Moderate De Research Articles

Collapse of spherical bubbles in Maxwell fluids

The collapse of a spherical cavitation bubble contained in a large body of upper convected Maxwell fluid is theoretically analyzed by using a variational principle approach in the Lagrangian frame for the K-BKZ rheological equation of state with potential functions. Based on the Rayleigh time scale for bubble collapse in ideal fluids, two parameters. Re and De, are identified. Using a finite element technique, a fully explicit numerical scheme is developed both for the pressure distribution calculation and for bubble surface tracking. The same problem is formulated also using the Galerkin-finite element method in the Lagrangian frame for the differential model of an upper convected Maxwell fluid. With the latter method, the viscoelastic stress can be determined explicitly. Even though the result is the same as far as the radius-time curve is concerned, each method has its own advantages. Highly oscillatory behaviors in bubble radius are observed for moderate Re and De. For large ReDe, the solution exhibits an asymptotic behavior. It is also observed that fluid elasticity accelerates the collapse in the early stage of collapse while in the later stages it retards the collapse. The retardation for a moderate range of Re is expected to be related to the reduced cavitation damage in viscoelastic fluids.

Numerically stable finite element techniques for viscoelastic calculations in smooth and singular geometries

Finite element calculations for viscoelastic flows are reported that use a restructured form of the equation of motion that makes explicit the elliptic character of this equation. We call this restructured equation the Explicity Elliptic Momentum Equation, and its use is illustrated for flow of an upper convected Maxwell (UCM) model between eccentric and concentric rotating cylinders and also for a modified upper convected Maxwell (MUCM) model in the stick-slip problem. Sets of mixed-order approximations for velocity, stress, and a modified pressure are used to test the algorithm in both problems. Both sets of calculations are shown to converge with mesh refinement and are limited at high values of Deborah number by the formation of elastic boundary layers that are identified in the momentum equation by the growth of low-order derivative terms that involve the local velocity gradient and divergence of stress. Similar convergence properties are observed for bilinear and biquadratic Lagrangian approximations to the stress components. However, calculations with the more accurate basis for stress converge to higher values of De and are sensitive to the weighted residual method used for the constitutive equation, particularly for the eccentric cylinder problem. Streamline-upwind Petroy-Galerkin (SUPG) and artificial diffusivity (AD) formulations of the constitutive equation are tested for solution of both problems by calculations of the stress fields with fixed kinematics and by solution of the coupled problem. The SUPG method improves the performance of the calculations with the biquadratic basis set for the eccentric cylinder problem. For the UCM model, adding artificial diffusion to the constitutive equation in the stick-slip problem changes the dominant balance for the stress field near the singularity, making it appear as an integrable stress approximation for fixed mesh. For the MUCM model the Newtonian-like behavior of the stress near this point is unaffected by the AD method and calculations converge to moderate De.

Biochemical Skeletal Muscle and Hematological Profiles of Moderate and Severely Iron Deficient and Anemic Adult Rats

The time course of the effects of moderate and iron anemia on tissue biochemistry of adult male rats was studied. Nine groups of adult male Sprague-Dawley rats were killed; eight of which had approximately 8 g hemoglobin (Hb)/dl (moderately anemic) for 3 to 360 days and one group which had approximately 4 g Hb/dl (severely anemic) for 30 days. Plasma iron was depressed in the mod erately and severely anemic groups at all times relative to nonanemic control rats. Moderately anemic rats at 360 days had a low mean corpuscu lar Hb and mean corpuscular volume and severely anemic rats at 210 days had a low mean corpuscular Hb, and elevated mean corpuscular Hb concen tration. Cytochrome oxidase activity of the white vastus lateralis, red vastus lateralis and soleus muscles were not different between the moderately anemic groups and their respective control groups at any time, but was significantly lower after 30 days of iron anemia. The myoglobin content and phosphofructokinase activity of these muscles showed no significant group differences with moderate or iron de ficiency. It is concluded that: 1) moderate iron anemia imposed no stimulus for alterations of the oxidative or glycolytic capacity of skeletal muscle, whereas iron anemia induced rapid select changes in a mitochondrial component of skeletal muscle, 2) the blood profiles in conjunction with the muscle biochemistry data in the moderately anemic group suggest a hypochromic, microcytic, manifest iron deficiency anemia, while those of the severely anemic group suggest a hypochromic, normo- cytic, severe iron deficiency anemia. J. Nutr. 108: 1306-1314, 1978.