1.1.1.4. Derivation of the Navier-Stokes Equations

1.1.1.4.1. Assumptions

  • Newtonian fluid: linear relationship between stress and strain, hence viscosity is constant

  • Flow is incompressible: density is constant

  • Flow is isothermal: temperature is constant

1.1.1.4.2. Normal Stresses

\[\sigma_{xx} = -p + 2 \mu {\partial u \over \partial x}\]
\[\sigma_{yy} = -p + 2 \mu {\partial v \over \partial y}\]
\[\sigma_{zz} = -p + 2 \mu {\partial w \over \partial z}\]

1.1.1.4.2.1. \(\nabla \cdot \vec V = 0\) for incompressible flow:

\[{1 \over 3} ( \sigma_{xx} + \sigma_{yy} + \sigma_{zz}) = -p\]

1.1.1.4.3. Shear Stresses

\[\tau_{xy} = \tau_{yx} = \mu \left ({\partial u \over \partial y} + {\partial v \over \partial x} \right )\]
\[\tau_{yz} = \tau_{zy} = \mu \left ({\partial v \over \partial z} + {\partial w \over \partial y} \right )\]
\[\tau_{zx} = \tau_{xz} = \mu \left ({\partial w \over \partial x} + {\partial u \over \partial z} \right )\]

1.1.1.4.4. x-direction Momentum Equation for a Newtonian, incompressible, isothermal fluid

1.1.1.4.4.1. From the Momentum Equation:

\[\rho g_x + {\partial \sigma_{xx} \over \partial x} + {\partial \tau_{yx} \over \partial y} + {\partial \tau_{zx} \over \partial z} = \rho \left ( {\partial u \over \partial t} + u{\partial u \over \partial x} + v {\partial u \over \partial y} + w {\partial u \over \partial z} \right )\]

1.1.1.4.4.2. Substitute in normal stresses and shear stresses:

\[\rho g_x -{\partial p \over \partial x}+2 \mu {\partial^2 u \over \partial x^2} + \mu \left ({\partial^2 u \over \partial y^2} + {\partial^2 v \over \partial y \partial x} \right ) + \mu \left ({\partial^2 w \over \partial z \partial x} + {\partial^2 u \over \partial z^2} \right ) = \rho \left ( {\partial u \over \partial t} + u{\partial u \over \partial x} + v {\partial u \over \partial y} + w {\partial u \over \partial z} \right )\]

1.1.1.4.4.3. Rewrite:

\[\rho g_x -{\partial p \over \partial x}+\mu \left ({\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} + {\partial^2 u \over \partial z^2} \right ) + \mu {\partial \over \partial x} \left ({\partial u \over \partial x} + {\partial v \over \partial y} + {\partial w \over \partial z} \right ) = \rho \left ( {\partial u \over \partial t} + u{\partial u \over \partial x} + v {\partial u \over \partial y} + w {\partial u \over \partial z} \right )\]

1.1.1.4.4.4. \(\nabla \cdot \vec V = 0\) for incompressible flow:

\[\rho g_x -{\partial p \over \partial x}+\mu \left ({\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} + {\partial^2 u \over \partial z^2} \right ) = \rho \left ( {\partial u \over \partial t} + u{\partial u \over \partial x} + v {\partial u \over \partial y} + w {\partial u \over \partial z} \right )\]

1.1.1.4.4.5. Vector Notation

\[\rho \vec g- \nabla p + \mu \nabla^2 \vec V = \rho \left ( {\partial \vec V \over \partial t} + \vec V (\nabla \cdot \vec V) \right )\]

1.1.1.4.5. Solutions

  • 3 Momentum Equations + Continuity = 4 Equations

  • Unknowns = u, v, w, p, \(\rho = 5\) Unknowns

  • Need an equation of state - to relate pressure and density

  • The Navier-Stokes Equations are time-dependent, non-linear, 2nd order PDEs - very few known solutions (parallel plates, pipe flow, concentric cylinders).

    • The Transient Term is \({\partial \vec V / \partial t}\)

    • The Convection Term is \(\vec V(\nabla \cdot \vec V)\). This is the non-linear term and is the cause most of the difficulty in solving these equations.

    • The Diffusion Term is \(\mu \nabla^2 \vec V\). This is the 2nd order term.

    • The Body Force Term is \(\rho \vec g\)

    • The Pressure Gradient Term is \(\nabla p\)

  • The only general approach is computational