Imaginary viscosity

In these cases the relative velocity of the shearing plates is not constant but varies in a sinusoidal manner so that the shear strain and the rate of shear strain are both cyclic, and the shear stress is also sinusoidal. For non-Newtonian fluids, the stress is out of phase with the rate of strain. In this situation a measured complex viscosity (rf) contains both the shear viscosity, or dynamic viscosity (t] ), related to the ordinary steady-state viscosity that measures the rate of energy dissipation, and an elastic component (the imaginary viscosity ij" that measures an elasticity or stored energy) ... 

Figure 9.1-3 shows a cutaway of a tube of radius a, length C, in which a fluid of viscosity q is flowing. The velocity goes to zero at the wall of the tube and reaches a maximum in the center in a parabolic shape. The flow is laminar (straight line, parallel to the axis), hence, an imaginary cylinder of radius r may be inserted as i... 

A roughness-viscosity model was proposed to interpret the experimental data. An effective viscosity /tef was introduced for this purpose as the sum of physical and imaginary = Mm(f) viscosities. The momentum equation is... 

The imaginary component, "(f), is the dilational viscosity modulus. This arises when the demulsifier in the monolayer is sufficiently soluble in the bulk liquid, so that the tension gradient created by an area compression/expansion can be short circuited by a transfer of demulsifiers to and from the surface. It is 90° out of phase with the area change. 

There are not a great number of studies on the viscoelastic behaviour of quasi-hard spheres. The studies of Mellema and coworkers13 shown in Figure 5.5 indicate the real and imaginary parts of the viscosity in a high-frequency oscillation experiment. Their data can be normalised to a characteristic time based on the diffusion coefficient given above. 

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

Figure 4 depicts the imaginary part of the frequency-dependent viscosity which clearly demonstrates the bimodality of the viscoelastic response. In the same figure the prediction from the Maxwell s relation have also been plotted. In the latter the relaxation time xs is calculated by the well-known... 

Figure 4. The imaginary part of the calculated viscosity is plotted as a function of the Fourier frequency at the triple point (solid line). Also shown is the prediction of the Maxwell viscoelastic model (dashed line), given by Eq. (239). The viscosity is scaled by a2/ /(mkBT) and the frequency is scaled by x l, where xsc = [ma2/kBT l/2. For more details see the text.

The surface rheological properties of the /3-lg/Tween 20 system at the macroscopic a/w interface were examined by a third method, namely surface dilation [40]. Sample data obtained are presented in Figure 24. The surface dilational modulus, (E) of a liquid is the ratio between the small change in surface tension (Ay) and the small change in surface area (AlnA). The surface dilational modulus is a complex quantity. The real part of the modulus is the storage modulus, e (often referred to as the surface dilational elasticity, Ed). The imaginary part is the loss modulus, e , which is related to the product of the surface dilational viscosity and the radial frequency ( jdu). 

The real and imaginary components of characteristic viscosity have been calculated according to equations (6.20) for zv = 2, (pi = 0.5, 0 = 0.5. The dashed curves depicts the alternation of the dependencies in the case when an internal relaxation process is taking into account, whereas equations (6.28) are used at r/2n = 10-5. 

Collision Cross-Section The model of gaseous molecules as hard, non-interacting spheres of diameter o can satisfactorily account for various gaseous properties such as the transport properties (viscosity, diffusion and thermal conductivity), the mean free path and the number of collisions the molecules undergo. It can be easily visualised that when two molecules collide, the effective area of the target is no1. The quantity no1 is called the collision cross-section of the molecule because it is the cross-sectional area of an imaginary sphere surrounding the molecule into which the centre of another molecule cannot penetrate. 

One-port FPW measurements, typically using a network analyzer, yield the input impedance of the transducer, the real and imaginary components of which can be used to determine the density and viscosity of a fluid contacting the device. 

Values of storage modulus G (a)) and loss modulus G"(m) can then be obtained by separating Eq. V-8 into its real and imaginary parts (Eq. V-2). Viscosity and recoverable compliance in the large N limit can be obtained from ... 

In the situation described above, the dynamic experiment was cturied out in dilation the resulting complex modulus was divided into a real ( elastic ) and an imaginary ( viscous ) part. As a counterpart, the experiment can also be carried out in shear, resulting in a complex surface shear viscosity G°, consisting of a real (viscous) part, the surface shear viscosity G° and the surface shear loss viscosity, G"" identical to the elasticity. This inversion of method is formally identical to measuring complex dielectric permittivities instead of complex conductivities, discussed in sec. I1.4.8a. In that case, flg. 3.26 is modified in that panel (b) describes G°, panel (c) G " and jianel (d) the sum, with - tan 0 = G" /G. ... 

The viscoelasticity in polymer solutions has been investigated for some time but the experimental techniques which enabled measurements at high dilutions have been developed since around 1948 (24). When a polymer solution is subject to a sinusoidally varying shearing stress its response can be expressed in terms of a complex intrinsic viscosity [j ], the imaginary part of which is the rigidity ... 

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