Tangential velocity within mass transfer boundary layer

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

TANGENTIAL VELOCITY COMPONENT vg WITHIN THE MASS TRANSFER BOUNDARY LAYER... 

The generalized form for the linear tangential velocity profile within the mass transfer boundary layer for a no-slip interface is... 

Tangential Velocity Component vg within the Mass Transfer Boundary Layer Creeping and Potential Flow around a Gas Bubble... 

Unlike creeping flow about a solid sphere, the r9 component of the rate-of-strain tensor vanishes at the gas-liquid interface, as expected for zero shear, but the simple velocity gradient (dvg/dr)r R is not zero. The fluid dynamics boundary conditions require that [(Sy/dt)rg]r=R = 0- The leading term in the polynomial expansion for vg, given by (11-126), is most important for flow around a bubble, but this term vanishes for a no-slip interface when the solid sphere is stationary. For creeping flow around a gas bubble, the tangential velocity component within the mass transfer boundary layer is approximated as... 

If T terface and Tbuik replace Ca, equilibrium and Ca, bulks respectively, in the definition of the dimensionless profile P, and the thermal diffusiv-ity replaces a. mix. then the preceding equation represents the thermal energy balance from which temperature profiles can be obtained. The tangential velocity component within the mass transfer boundary layer is calculated from the potential flow solution for vg if the interface is characterized by zero shear and the Reynolds number is in the laminar flow regime. Since the concentration and thermal boundary layers are thin for large values of the Schmidt and Prandtl... 

The tangential component of the dimensionless equation of motion is written explicitly for steady-state two-dimensional flow in rectangular coordinates. This locally flat description is valid for laminar flow around a solid sphere because it is only necessary to consider momentum transport within a thin mass transfer boundary layer at sufficiently large Schmidt numbers. The polar velocity component Vo is written as Vx parallel to the solid-liquid interface, and the x direction accounts for arc length (i.e., x = R9). The radial velocity component Vr is written... 

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