Studies on melt spinning. III. Velocity field within the thread

The velocity field within a molten spinning thread was analyzed quantitatively by solving the equations of continuity and momentum for Newtonian liquids. In solving the equations, the viscosity was assumed known and was given by the expression\documentclass{article}\pagestyle{empty}\begin{document}$ \mu = \mu _0 e^{\beta x} \left( {1 + cr^2 } \right) $\end{document} where x and r are distances in cylindrical coordinates. A series solution in velocity v having the expression\documentclass{article}\pagestyle{empty}\begin{document}$ \nu = \nu _0 e^{\alpha x} \left( {1 + a_2 r^2 + a_4 r^4 + a_6 r^6 + \cdots } \right) $\end{document} was obtained when several simplifying assumptions were made on the equations. The series solution was found to converge when cr 2 < 1 is satisfied. μ 0 e β x and ν 0 e α x above are tangents on semilog paper at x = x to the macroscopic viscosity and velocity profiles μ( x ) and ν( x ) which were computed separately by means of a technique developed previously by the author. 1,2 The value c was derived from the temperature profile across the thread at x = x computed separately using another technique developed by the author. 3 The above series solution showed numerically that under most conceivable spinning conditions the velocity field within the thread is for practical purposes flat across the thread and, in addition, purely extensional.