.unige.it/rrepetto/
                                                               skype contact: rodolfo-repetto
                                                                January 13, 2016
        Rodolfo Repetto (University of Genoa)                         Fluid dynamics                                       January 13, 2016     1 / 161
    Table of contents I
    1  Acknowledgements
    2  Stress in fluids
          The continuum approach
          Forces on a continuum
          The stress tensor
          Tension in a fluid at rest
    3  Statics of fluids
          The equation of statics
          Implications of the equation of statics
          Statics of incompressible fluids in the gravitational field
          Equilibrium conditions at interfaces
          Hydrostatic forces on flat surfaces
          Hydrostatic forces of curved surfaces
    4  Kinematics of fluids
          Spatial and material coordinates
          The material derivative
          Definition of some kinematic quantities
          Reynolds transport theorem
          Principle of conservation of mass
          The streamfunction
          The velocity gradient tensor
          Physical interpretation of the rate of deformation tensor D
          Physical interpretation of the rate of rotation tensor Ω
       Rodolfo Repetto (University of Genoa)                 Fluid dynamics                                January 13, 2016   2 / 161
    Table of contents II
    5  Dynamics of fluids
          Momentum equation in integral form
          Momentum equation in differential form
          Principle of conservation of the moment of momentum
          Equation for the mechanical energy
    6  The equations of motion for Newtonian incompressible fluids
          Definition of pressure in a moving fluid
          Constitutive relationship for Newtonian fluids
          The Navier-Stokes equations
          The dynamic pressure
    7  Initial and boundary conditions
          Initial and boundary conditions for the Navier-Stokes equations
          Kinematic boundary condition
          Continuity of the tangential component of the velocity
          Dynamic boundary conditions
          Two relevant cases
    8  Scaling and dimensional analysis
          Units of measurement and systems of units
          Dimension of a physical quantity
          Quantities with independent dimensions
          Buckingham’s Π theorem
          Dimensionless Navier-Stokes equations
       Rodolfo Repetto (University of Genoa)                 Fluid dynamics                                January 13, 2016   3 / 161
    Table of contents III
    9  Unidirectional flows
          Introduction to unidirectional flows
          Some examples of unidirectional flows
          Unsteady unidirectional flows
          Axisymmetric flow with circular streamlines
   10  Low Reynolds number flows
          Introduction to low Reynolds number flows
          Slow flow past a sphere
          Lubrication Theory
   11  High Reynolds number flows
          The Bernoulli theorem
          Vorticity equation and vorticity production
          Irrotational flows
          Bernoulli equation for irrotational flows
   12  Appendix A: material derivative of the Jacobian
          Determinants
          Derivative of the Jacobian
   13  Appendix B: the equations of motion in different coordinates systems
          Cylindrical coordinates
          Spherical polar coordinates
   14  References
       Rodolfo Repetto (University of Genoa)                 Fluid dynamics                                January 13, 2016   4 / 161