On a modified form of navier-stokes equations for three-dimensional flows.

A rephrased form of Navier-Stokes equations is performed for incompressible, three-dimensional, unsteady flows according to Eulerian formalism for the fluid motion. In particular, we propose a geometrical method for the elimination of the nonlinear terms of these fundamental equations, which are expressed in true vector form, and finally arrive at an equivalent system of three semilinear first order PDEs, which hold for a three-dimensional rectangular Cartesian coordinate system. Next, we present the related variational formulation of these modified equations as well as a general type of weak solutions which mainly concern Sobolev spaces.

pH-Specific structural speciation of the ternary V(V)-peroxido-betaine system: a chemical reactivity-structure correlation.

Vanadium involvement in cellular processes requires deep understanding of the nature and properties of its soluble and bioavailable forms arising in aqueous speciations of binary and ternary systems. In an effort to understand the ternary vanadium-H(2)O(2)-ligand interactions relevant to that metal ion's biological role, synthetic efforts were launched involving the physiological ligands betaine (Me(3)N(+)CH(2)CO(2)(-)) and H(2)O(2). In a pH-specific fashion, V(2)O(5), betaine, and H(2)O(2) reacted and afforded three new, unusual, and unique compounds, consistent with the molecular formulation K(2)[V(2)O(2)(O(2))(4){(CH(3))(3)NCH(2)CO(2))}]·H(2)O (1), (NH(4))(2)[V(2)O(2)(O(2))(4){(CH(3))(3)NCH(2)CO(2))}]·0.75H(2)O (2), and {Na(2)[V(2)O(2)(O(2))(4){(CH(3))(3)NCH(2)CO(2))}(2)]}(n)·4nH(2)O (3). All complexes 1-3 were characterized by elemental analysis; UV/visible, FT-IR, Raman, and NMR spectroscopy in solution and the solid state; cyclic voltammetry; TGA-DTG; and X-ray crystallography. The structures of 1 and 2 reveal the presence of unusual ternary dinuclear vanadium-tetraperoxido-betaine complexes containing [(V(V)═O)(O(2))(2)] units interacting through long V-O bonds. The two V(V) ions are bridged through the oxygen terminal of one of the peroxide groups bound to the vanadium centers. The betaine ligand binds only one of the two V(V) ions. In the case of the third complex 3, the two vanadium centers are not immediate neighbors, with Na(+) ions (a) acting as efficient oxygen anchors and through Na-O bonds holding the two vanadium ions in place and (b) providing for oxygen-containing ligand binding leading to a polymeric lattice. In 1 and 3, interesting 2D (honeycomb) and 1D (zigzag chains) topologies of potassium nine-coordinate polyhedra (1) and sodium octahedra (3), respectively, form. The collective physicochemical properties of the three ternary species 1-3 project the chemical role of the low molecular mass biosubstrate betaine in binding V(V)-diperoxido units, thereby stabilizing a dinuclear V(V)-tetraperoxido dianion. Structural comparisons of the anions in 1-3 with other known dinuclear V(V)-tetraperoxido binary anionic species provide insight into the chemical reactivity of V(V)-diperoxido systems and their potential link to cellular events such as insulin mimesis and anitumorigenicity modulated by the presence of betaine.

Aqueous V(V)-peroxo-amino acid chemistry. Synthesis, structural and spectroscopic characterization of unusual ternary dinuclear tetraperoxo vanadium(V)-glycine complexes.

Vanadium participation in cellular events entails in-depth comprehension of its soluble and bioavailable forms bearing physiological ligands in aqueous distributions of binary and ternary systems. Poised to understand the ternary V(V)-H(2)O(2)-amino acid interactions relevant to that metal ion's biological role, we have launched synthetic efforts involving the physiological ligands glycine and H(2)O(2). In a pH-specific fashion, V(2)O(5), glycine, and H(2)O(2) reacted and afforded the unusual complexes (H(3)O)(2)[V(2)(O)(2)(mu(2):eta(2):eta(1)-O(2))(2)(eta(2)-O(2))(2)(C(2)H(5)NO(2))] x 5/4 H(2)O (1) and K(2)[V(2)(O)(2)(mu(2):eta(2):eta(1)-O(2))(2)(eta(2)-O(2))(2)(C(2)H(5)NO(2))] x H(2)O (2). 1 crystallizes in the triclinic space group P1, with a = 7.805(4) A, b = 8.134(5) A, c = 12.010(7) A, alpha = 72.298(9) degrees, beta = 72.991(9) degrees, gamma = 64.111(9) degrees, V = 641.9(6) A(3), and Z = 2. 2 crystallizes in the triclinic space group P1, with a = 7.6766(9) A, b = 7.9534(9) A, c = 11.7494(13) A, alpha = 71.768(2) degrees, beta = 73.233(2) degrees, gamma = 65.660(2) degrees, V = 610.15(12) A(3), and Z = 2. Both complexes 1 and 2 were characterized by UV/visible, LC-MS, FT-IR, Raman, NMR spectroscopy, cyclic voltammetry, and X-ray crystallography. The structures of 1 and 2 reveal the presence of unusual ternary dinuclear vanadium-tetraperoxo-glycine complexes containing [(V(V)=O)(O(2))(2)](-) units interacting through long V-O bonds and an effective glycinate bridge. The latter ligand is present in the dianionic assembly as a bidentate moiety spanning both V(V) centers in a zwitterionic form. The collective physicochemical properties of the two ternary species 1 and 2 project the chemical role of the low molecular mass biosubstrate glycine in binding V(V)-diperoxo units, thereby stabilizing a dinuclear V(V)-tetraperoxo dianion. Structural comparisons of the anions in 1 and 2 with other known dinuclear V(V)-tetraperoxo binary anionic species provide insight into the chemical reactivity of V(V)-diperoxo species in key cellular events such as insulin mimesis and antitumorigenicity, potentially modulated by the presence of glycinate and hydrogen peroxide.

Probing for missing links in the binary and ternary V(V)-citrate-(H2O2) systems: synthetic efforts and in vitro insulin mimetic activity studies.

In a pH-specific fashion, V(2)O(5) and citric acid in the absence and presence of H(2)O(2) reacted and afforded, in the presence of NaOH and (CH(6)N(3))(2)CO(3), two new dinuclear V(V) binary non-peroxo (CH(6)N(3))(6)[V(2)O(4)(C(6)H(4)O(7))(2)].2H(2)O (1) and ternary peroxo (CH(6)N(3))(4)[V(2)O(2)(Omicron(2))(2)(C(6)H(5)O(7))(2)].6Eta(2)Omicron (2) species, respectively. Complexes 1 and 2 were further characterized by elemental analysis, UV/Vis, FT-IR, NMR (solution and solid state Cross Polarization-Magic Angle Spinning (CP-MAS)) and Raman spectroscopies, cyclic voltammetry, and X-ray crystallography. Both 1 and 2 are members of the family of dinuclear V(V)-citrate species bearing citrate with a distinct coordination mode and degree of deprotonation, with 2 being the missing link in the family of pH-structural variants of the ternary V(V)-peroxo-citrate system. Given that 1 and 2 possess distinct structural features, relevant binary V(III), V(IV) and V(V), and ternary V(V) species bearing O- and N-containing ligands were tested in in vitro cell cultures to assess their cellular toxicity and insulin mimetic capacity. The results project a clear profile for all species tested, earmarking the importance of vanadium oxidation state and its ligand environment in influencing further binary and ternary interactions of vanadium arising with variable mass cellular targets, ultimately leading to a specific (non)toxic phenotype and glucose uptake ability.