We construct a new method for inextensible flows of timelike curves in Minkowski space-time . Using the Frenet frame of the given curve, we present partial differential equations. We give some characterizations for curvatures of a timelike curve in Minkowski space-time . 1. Introduction Numerous processing operations of complex fluids involve free surface deformations; examples include spraying and atomization of fertilizers and pesticides, fiber-spinning operations, paint application, roll-coating of adhesives, and food processing operations such as container- and bottle-filling. Systematically understanding such flows can be extremely difficult because of the large number of different forces that may be involved, including capillarity, viscosity, inertia, gravity, and the additional stresses resulting from the extensional deformation of the microstructure within the fluid. Consequently many free-surface phenomena are described by heuristic and poorly quantified words such as “spinnability,” “tackiness,” and “stringiness.” Additional specialized terms used in other industries include “pituity” in lubricious aqueous coatings, “body” and “length” in the printing ink business, “ropiness” in yogurts, and “long/short textures” in starch processing [1]. The flow of a curve or surface is said to be inextensible if, in the former case, the arc length is preserved, and, in the latter case, if the intrinsic curvature is preserved [2–7]. Physically, inextensible curve and surface flows are characterized by the absence of any strain energy induced from the motion. Kwon investigated inextensible flows of curves and developable surfaces in . Necessary and sufficient conditions for an inextensible curve flow first are expressed as a partial differential equation involving the curvature and torsion. Then, they derived the corresponding equations for the inextensible flow of a developable surface and showed that it suffices to describe its evolution in terms of two inextensible curve flows [8]. Additionally, there are many works related with inextensible flows [1, 8–15]. In the past two decades, for the need to explain certain physical phenomena and to solve practical problems, geometers and geometric analysis have begun to deal with curves and surfaces which are subject to various forces and which flow or evolve with time in response to those forces so that the metrics are changing. Now, various geometric flows have become one of the central topics in geometric analysis. Many authors have studied geometric flow problems [1, 12, 16, 17]. This study is organised as
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