Boundary Value Problems and Boundary Conditions

The differential equations for the conservation laws require boundary conditions for proper solution. Specifically, the Navier-Stokes momentum equation requires the specification of the velocity vector on all surfaces bounding the flow domain. For an external flow, one that is not contained by walls or surfaces at specified locations, the fluid’s velocity vector and the thermodynamic state must be specified on a closed distant surface.

\[ \rho \left( \frac{\partial u_j}{\partial t} + u_i \frac{\partial u_j}{\partial x_i} \right) = - \frac{\partial p}{\partial x_j} + \rho g_j + \frac{\partial}{\partial x_i} \left[ \mu \left( \frac{\partial u_j}{\partial x_i} + \frac{\partial u_i}{\partial x_j} \right) + \left( \mu_v - \frac{2}{3} \mu \right) \frac{\partial u_m}{\partial x_m} \delta_{ij} \right] \] This is the Navier-Stokes momentum equation
\begin{equation} \underbrace{\frac{\partial \mathbf{u}}{\partial t}}_{\text{Local acceleration}} + \underbrace{(\mathbf{u} \cdot \nabla) \mathbf{u}}_{\text{Advection}} = \underbrace{-\frac{1}{\rho} \nabla p}_{\text{Pressure Gradient}} + \underbrace{g\mathbf{k}}_{\text{Gravity}} + \underbrace{\nu \nabla^2 \mathbf{u}}_{\text{Viscous Term}} \end{equation} \begin{equation} \underbrace{\nabla \cdot \mathbf{u} = 0}_{\text{Mass conservation (Incompressibility)}} \end{equation}

1 \(\rho \frac{D \mathbf{u}}{D t} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u} \quad \text{(incompressible)} \Rightarrow \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho} \nabla p + g\mathbf{k} + \nu \nabla^2 \mathbf{u}\)

2 The form of the relation between shear stress and rate of strain depends on a fluid, and most common fluids obey Newton's law of viscosity, which states that the shear stress is proportional to the strain rate \( \tau = \mu \frac{d\gamma}{dt} \) Such fluids are called Newtonian fluids. The coefficient of proportionality \(\mu\) is known as dynamic viscosity and its value depends on the particular fluid. The ratio of dynamic viscosity to density is called kinematic viscosity \( \nu = \frac{\mu}{\rho} \)

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