Quasi-Steady Analysis of Mold Filling

We assume isothermal flow and a Newtonian fluid. (Isothermality is the more serious of the two assumptions. Even if the physical properties of the melt are independent of temperature, we must deal with the possibility of solidification of the melt near the cold mold faces during filhng.) For radial flow we have a single velocity component in cylindrical coordinates, Vr, and the continuity equation from Table 2.1 is 

The components of the creeping flow equations are obtained from Table 2.4 by setting /o = 0 with the velocity of the form Vr = f(z)/r, vg = v = 0, we obtain 

(Pis a function only of r (and perhaps of time, which does not appear explicitly). Equation 6.3a can be rewritten as 

Since z and r are independent variables, the function of z on the left of Equation 6.4 can equal the function of r on the right for all r and z only if both are constant. 

The function /(z) must be a quadratic, since its second derivative is a constant. After two integrations we obtain 

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