Unsteady Slow Flows Over a Cooled Flat Plate

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Abstract
The unsteady flow of a hot viscous fluid with temperature-dependent viscosity into a cold channel with parallel sides is used as a simple model for injection moulding. Two thermal boundary layers occur on each channel wall, one trailing behind the moving fluid front, and the other, an “entry” layer, attached to the leading edge of the channel. This double “contained” boundary layer is analysed in appropriate similarity variables which, in the case of constant viscosity, reduce the problem to a diffusion equation with a diffusion coefficient which changes sign in the domain of interest. This “singular” parabolic problem is compared with other known examples, and an extension of a numerical finite difference method suggested by Ingham is used to solve it in the cases of constant entry flow and constant entry pressure. To obtain sufficient accuracy an asymptotic solution has to be incorporated into the numerical procedure. The problem with viscosity varying exponentially with temperature leads to a singular parabolic system which is also solved by a similar numerical procedure. The results show that variable viscosity can cause a dramatic increase in the growth of the entry boundary layer but that variations in the prescribed entry flow or pressure are much less significant.

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