Tangential flow, definition

This brief overview describes some experiences using tangential-flow and dead-end ultrafiltration techniques for concentration of eukaryotic cells, proteins and virus. The data and conclusions presented here have been drawn from process development work employing available apparatus and should be considered preliminary, rather than definitive or exhaustive. Previous ultrafiltration systems have been described (1-14) for both bench and pilot scale separations of proteins and virus. This paper primarily summarizes work on cartridge and sheet filter systems and their application to processes requiring sterilizable and contained systems. 

By Newton s definition the viscosity or, more appropriately, the viscosity coefficient, jj, of a fluid in a laminar steady-state flow is expressed as the tangential force, F, per unit area. A, required to maintain a unit rate of shear (or velocity gradient), G, in the liquid. If the liquid fills the space between two parallel planes of area. A, one of which moves at a constant distance, r, from the other with a relative velocity, u, then we have... 

In deriving the equations for the flow of a simple viscous fluid the theoretical physicist uses a definition of viscosity based on a mathematical statement rather than a physical model. Let us define viscosity from a physical point of view. Consider two planes in the body of a fluid a distance dy apart, as shown in Fig. 4-1. If we apply tangential stress along one of these planes and observe a rate of shear yi then... 

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... 

If the tangential branch pipe is designed in such a way that nap ap = then e = 0, and the calculation of CE for the tangential branch pipe can be performed in the same way as for the axial one. The only difference is in the definition of the flow spin angle. 

For the pendant drop in figure I, the equations above are subjected to the no-slip boundary conditions at solid surfaces and the kinematic condition on the free surfaces. The kinematic condition implies that there is no liquid crossing the boundary into the gas phase, or in other words forms a definite boundary between the phases. For creeping flows into an atmosphere of gas with minimal velocities there will be no interfacial shear stress tangential to the surface and the normal stress inside the fluid is balanced by the surface tension as described by the famous Young-Laplace equation. 

First, let us consider the kinematic definition of a streamline. A streamline is a flow trajectory across which fluid motion is absent fluid moves tangentially to it. Thus, its local slope must be equal to the ratio of the vertical to the horizontal velocities,... 

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