Similarity and Transport Phenomena in Fluid

Chapter 1: The concept of similarity

•          Geometrical similarity

•          Invariance by affine transformation, rotation, translation

•          Fractal similarity

•          Scaling law

•          Physical similarity

•          Complete similarity: drag force

•          Incomplete similarity: flow resistance

Chapter 2:  Transport phenomena in fluid dynamics

•          Transport phenomena

•          Advection

•          Diffusion Heat equation

•          Wave

•          Shocks and conservation equations

•          Boundary problems: fixed boundary, boundary layer, free boundary problem

•          Classification of partial differential equations

•          First-order equation, characteristic form

•          Second order equation, hyperbolic, elliptic, parabolic

Chapter 3: One-parameter groups, Lie groups

•          Groups of transformation

•          Group invariants

•          Invariant curves

•          Transformation of derivative

Chapter 4: First-order differential equations

•          Phase portrait

•          Singular point

•          Separatrix

•          Integrating factor

•          Invariant integral curves

•          Singular solution

•          Change of variables

Chapter 5: Second-order differential equations

•          Invariant differential equations

•          Lie’s reduction theorem

•          Stretching group

•          Singularities

Chapter 6: Similarity solutions to partial differential equation

•          Similarity solutions

•          Associated stretching group

•          Asymptotic behavior

•          Determining equations

Chapter 5: Travelling wave solution

•          Translation groups

•          Example: diffusion with source

•          Propagation velocity

•          Approach to travelling waves

Chapter 8: Hyperbolic problems

  Hyperbolic problems

•          One dimensional problems

•             Characteristic equations

•             Shock formation

•            The Riemann problem

  Generalization to multidimensional problems

•             Linear systems

•             Nonlinear systems

•            The shallow-water equations

Chapter 9: Parabolic problems

•          Linear diffusion

•          Nonlinear diffusion

•          Stefan problem

•          Boundary layer problem