Constant-Viscosity Momentum
Balances in Terms of Velocity Gradients
Verify the first of the three momentum balances of Eqn. (5.67) by starting from the x momentum balance of Eqn. (5.54) and substituting for the stresses \sigma_{x x}, \tau_{y x}, and \tau_{z x} from Eqns. (5.60) and (5.61).
\begin{array}{c}+\rho g_{x}+\frac{1}{3} \mu \frac{\partial}{\partial x} \nabla \cdot \mathbf{v}, \\ \rho\left(\frac{\partial v_{y}}{\partial t}+v_{x} \frac{\partial v_{y}}{\partial x}+v_{y} \frac{\partial v_{y}}{\partial y}+v_{z} \frac{\partial v_{y}}{\partial z}\right)=-\frac{\partial p}{\partial y}+\mu\left(\frac{\partial^{2} v_{y}}{\partial x^{2}}+\frac{\partial^{2} v_{y}}{\partial y^{2}}+\frac{\partial^{2} v_{y}}{\partial z^{2}}\right)\\ +\rho g_{y}+\frac{1}{3} \mu \frac{\partial}{\partial y} \nabla \cdot \mathbf{v}, \\ \rho\left(\frac{\partial v_{z}}{\partial t}+v_{x} \frac{\partial v_{z}}{\partial x}+v_{y} \frac{\partial v_{z}}{\partial y}+v_{z} \frac{\partial v_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left(\frac{\partial^{2} v_{z}}{\partial x^{2}}+\frac{\partial^{2} v_{z}}{\partial y^{2}}+\frac{\partial^{2} v_{z}}{\partial z^{2}}\right)\\+\rho g_{z}+\frac{1}{3} \mu \frac{\partial}{\partial z} \nabla \cdot \mathbf{v}. (5.67) \end{array}\begin{aligned} \rho \frac{\mathcal{D} v_{x}}{\mathcal{D} t} &=\frac{\partial \sigma_{x x}}{\partial x}+\frac{\partial \tau_{y x}}{\partial y}+\frac{\partial \tau_{z x}}{\partial z}+\rho g_{x}, \\ \rho \frac{\mathcal{D} v_{y}}{\mathcal{D} t} &=\frac{\partial \tau_{x y}}{\partial x}+\frac{\partial \sigma_{y y}}{\partial y}+\frac{\partial \tau_{z y}}{\partial z}+\rho g_{y}, \\ \rho \frac{\mathcal{D} v_{z}}{\mathcal{D} t} &=\frac{\partial \tau_{x z}}{\partial x}+\frac{\partial \tau_{y z}}{\partial y}+\frac{\partial \sigma_{z z}}{\partial z}+\rho g_{z}. (5.54) \end{aligned}
\begin{aligned}\tau_{x y} &=\tau_{y x}=\mu\left(\frac{\partial v_{x}}{\partial y}+\frac{\partial v_{y}}{\partial x}\right), \\ \tau_{y z} &=\tau_{z y}=\mu\left(\frac{\partial v_{y}}{\partial z}+\frac{\partial v_{z}}{\partial y}\right), \\ \tau_{z x} &=\tau_{x z}=\mu\left(\frac{\partial v_{z}}{\partial x}+\frac{\partial v_{x}}{\partial z}\right). (5.60)\end{aligned}
\begin{aligned} \sigma_{x x} &=-p+2 \mu \frac{\partial v_{x}}{\partial x}-\frac{2}{3} \mu \nabla \cdot \mathbf{v}, \\ \sigma_{y y} &=-p+2 \mu \frac{\partial v_{y}}{\partial y}-\frac{2}{3} \mu \nabla \cdot \mathbf{v}, \\ \sigma_{z z} &=-p+2 \mu \frac{\partial v_{z}}{\partial z}-\frac{2}{3} \mu \nabla \cdot \mathbf{v}. (5.61) \end{aligned}
