Bubble wall friction force

The flow of the continuous phase is considered to be initiated by a balance between the interfacial particle-fluid coupling - and wall friction forces, whereas the fluid phase turbulence damps the macroscale dynamics of the bubble swarms smoothing the velocity - and volume fraction fields. Temporal instabilities induced by the fluid inertia terms create non-homogeneities in the force balances. Unfortunately, proper modeling of turbulence is still one of the main open questions in gas-liquid bubbly flows, and this flow property may significantly affect both the stresses and the bubble dispersion [141]. 

The radial velocity distribution inside a bubble column has been given by Eq. (3-15). Differentiating this with respect to 4>, one finds the velocity gradient at the column wall, du/d(t>) i, to be zero, which means that peripherally descending liquid flow slides freely along the column wall. In other words, the column wall keeps the downflow in a cylindrical form at the periphery, but exerts only a negligible frictional force. 

The wall might also exert a friction force on the bubbles [97]. A wall boundary condition for the gas momentum equations that was formulated by Lopez de Bertodano [94] is thus sometimes used ... 

Another development was to coat the tubing inner wall with a chemically inert liquid species, which was originally perfluorocarbon [20]. This reduces the thickness of the liquid film on the tubing inner wall, minimises the mass transference between successive liquid segments and reduces the frictional forces at the solid—liquid interface. Consequently, carryover is also reduced. The integrity of the air bubbles is better... 

An additional wall lift force that pushes the dispersed phase away from the wall was suggested by Antal et al. [2]. Recently, the defect in the original model that a bubble located far from the wall is attracted to the wall, has been removed by Politano et al. [127]. The modified wall lift force model used by Jakobsen et al. [71] is given by (5.75). The wall might also exert a friction force on the bubbles [107]. A wall boundary condition for the gas momentum equations that was formulated by Lopez de Bertodano [103] is thus sometimes used ... 

If the adhesion or friction force at interface between mold wall and melt is high enough it results in an assumption of the zero velocity for the melt in this interface. Then, the interface roughness mainly results from a fountain flow. Fig. 1 shows the theoretical bubble distribution in the thickness direction during mold filling. Any bubble away from the center layer of flow channel will be sheared more or less. As the result of this shearing, the... 

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