\[\newcommand{\evec}[1]{\mathbf{e}_{#1}}\]

  • Streamline: a line along which the velocity is constant: \(\frac{dx}{v_x} = \frac{dy}{v_y} = \frac{dy}{v_y}\)

  • Stream function : a value that is constant along a streamline

    \[\text{2D}: v_x = \frac{\partial phi}{\partial y}, v_y = -\frac{\partial phi}{\partial x}\]

  • Steady flow: $\frac{\partial v}{\partial t} = 0$

  • Inviscid flow: $\nabla \tau = \nabla\cdot(-p\mathbf{I}) = -\nabla$

  • Incompressible flow: $\frac{-1}{\rho}\frac{D\rho}{Dt} = 0$

  • Volumetric strain rate: $\nabla \cdot \mathbf{v}$

  • Conservation of mass:

    \[\frac{-1}{\rho}\frac{D\rho}{Dt} = \nabla \cdot \mathbf{v}\]

  • Conservation of mass, incompressible flow

    \[\frac{-1}{\rho}\frac{D\rho}{Dt} = 0 \Longleftrightarrow \nabla \cdot \mathbf{v} = 0\]

  • Conservation of momentum

    \[\rho\frac{D\rho}{Dt} = \nabla \cdot \tau + \rho g\]

  • Conservation of momentum, inviscid flow (Euler equation)

    \[\nabla \tau = \nabla\cdot(-p\mathbf{I}) = -\nabla \\ \rho\frac{D\rho}{Dt} = -\nabla p + \rho g\]

  • Euler equation, hydrostatics

    \[\frac{D\rho}{Dt} = 0 \\ \nabla p = \rho g\]

  • Archimedes principle (can be derived using Euler equation for hydrostatics)

    \[F_B = -\text{mass of displaced fluid} \times \mathbf{g}\]

  • Bernouilli equation, steady flow, invisicid flow

    • Streamline coordinates, integral: $\evec{s}$
      • Integral form (usually seen)
      \[P_1 + \frac{1}{2}\rho v_1^2 + \rho g z_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g z_2\]
      • Differential form
      \[\frac{\partial}{\partial s}\left({P + \frac{1}{2}\rho v^2 + \rho g z}\right) = 0\]
    • Normal to streamline: $\evec{n}$:
    \[\text{Radius of curvature}: R \\ \frac{\partial}{\partial n}\left(p + \rho g z\right) = \frac{\rho v_s^2}{R}\]
    • Binormal to streamline: $\evec{l} = \evec{n} \times \evec{s}$
    \[\frac{\partial}{\partial l}\left({P+ \rho g z}\right) = 0\]