The Steady Flow of a Viscous Fluid through a Conical Nozzle

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 through the vertex of the cone, that is, the end effect is neglected; viii) the motion is so slow 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 byvr=3A(cos2θ-cos2α)/4r2, where α is the semi-angle of the cone, and A is a constant. The expression for the pressure is obtained as follows:p=p0+3/2Aη[cos2θ+1/3/r3-cos2α+1/3/r13], where η is the coefficient of viscosity, and p0 the atmospheric pressure. Thus the average pressure gradient taken over the spherical surface of radius r is given by(∂p/∂r)=-3/2Aη/r4(2+cosα+cos2α).On the other hand, the volume of flow in unit time is given byQ=-πA(1-cosα)2(1+2cosα).Eliminating A from the above two equations, we getQ=π/8R4/η(∂p/∂r)F(α), where R is equal to rα, and F(α) is given byF(α)=16/3(1-cosα)2(1+2cosα)/α4(2+cosα+cos2α).Since limα→0F(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 vr we can calculate stress components of the fluid. Especially, the tangential stress σrθ is given byσrθ=-3/2Aηsin2θ/r3.For a given value of r, σrθ 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.