This paper discusses the relationship between natural coordinates in fluid mechanics and orthogonal curvilinear coordinates. Since orthogonal curvilinear coordinates have some excellent mathematical properties, natura...This paper discusses the relationship between natural coordinates in fluid mechanics and orthogonal curvilinear coordinates. Since orthogonal curvilinear coordinates have some excellent mathematical properties, natural coordinates can be applied more widely if they can be transformed to orthogonal curvilinear coordinates. Frenet formulas which describe the differential properties of natural coordinates were compared with the derivative formulas of orthogonal curvilinear coordinates to show that natural coordinates are not generally orthogonal curvilinear coordinates. A method was introduced to transform natural coordinormal planes of the natural coordinates about the streamlines. The transformation is true as long as the natural coordinates satisfy several equations. Vorticity decomposition in the natural coordinates is used to show that these conditional equations are satisfied only if the streamlines are perpendicular to the vortexlines on every given point in the flow field. These equations apply in both planar flows and axisymmetric flows without a circumferential velocity component, but do not apply in some 3-D flows such as Beltrami flow.展开更多
In this article, we study the flows of curves in the Galilean 3-space and its equiform geometry without any constraints. We find that the Frenet equations and the intrinsic quantities of the inelastic flows of curves ...In this article, we study the flows of curves in the Galilean 3-space and its equiform geometry without any constraints. We find that the Frenet equations and the intrinsic quantities of the inelastic flows of curves are independent of time. We show that the motion of curves in the Galilean 3-space and its equiform geometry are described by the inviscid and viscous Burgers＇ equations.展开更多
The differential geometry of curves on a hypersphere in the Euclidean space reflects instantaneous properties of spherecal motion. In this work, we give some results for differential geometry of spacelike curves in 3-...The differential geometry of curves on a hypersphere in the Euclidean space reflects instantaneous properties of spherecal motion. In this work, we give some results for differential geometry of spacelike curves in 3-dimensional de-Sitter space. Also, we study the Frenet reference frame, the Frenet equations, and the geodesic curvature and torsion functions to analyze and characterize the shape of the curves in 3-dimensional de-Sitter space.展开更多
文摘This paper discusses the relationship between natural coordinates in fluid mechanics and orthogonal curvilinear coordinates. Since orthogonal curvilinear coordinates have some excellent mathematical properties, natural coordinates can be applied more widely if they can be transformed to orthogonal curvilinear coordinates. Frenet formulas which describe the differential properties of natural coordinates were compared with the derivative formulas of orthogonal curvilinear coordinates to show that natural coordinates are not generally orthogonal curvilinear coordinates. A method was introduced to transform natural coordinormal planes of the natural coordinates about the streamlines. The transformation is true as long as the natural coordinates satisfy several equations. Vorticity decomposition in the natural coordinates is used to show that these conditional equations are satisfied only if the streamlines are perpendicular to the vortexlines on every given point in the flow field. These equations apply in both planar flows and axisymmetric flows without a circumferential velocity component, but do not apply in some 3-D flows such as Beltrami flow.
文摘In this article, we study the flows of curves in the Galilean 3-space and its equiform geometry without any constraints. We find that the Frenet equations and the intrinsic quantities of the inelastic flows of curves are independent of time. We show that the motion of curves in the Galilean 3-space and its equiform geometry are described by the inviscid and viscous Burgers＇ equations.
文摘The differential geometry of curves on a hypersphere in the Euclidean space reflects instantaneous properties of spherecal motion. In this work, we give some results for differential geometry of spacelike curves in 3-dimensional de-Sitter space. Also, we study the Frenet reference frame, the Frenet equations, and the geodesic curvature and torsion functions to analyze and characterize the shape of the curves in 3-dimensional de-Sitter space.
