Solutions of one dimensional steady flow of dusty gas in an anholonomic co-ordinate system
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Solutions of one dimensional steady flow of dusty gas in an anholonomic co-ordinate system C S BAGEWADI and A N SHANTHARAJAPPA* Department of Studies in Mathematics, Kuvempu University, Shankaragatta 577 451, India *Department of Mathematics, J.N.N. College of Engineering, Shimoga 577 204, India MS received 10 November 1997; revised 13 July 1999 Abstract. We study the geometry of one-dimensional (i.e. unidirectional) incompressible steady dusty gas flow in Frenet frame field system (anholonomic co-ordinate system) by assuming the paths of velocities of dust and fluid phases to be in the same direction. The intrinsic decompositions of the basic equation are carried out and solutions for velocity of fluid phase u, velocity of dust phase v and pressure of the fluid are obtained in terms of spin coefficients, i.e. geometrical parameters like curvatures and torsions of the streamline when the flow is
(i) parallel straight line i.e. ks = 0 (ii) parallel and ks ~ O, under the assumption that, the sum of the deformations at a point of the fluid surface along the stream line, its principal normal and binormal is constant. Further, we have proved a result, which is an extension of Barron, and a graph ofp against s is plotted (figure 1). Keywords.
Frenet frame field; dusty gas; velocities of dusty gas and fluid phases.
1. Introduction
We study two phase flow of a dusty gas because it is useful in lunar ash flow, which explains many features of lunar soil. By ash flow, we mean the flow of a mixture of gas and small particles (ash or fine powder) such that the ash particles are fluidized and they behave like a pseudo fluid. Thus, from a fluid dynamical point of view, the ash flow is a two phase flow of a gas and a pseudo fluid of solid particles. We consider gas flow in which spherical solid dusty particles are uniformly distributed. The study of such flow is of interest in a wide variety of areas of technical importance like environmental pollution, formation of raindrops and blood flow etc. The introduction of geometric theories in the study of fluid flows simplifies the mathematical complexities to a great extent and furnishes information regarding flow fields in a more general way. During the second part of the 20th century some authors like Truesdell [10], Kanwal [5,6], Indrasena [4, 8], Purushotham [7, 8] and Bagewadi and Prasanna Kumar [1, 2] etc., have tried to study fluid flow by solving partial differential equations without using boundary conditions. They have prescribed some analytic and geometric conditions, which are valid not only on the boundary but also in the whole region. It seems that not much work has been done on the lines of the above authors. 435
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C S Bagewadi and A N Shantharajappa
Barron [3] has obtained solutions in steady plane flow of a viscous dusty fluid with parallel velocity fields in orthogonal curvilinear coordinates, i.e. by taking co-ordinate axes as fluid streamlines ~ = constant and their orthogonal trajectories ~ = constant as co-ordinate curves. He has proved the following the
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