Thermal Conduction in Higher-Dimensional Sphere and Application to Fluid Rotation

Abstract

When two parallel disks, a cylinder, and a sphere having a static fluid within start to rotate, the increase in the angular velocity of the laminar fluid layers starting coaxial rotation was found to be mathematically equivalent to the thermal conduction of one-, four-, and five-dimensional spheres heated from their surfaces, respectively. The thermal conduction of an n-dimensional sphere for 4 ≤ n ≤ 100 was numerically simulated using a time step t and radial division r. The growth period required to achieve a final flat temperature profile was defined as the time when the center temperature reached 99/100 of the surface temperature. The growth period was numerically determined to be proportional to n -1.25 , which was analytically supported by the thermal conduction of n-dimensional cubes inscribed within and circumscribing an n-dimensional sphere. Because the exponential time change in the surface temperature gradient was almost similar for the spherical heating of 1-8-dimensional spheres, the entire thermal transport in the n-dimensional sphere for 1 ≤ n ≤ 8 was numerically determined to be similar with respect to the growth period, which was supported by the analytical solutions for spheres with 1 ≤ n ≤ 3 by regarding slab and cylindrical thermal conduction as the heating modes of one-and two-dimensional spheres, respectively. The observed attenuation periods of fluid rotation in cylinder and spherical flasks were supported by the numerically determined growth periods of the thermal conduction of four-and five-dimensional spheres, where the rotation in the cylinder should be treated as disk rotation when the height is less than the radius. Heat Transfer between Adjacent n-DimensionalSpherical Shells: The inward thermal conduction of a sphere heated from the surface shown in Figure 1a is considered. The entire volume n V and surface area n S of the n-dimensional sphere with a radius of a and the relation between n V and n S are given by [17].

Relaxivity of Gd‐Based MRI Contrast Agents in Crosslinked Hyaluronic Acid as a Model for Tissues

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“…[42][43][44] As expected, the values of D are somewhat larger than the diffusion coefficients of the water molecules because of the nonnegligible contribution from the diffusion coefficient of Gd-HPDO3A. [45]…”