Fluid Mechanics

Fluid mechanics is a field that deals with the motion of fluids and the resulting transfer of energy and mass, among other things. Interestingly, those majoring in physics itself rarely study it deeply, but it is a subject with very broad applications and is of great importance. Rather than from an engineer’s perspective, this is organized for ease of study from the perspective of those majoring in mathematics or physics. Particularly for graduate students and above, it is recommended to follow the derivation of the Navier-Stokes equations directly.

Introduction to Fluid Mechanics

Properties

• Definition of Fluids and Fluid Mechanics
  • Definition of Viscous Fluid
  • Definition of Compressible Fluid
• Definition of Conductive Fluids and Magnetohydrodynamics
  • Definition of Magnetic Fluids
• Definition of Convection, Diffusion, and Advection

Laws

• Newton’s Law of Viscosity and Newtonian Fluids
• Porous Media and Darcy’s Law
• Fick’s Law and Mass Diffusivity $D$
• Stokes’ Law, Drag Force, and Terminal Velocity
• 🔒(26/06/03)Newton’s Law of Cooling and Heat Transfer Coefficient $h$
• 🔒(26/06/11)Fourier’s Law of Heat Conduction

Pressure

• Pressure of a Fluid $P = F / A$
• Pressure of a Fluid with Depth $P_{h} = P_{0} + \rho g h$
  • Pressure of a Fluid with Depth When an Object is Placed on the Fluid
• Flow Rate and Continuity Equation $Q = AU$
• Bernoulli’s Equation
  • Proof of Torricelli’s Theorem $Q = A_{2} \sqrt{2 g h}$
• Proof of Hagen-Poiseuille Law

Computational Fluid Dynamics

Governing Equations

• Fluid Velocity and Steady Flow $\mathbf{u}$
• Eulerian and Lagrangian Descriptions
• Material Derivative $D / D t$
• Cauchy Stress Tensor $\sigma$
  • Lamé Parameters $\mu$, $\lambda$
  • Bulk Viscosity $\zeta$
• Derivation of Euler Equations
• Derivation of Navier-Stokes Equations
  • Derivation of Compressible Navier-Stokes Equations
• 🔒(26/06/23)Derivation of Darcy-Brinkman-Forchheimer Equation

Dimensionless Numbers

• Reynolds Number: Distinguishing Laminar and Turbulent Flow $\mathrm{Re}$
• Definition of Hartmann Number $\mathrm{Ha}$
• Definition of Darcy Number $\mathrm{Da}$
• Definition of Prandtl Number $\mathrm{Pr}$
• Definition of Schmidt Number $\mathrm{Sc}$
• Definition of Péclet Number $\mathrm{Pe}$
• Definition of Rayleigh Number $\mathrm{Ra}$
• 🔒(26/05/30)Definition of Lewis Number $\mathrm{Le}$
• 🔒(26/06/07)Definition of Biot Number $\mathrm{Bi}$
• 🔒(26/06/15)Definition of Nusselt Number $\mathrm{Nu}$
• 🔒(26/06/19)Definition of Sherwood Number $\mathrm{Sh}$