Dimensionless deformation time

The droplet deformation is also time-dependent. A practicable dimensionless breakup time can be defined as ... 

Ratio between characteristic diffusion and deformation time Dimensionless parameter Deformation rate of a filament (1/s) Solubility parameter [(J/m ) ]... 

In steady-state uniform shear flow at low concentrations and stresses, the drop deformation can be expressed using the three dimensionless parameters the capillarity nrunber, k, the viscosity ratio, k, and the reduced deformation time, t [5] ... 

A GW gives rise to a quadrupolar deformation normal to the direction of propagation. The deformation can be described by means of a dimensionless strain amplitude h = AL/L, where AL is the deformation of a region of space-time separated by a distance L. For example, a supernova explosion, with a mass conversion into GWs of 1% of the total mass, at a distance of 10 kpc (roughly in the centre of our galaxy), would cause a strain on earth of h 3 x 10-18 [50],... 

The dimensionless number in rheology that compares relative importance of the time scale of the deformation process tD over the observation time tQ is called the Deborah number (De) ... 

While the Deborah number is often used to compare the time for deformation with the time of observation in experiments, it also inspires us to identify and formulate other dimensionless groups that compare the various characteristic times and forces relevant in colloidal phenomena. We discuss some of the important ones. 

Assuming a characteristic viscosity of fj = me aTlA/n, where the characteristic rate of deformation is taken as 7 = u/b, where t f is the fill time and u = i 2/t / the characteristic velocity, we can write the viscosity in dimensionless form as... 

Whether a polymer exhibits elastic as well as viscous behavior depends in part on the time scale of the imposition of a load or deformation compared to the characteristic response time of the matei ial. This concept is expressed in the dimensionless Deborah number ... 

A unique dimensionless number in rheology is the Deborah (De) number (named after the song of Deborah, Book of Judges, the Old Testament) that compares the time scale of the deformation process ((d) to the observation time (to) ... 

Predicted Deformation Mechanisms. Recent work has developed maps of the deformation mechanisms expected in films with different properties. Two dimensionless groups were found to determine which of the deformation mechanism occurs. The first is the time for particle deformation compared to the time for evaporation, captured in 1 = ERt]o/yH, where E is the evaporation rate, t]o is the polymer viscosity, and y is the water-air surface tension. The second dimensionless group is the Peclet number, which determines the vertical homogeneity in the film, Pe = 6nt] R H E/kT. The deformation regimes are shown in Fig. 9. 

Figure 2-15. Photographs of the relaxation of a pair of initially deformed viscous drops back to a sphere under the action of surface tension. The characteristic time scale for this surface-tension-driven flow is tc = fiRi 1 + X)/y. The properties of the drop on the left-hand side are X = 0.19, /id = 5.5 Pa s, ji = 29.3 Pa s, y = 4.4 mN/rn, R = 187 /an, and this gives tc = 1.48 s. For the drop on the right-hand side, X = 6.8, lid = 199 Pa s, //. = 29.3 Pa s, y = 4.96 mN/m, R = 217 /an, and tc = 9.99 s. The photos were taken at the times shown in the figure. When compared with the characteristic time scales these correspond to exactly equal dimensionless times (/ = t/tc) (a) t = 0.0, (b) t = 0.36, (c) t = 0.9, (d) t = 1.85, (e) t = 6.5. It will be noted that the drop shapes are virtually identical when compared at the same characteristic times. This is a first illustration of the principle of dynamic similarity, which will be discussed at length in subsequent chapters.

Figure 8-3. Drop deformation versus time for relaxation of the pair of drops shown in Fig. 2.15. The figure on the right-hand side shows the relaxation data as a function of the actual time. The data on the right-hand side are shown plotted versus a dimensionless time scale, t = t/t, where t = M.

Although Ca 1, we do not neglect Ca (db /dt), which may, in fact, be 0(1). In the absence of flow, this result shows that the relaxation from a deformed elliptic shape will occur on a dimensionless time-scale... 

We have already shown visual evidence in Chap. 2 (Fig. 2 15), that this is the correct time scale for the relaxation process. Further quantitative evidence is offered in Fig. 8 3. In this figure is plotted the measured deformation from Fig. 2 15, specified in terms of the dimensionless parameter Df = L — B)/ L + B) where L is the length of the drop and B is its breadth at the equator. It can be seen that when time is scaled with t, the deformation curves for the two different viscosity ratio fluids are reduced approximately to a single curve. 

When discussing the morphology it is useful to use the microrheology as a guide. At low stresses in a steady uniform shear flow, the deformation can be expressed by means of three dimensionless parameters the viscosity ratio, the capillarity number, and the reduced time, respectively ... 

Red cell deformation takes place under two important constraints fixed surface area and fixed volume. The constraint of fixed volume arises from the impermeability of the membrane to cations. Even though the membrane is highly permeable to water, the inability of salts to cross the membrane prevents significant water loss because of the requirement for colloidal osmotic equilibrium [Lew and Bookchin, 1986). The constraint of fixed surface area arises from the large resistance of bilayer membranes to changes in area per molecule [Needham and Nunn, 1990]. These two constraints place strict limits on the kinds of deformations that the cell can undergo and the size of the aperture that the cell can negotiate. Thus, a major determinant of red cell deformability is its ratio of surface area to volume. One measure of this parameter is the sphericity, defined as the dimensionless ratio of the two-thirds power of the cell volume to the cell area times a constant that makes its maximum value 1.0 ... 

Fig. 3.9 Evolution of the dimensionless sinface deformation y as a function of dimensionless distance j for We = 40, = 0.001, k = 0.02, and fo = 0.1. The dimensionless time t is speeified on the figure [41 fig. 3] (Courtesy of Cambridge University Press)...

At low concentration of a second polymer, blends have dispersed-phase morphology of a matrix and discrete second phase. As the concentration increases, at the percolation threshold volume fraction of the dispersed phase, (f>c 0.16, the blends structure changes into co-continuous. Maximum co-continuity is achieved at the phase inversion concentration, (py. The morphology as well as the level of stress leads to different viscosity-composition dependencies. The deformation and dispersion processes are best described by microrheology, using the three dimensionless parameters the viscosity ratio (2), the capillarity number (k), and the reduced time (f ), respectively (Taylor 1932) ... 

Dimensionless standard deviation Deformation angle Number of fully filled chambers Initial orientation Dimensionless time... 

DiStribUtiVG Mixing. The extent of distributive mixing can be determined from the total shear deformation of the plastic melt in the melt conveying zone. The total shear deformation is also called the total shear strain-, it is determined by the product of the shear rate and the length of time that the fluid is exposed to the shear rate. For instance, if the plastic melt is exposed to a shear rate of 100 s for 15 s, the resulting shear strain is 100 x 15 = 1500. The units of shear strain are dimensionless. The shear rate is determined by the velocity profiles in the extruder. 

Lp = deflection lag factor compensating for the time dependence of soil deformation, dimensionless... 

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