The homotheties of spherically symmetric space-times admitting maximal isometry groups larger than SO͑3͒ are found along with their metrics, using the homothety equations and without imposing any restriction on the stress-energy tensor. It turns out that there are either 11 or 7 or 5 homotheties. For the space-times with SO͑3͒ as a maximal group, solution is provided in the form of derivatives of metric coefficients, which then requires a further classification, for example, according to different types of stress-energy tensor, as has been done by Eardley ͓Commun.

A numerical investigation of three-dimensional hybrid nanomaterial micropolar fluid flow across an exponentially stretched sheet is performed. Recognized similarity transformations are adopted to convert governing equations from PDEs into the set ODEs. The dimensionless system is settled by the operating numerical approach bvp4c. The impacts of the nanoparticle volume fraction, dimensionless viscosity ratio, stretching ratio parameter, and dimensionless constant on fluid velocity, micropolar angular velocity, fluid temperature, and skin friction coefficient in both x-direction and y-direction are inspected. Graphical outcomes are shown to predict the features of the concerned parameters into the current problem. These results are vital in the future in the branches of technology and industry. The micropolar function  Rη increases for higher values of the micropolar parameter and nanoparticle concentration. Micropolar function Rη declines for higher values of the micropolar parameter and nanoparticle concentration. Temperature function is enhanced for higher values of solid nanoparticle concentration. Temperature function declines for higher values of the micropolar parameter. The range of the physical parameters are presented as: 0.005<ϕ2<0.09, Pr=6.2, 0<K<2, 0<a<2.0, ϕ1=0.1, and 0<c<1.5.

In this paper we present an algorithm to reduce the area of a surface spanned by a finite number of boundary curves by initiating a variational improvement in the surface. The ansatz we suggest consists of original surface plus a variational parameter t multiplying the numerator H0 of mean curvature function defined over the surface. We point out that the integral of the square of the mean curvature with respect to the surface parameter becomes a polynomial in this variational parameter. Finding a zero, if there is any, of this polynomial would end up at the same (minimal) surface as obtained by minimizing more complicated area functional itself. We have instead minimized this polynomial. Moreover, our minimization is restricted to a search in the class of all surfaces allowed by our ansatz. All in all, we have not yet obtained the exact minimal but we do reduce the area for the same fixed boundary. This reduction is significant for a surface (hemiellipsoid) for which we know the exact minimal surface. But for the bilinear interpolation spanned by four bounding straight lines, which can model the initial and final configurations of re-arranging strings, the decrease remains less than 0.8 percent of the original area. This may suggest that bilinear interpolation is already a near minimal surface.

No abstract

In surface modeling a surface frequently encountered is a Coons patch that is defined only for a boundary composed offouranalytical curves. In this paper we extend the range of applicability of a Coons patch by telling how to write it for a boundary composed of an arbitrary number of boundary curves. We partition the curves in a clear and natural way into four groups and then join all the curves in each group intooneanalytic curve by using representations of the unit step function including one that isfully analytic. Having a well-parameterized surface, we do some calculations on it that are motivated by differential geometry but give a better optimized and possibly more smooth surface. For this, we use an ansatz consisting of the original surface plus a variational parameter multiplying the numerator part of its mean curvature function and minimize with the respect to it the rms mean curvature and decrease the area of the surface we generate. We do a complete numerical implementation for a boundary composed of five straight lines, that can model a string breaking, and get about 0.82 percent decrease of the area. Given the demonstrated ability of our optimization algorithm to reduce area by as much as 23 percent for a spanning surface not close of being a minimal surface, this much smaller fractional decrease suggests that the Coons patch we have been able to write is already close of being a minimal surface.