円錐系ノズル内での定常粘性流動 : 分散系のレオロジー,その他

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The purpose of this paper is to present a theory of the steady flow of Newtonian liquid through a conical nozzle. The equation of motion of viscous liquid has been treated on the following assumptions : i) the liquid is incompressible ; ii) the motion of liquid is not turbulent ; iii) the motion is steady ; iv) no body force acts on the liquid ; v) the motion has an axial symmetry ; vi) there is no slip at the wall ; vii) the stream lines are the straight lines passing that the inertia term can be neglected. We have taken a spherical coordinate system r, θ, and φ whose origin is at the vertex of the cone. Then the velocity is given by v_r=3A(cos2θ-cos2α)/4r^2, where α is the semi-angle of the cone, and A is a constant. The expression for the pressure is obtained as follows : [numerical formula], where η is the coefficient of viscosity, and p_0 the atmospheric pressure. Thus the average pressure gradient taken over the spherical surface of radius r is given by [numerical formula]. On the other hand, the volume of flow in unit time is given by [numerical formula]. Eliminating A from the above two equations, we get [numerical formula], where R is equal to r_α, and F(α) is given by [numerical formula]. Since lim__<α→0>F(α)=1, the above equation may be reduced to Poiseuille's equation for a tube of uniform cross section. From the expression for the velocity v_r we can calculate stress components of the fluid. Especially, the tangential stress σ_<rθ> is given by [numerical formula]. For a given value of r,σ_<rθ> varies as sinθ does in contrast with the case of a tube of uniform cross section where tangential stress decreases linearly from a maximum to zero with decreasing distance from the axis of the tube.
社団法人日本材料学会の論文
1963-05-15

円錐系ノズル内での定常粘性流動 : 分散系のレオロジー,その他

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