Interestingly, a requirement on the tangential fluid velocity components at an interface cannot be developed from the equations of motion. Fortunately, the no-slip condition provides a simple experimentally verified result that addresses this analytical insufficiency. The simplest statement of the no-slip condition is that tangential velocity components must match at the interface \[ \mathbf{u}_1 \cdot \mathbf{t}' = \mathbf{u}_2 \cdot \mathbf{t}' \quad \text{and} \quad \mathbf{u}_1 \cdot \mathbf{t}'' = \mathbf{u}_2 \cdot \mathbf{t}'' \]

The no-slip condition is widely accepted as an experimental fact in macroscopic flows of ordinary fluids. For the simplest case of a viscous fluid (medium 2) moving with respect to an impermeable solid (medium 1), \(\boxed{\dot{m}_s = \rho_1 (\mathbf{u}_1 - \mathbf{u}_s) \cdot \mathbf{n} = \rho_2 (\mathbf{u}_2 - \mathbf{u}_s) \cdot \mathbf{n}}\) and \(\boxed{\mathbf{u}_1 \cdot \mathbf{t}' = \mathbf{u}_2 \cdot \mathbf{t}', \mathbf{u}_1 \cdot \mathbf{t}'' = \mathbf{u}_2 \cdot \mathbf{t}''}\) all together reduce to the interface condition \[ \mathbf{u}_2 = \mathbf{u}_1 \]

Known violations of the no-slip boundary condition occur in rarefied gases and for superfluid helium at or below 2.17 K, where it has an immeasurably small (essentially zero) viscosity Slip has also been observed in microscopic flows, and on micro-patterned and super-hydrophobic surfaces
For the control volume, the tangential momentum conservation results from conservative body forces \(\boxed{\mathbf{g} = - \nabla \Phi \quad \text{or} \quad g_j = - \frac{\partial \Phi}{\partial x_j}} \) and \(\boxed{\mathbf{u}_1 \cdot \mathbf{t}' = \mathbf{u}_2 \cdot \mathbf{t}', \mathbf{u}_1 \cdot \mathbf{t}'' = \mathbf{u}_2 \cdot \mathbf{t}''}\) are \[ 0 = + \left( (n_i \tau_{ij})_2 - (n_i \tau_{ij})_1 \right) t'_j + \left( \partial \sigma / \partial x_j \right) t'_j \quad \text{and} \quad 0 = + \left( (n_i \tau_{ij})_2 - (n_i \tau_{ij})_1 \right) t''_j + \left( \partial \sigma / \partial x_j \right) t''_j \] where \( \tau_{ij} \) is given by the general form of the viscous stress tensor \( \tau_{ij} \) for a compressible Newtonian fluid \(\tau_{ij} = \mu \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) + \left( \mu_v - \frac{2}{3} \mu \right) \frac{\partial u_m}{\partial x_m} \delta_{ij}\). These conditions include surface tension gradients and are statements of tangential stress matching at fluid–fluid interfaces. In general, \(\boxed{\dot{m}_s = \rho_1 (\mathbf{u}_1 - \mathbf{u}_s) \cdot \mathbf{n} = \rho_2 (\mathbf{u}_2 - \mathbf{u}_s) \cdot \mathbf{n}}\)
\(\boxed{\dot{m}_s (\mathbf{u}_2 - \mathbf{u}_1) \cdot \mathbf{n} = - (p_2 - p_1) + \left( (n_i \tau_{ij})_2 - (n_i \tau_{ij})_1 \right) n_j + \sigma \left( \frac{1}{R'} + \frac{1}{R''} \right)}\)
\(\boxed{\mathbf{u}_1 \cdot \mathbf{t}' = \mathbf{u}_2 \cdot \mathbf{t}', \mathbf{u}_1 \cdot \mathbf{t}'' = \mathbf{u}_2 \cdot \mathbf{t}''}\) and \(\boxed{\begin{aligned} 0 &= + \left( (n_i \tau_{ij})_2 - (n_i \tau_{ij})_1 \right) t'_j + \left( \frac{\partial \sigma}{\partial x_j} \right) t'_j, \\ 0 &= + \left( (n_i \tau_{ij})_2 - (n_i \tau_{ij})_1 \right) t''_j + \left( \frac{\partial \sigma}{\partial x_j} \right) t''_j \end{aligned}}\) are required for analyzing multiphase flows with phase change.

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