Since \( \nu \) and \( \kappa \) have the units of (length)\(^2\)/time, the kinematic viscosity \( \nu \) is sometimes called the momentum diffusivity, in analogy with \( \kappa \), the thermal diffusivity

However, velocity cannot be simply regarded as being diffused and advected in a flow because of the presence of the pressure gradient in \(D \mathbf{u} / Dt = - (1 / \rho) \nabla p + \nu \nabla^2 \mathbf{u}\)

For laminar flows in which viscous effects are important throughout the flow, the primary field equations will be \( \nabla \cdot \mathbf{u} = 0 \) and \(D \mathbf{u} / Dt = - (1 / \rho) \nabla p + \nu \nabla^2 \mathbf{u}\) or the version that includes a body force:

\[\rho \frac{D\mathbf{u}}{Dt} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{u} \quad \text{(incompressible)}\]

The velocity boundary conditions on a solid surface are simplified versions of

\[\dot{m}_s = \rho_1 (\mathbf{u}_1 - \mathbf{u}_s) \cdot \mathbf{n} = \rho_2 (\mathbf{u}_2 - \mathbf{u}_s) \cdot \mathbf{n}, \quad \mathbf{u}_1 \cdot \mathbf{t}' = \mathbf{u}_2 \cdot \mathbf{t}', \quad \text{and} \quad \mathbf{u}_1 \cdot \mathbf{t}'' = \mathbf{u}_2 \cdot \mathbf{t}''\]

\[\mathbf{n} \cdot \mathbf{u}_s = (\mathbf{n} \cdot \mathbf{u})_{\text{on the surface}} \quad \text{and} \quad \mathbf{t} \cdot \mathbf{u}_s = (\mathbf{t} \cdot \mathbf{u})_{\text{on the surface}}\]

where \( \mathbf{u}_s \) is the velocity of the surface, \( \mathbf{n} \) is the normal to the surface, and \( \mathbf{t} \) is the tangent to the surface in the plane of interest

Here fluid density will be assumed constant, and the frame of reference will be inertial. Thus, gravity can be dropped from the momentum equation as long as no free surface is present

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