Dimensionless tension

It is a function of the dimensionless tension t defined by the work fa to elongate the chain by the fundamental length unit a divided by the thermal energy k T ... 

By expanding the Laplace transformed partition function (1.14) in powers of the dimensionless tension, we find... 

Fig. 4.20 Ratio X for a uniaxial elongation under a constant tension plotted against the reduced temperature. The dimensionless tension is varied from curve to curve.

Example 9.2. Dimensionless Tension, /e, for Newtonian and Power-Law Fluids... 

Molecular dynamics and density functional theory studies (see Section IX-2) of the Lennard-Jones 6-12 system determine the interfacial tension for the solid-liquid and solid-vapor interfaces [47-49]. The dimensionless interfacial tension ya /kT, where a is the Lennard-Jones molecular size, increases from about 0.83 for the solid-liquid interface to 2.38 for the solid-vapor at the triple point [49], reflecting the large energy associated with a solid-vapor interface. 

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

Where surface-active agents are present, the notion of surface tension and the description of the phenomena become more complex. As fluid flows past a circulating drop (bubble), fresh surface is created continuously at the nose of the drop. This fresh surface can have a different concentration of agent, hence a different surface tension, from the surface further downstream that was created earlier. Neither of these values need equal the surface tension developed in a static, equiUbrium situation. A proper description of the flow under these circumstances involves additional dimensionless groups related to the concentrations and diffusivities of the surface-active agents. 

O = Standard deviation (statistics), or interfacial tension x = Torque on shaft, consistent units, FL or ML2/t2 = Np = P0 = Power number, dimensionless 3> = Power number, POJ or ratio of power number to Fioude number, Njrr, to exponential power, n... 

Roy et al. (R3) define the critical solids holdup as the maximum quantity of solids that can be held in suspension in an agitated liquid. They present measurements of this factor for various values of gas velocity, gas distribution, solid-particle size, liquid surface tension, liquid viscosity, and a solid-liquid wettability parameter, and they propose the following two correlations in terms of dimensionless groups containing these parameters ... 

For adiabatic, steady-state, and developed gas-liquid two-phase flow in a smooth pipe, assuming immiscible and incompressible phases, the essential variables are pu, pG, Pl, Pg, cr, dh, g, 9, Uls, and Uas, where subscripts L and G represent liquid and gas (or vapor), respectively, p is the density, p is the viscosity, cr is the surface tension, dh is the channel hydraulic diameter, 9 is the channel angle of inclination with respect to the gravity force, or the contact angle, g is the acceleration due to gravity, and Uls and Ugs are the liquid and gas superficial velocities, respectively. The independent dimensionless parameters can be chosen as Ap/pu (where Ap = Pl-Pg), and... 

The dimensionless constants b and 2 were introduced by Leikin [78] they account for the fact that contributions from each monolayer to the bending modulus and the surface tension can differ for the two modes considered. Such effects are probably small so that bi bj I [78]. We note that the electrical potential enters Eqs. (69) and (70) only through the parameter z. 

Figure 20. (a) The (dimensionless) lateral compressibility (dilatational modulus, elastic area expansion modulus) (left ordinate) and the dimensionless area per molecule (right ordinate) as a function of the tail length (t) of the PC lipids in equilibrium bilayer membranes. The conversion to real compressibilities and areas per molecule is discussed in the text, (b) The (dimensionless) surface tension and the (dimensionless) lateral compressibility as a function of the relative expansion for the C PC lipid... 

The lateral compressibility, i.e. the relative area change upon an imposed membrane tension, decreases slightly more than linearly with the chain length. This means that it is more difficult to expand the membrane surface area of a long-chained lipid than a shorter one. In Figure 20 dimensionless units are used, which means that the surface tension is given in units kT/as. 

For steady injection of a liquid through a single nozzle with circular orifice into a quiescent gas (air), the mechanisms of jet breakup are typically classified into four primary regimes (Fig. 3 2)[4°][41][22°][227] according to the relative importance of inertial, surface tension, viscous, and aerodynamic forces. The most commonly quoted criteria for the classification are perhaps those proposed by Ohnesorge)40] Each regime is characterized by the magnitudes of the Reynolds number ReL and a dimensionless number Z ... 

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