Power Law Stretching of Associating Polymers in Steady-State Extensional Flow

The published morphometric data from human, cat, and dog lungs suggest that the power-law relationships between the numbers (Na and Nv) and diameters (Da and Dv) of arteries and veins and between the lengths (La and Lv) and diameters of the arteries and veins could be used as scaling rules for assigning dimensions and numbers to the intrapulmonary vessels of the arterial and venous trees of the dog lung. These rules, along with the dimensions of the extrapulmonary arteries and capillary sheet and the distensibility coefficients of the vessels obtained from the literature, were used to construct a steady-state hemodynamic model of the dog lung vascular bed. The model can be characterized approximately by 15 orders of arteries with Na approximately 2.07 Da-2.58 and 13 orders of veins with Nv approximately 2.53 Dv-2.61. For the intrapulmonary vessels (orders 1–12), La approximately 4.85 Da1.01, and Lv approximately 6.02 Da1.07. The average ratio of the numbers of vessels in consecutive orders is approximately 3.2 for the arteries and veins. These arterial and venous trees are connected by the capillary sheet with an undistended thickness of approximately 3.5 microns and an area of 33 m2. The average distensibility (% increase in diameter over the undistended diameter/Torr increase in transmural pressure) for the model arteries and veins is approximately 2.4%/Torr, and the distensibility of the capillary sheet (% increase in thickness over the undistended thickness/Torr increase in transmural pressure) is approximately 3.6%/Torr. The calculated arterial-capillary-venous volumes and compliances of the model agree well with experimental estimates of these variables in dogs. In addition, the model appears consistent with certain aspects of the pressure-flow relationships measured in dog lungs. The model appears to be a useful summary of some of the available data on pulmonary morphometry and vessel properties. It is anticipated that the model will provide the basis for dynamic modeling of the dog lung in the future.

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A cluster approach to the structure of imperfect materials and their relaxation spectroscopy

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1983.0125 ◽

1983 ◽

Vol 390 (1798) ◽

pp. 131-180 ◽

Cited By ~ 262

Keyword(s):

Steady State ◽

Power Law ◽

Structural Organization ◽

Configurational Entropy ◽

Binding Energies ◽

Coarse Grained ◽

Dielectric Relaxation Spectroscopy ◽

Relaxation Dynamics ◽

Steady State Distribution ◽

State Distribution

A new theory is proposed for the explanation of observed relaxation phenomena, which differs significantly from theories suggested by the authors before. The theory is based on a model of structural organization of macroscopically sized samples of imperfectly structured materials, both solids and liquids, and is intermediate in character. In terms of the model, a microscopic structure is maintained over a cluster containing a number of microscopic units, with an array of clusters described by a steady-state distribution completing the macroscopic picture. The structural regularity of each level of morphological organization is precisely defined by a coarse-grained index, which is given a thermodynamic interpretation in terms of binding energies and configurational entropy. The limiting cases of an ideal liquid and a perfect crystal are recovered as asymptotic extremes in terms of this definition. The consequences of this model for the relaxation dynamics of the structure are examined and it is shown that prepared fluctuations decay in a time-power law manner as coupled zero-point motions evolve either within clusters or between clusters, with a power determined by the relevant regularity index. As a result, the origin of power law noise in materials is explained in terms of configurational entropy, and its relation with gaussian and white noise, which appear as asymptotic limits, outlined. The shape of the steady-state distribution of the array of clusters is also determined without any a priori assumptions, and it is shown to range from an unbounded form to a δ function as the regularity of the array superstructure increases. Experimental examples of dielectric relaxation spectroscopy have been used to illustrate these structural concepts and outline the way in which this technique can be used to deduce the structural organization of the sample. Finally, a short description is given of some commonly observed forms of response and their structural interpretation.

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Non-Newtonian slender drops in a nonlinear extensional flow

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.626 ◽

2019 ◽

Vol 877 ◽

pp. 561-581 ◽

Cited By ~ 1

Author(s):

Moshe Favelukis

Keyword(s):

Stability Analysis ◽

Power Law ◽

Analytical Solutions ◽

Viscosity Ratio ◽

Extensional Flow ◽

Shear Thinning ◽

Newtonian Liquid ◽

Creeping Flow ◽

Stationary States ◽

Power Law Index

In this theoretical report we explore the deformation and stability of a power-law non-Newtonian slender drop embedded in a Newtonian liquid undergoing a nonlinear extensional creeping flow. The dimensionless parameters describing this problem are: the capillary number $(Ca\gg 1)$, the viscosity ratio $(\unicode[STIX]{x1D706}\ll 1)$, the power-law index $(n)$ and the nonlinear intensity of the flow $(|E|\ll 1)$. Asymptotic analytical solutions were obtained near the centre and close to the end of the drop suggesting that only Newtonian and shear thinning drops $(n\leqslant 1)$ with pointed ends are possible. We described the shape of the drop as a series expansion about the centre of the drop, and performed a stability analysis in order to distinguish between stable and unstable stationary states and to establish the breakup point. Our findings suggest: (i) shear thinning drops are less elongated than Newtonian drops, (ii) as non-Newtonian effects increase or as $n$ decreases, breakup becomes more difficult, and (iii) as the flow becomes more nonlinear, breakup is facilitated.

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Steady-state power-law creep in “inclusion matrix” composite materials

Acta Metallurgica et Materialia ◽

10.1016/0956-7151(95)00100-a ◽

1995 ◽

Vol 43 (11) ◽

pp. 4027-4034 ◽

Cited By ~ 10

Author(s):

E. Herve ◽

R. Dendievel ◽

G. Bonnet

Keyword(s):

Steady State ◽

Composite Materials ◽

Matrix Composite ◽

Power Law ◽

State Power ◽

Power Law Creep ◽

Inclusion Matrix

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The shape of submarine levees: exponential or power law?

Journal of Fluid Mechanics ◽

10.1017/s0022112008004862 ◽

2009 ◽

Vol 619 ◽

pp. 367-376 ◽

Cited By ~ 18

Author(s):

V. K. BIRMAN ◽

E. MEIBURG ◽

B. KNELLER

Keyword(s):

Steady State ◽

Shallow Water ◽

Power Law ◽

Sediment Supply ◽

Water Model ◽

Navier Stokes ◽

Entrainment Rate ◽

Quasi Steady State ◽

Shallow Water Model ◽

Power Law Decay

Field observations indicate that the height of submarine levees decays with distance from the channel either exponentially or according to a power law. This investigation clarifies the flow conditions that lead to these respective shapes, via a shallow water model for the overflow currents that govern the levee formation. The model is based on a steady state balance of sediment supply by the turbidity current, and sediment deposition onto the levee, with the settling velocity and the entrainment rate appearing as parameters. It demonstrates that entrainment of ambient fluid is the determining factor for the levee shape. For negligible entrainment rates, levee shapes tend to exhibit exponential profiles, while constant rates of entrainment or detrainment result in power law shapes. Interestingly, whether a levee has an exponential or a power law shape is determined by kinematic considerations only, viz. the balance laws for sediment mass and fluid volume. We find that the respective coefficients governing the exponential or power law decay depend on the settling speeds of the sediment grains, which in turn is a function of the grain size. Two-dimensional, unsteady Navier–Stokes simulations confirm the emergence of a quasi-steady state. The depositional behaviour of this quasi-steady state is consistent with the predictions of the shallow water model, thus validating the assumptions underlying the model, and demonstrating its predictive abilities.

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