Equation 2-24
\tau_{xy} = \tau_{11}l_1l_2 + \tau_{22}m_1m_2 + \tau_{33}n_1n_2
\epsilon_{xy} = \epsilon_{11}l_1l_2 + \epsilon_{22}m_1m_2 + \epsilon_{33}n_1n_2
Equation 2-25
\tau_{xx} = -p + \mathbf{K}\epsilon_{xx} + C_2 \ \text{div} \mathbf{V}
Equation 2-26
\tau_{xy} = \mathbf{K} \epsilon_{xy}
Equation 2-27: General deformation law for a newtionian viscous fluid
\tau_{ij} = -p\delta_{ij} + \mu (\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}) + \delta_{ij}\lambda \text{div} \mathbf{V}
\overline{p} = -\frac{1}{3} (\tau_{xx}+\tau_{yy}+ \tau_{zz}) = p - (\lambda + \frac{2}{3}\mu) \text{div} \mathbf{V}
Stoke’s hypothesis (1845)
\lambda + \frac{2}{3}\mu
Cartesian Coordinates
\nabla \phi = (\frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y},\frac{\partial \phi}{\partial z})
\text{div} \mathbf{V} = \frac{\partial u}{\partial x}+ \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z}
\text{curl}\mathbf{V} = (\frac{\partial w}{\partial y} - \frac{\partial v}{\partial z}, \frac{\partial u}{\partial z} - \frac{\partial w}{\partial x},\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y})
\mathbf{V}\cdot\mathbf{\nabla}=u\frac{\partial}{\partial x} + v\frac{\partial}{\partial y} +w\frac{\partial}{\partial z}
\mathbf{\nabla}^2 \phi = \frac{\partial^2 \phi}{\partial x^2}+ \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}