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U0U0=22U0 The initial and boundary conditions (10) and (11) are transformed to U0(,0)U0(0,)U0(,)=0,=1,0. This set of equations can be solved using the method of Laplace transforms . However, some insight can be gained by examining large time behaviour. Thus we look for the "quasi-equilibrium" solution, obtained by setting the derivative with respect to to zero. This corresponds to the large time solution in the moving coordinate system (,). Setting U0/=0 in (14) gives a simple constant-coefficient differential equation to solve. The solution (see exercises, Question 16), which also satisfies the boundary conditions (16) and (17) is U0()=e Reverting to our original coordinates (X,T), using (13), the zerothorder solution can be written as U0=e(XT). This quasi-equilibrium solution has neglected transients relative to the moving boundary (to zeroth-order). The solution predicts that the dimensionless temperature dies off very quickly a short distance from the moving boundary. This is expected since we are supplying a large amount of energy to the moving boundary It is also possible to obtain the time dependent solution using the method of Laplace transforms, see Andrews and McLone (1976) and Bedding (1994). This solution is U0(,)=21e[1erf(4)]+21[1erf(4+)]. Note that the limit corresponds to the quasi-equilibrium solution. U0U0=22U0 The initial and boundary conditions (10) and (11) are transformed to U0(,0)U0(0,)U0(,)=0,=1,0. This set of equations can be solved using the method of Laplace transforms . However, some insight can be gained by examining large time behaviour. Thus we look for the "quasi-equilibrium" solution, obtained by setting the derivative with respect to to zero. This corresponds to the large time solution in the moving coordinate system (,). Setting U0/=0 in (14) gives a simple constant-coefficient differential equation to solve. The solution (see exercises, Question 16), which also satisfies the boundary conditions (16) and (17) is U0()=e Reverting to our original coordinates (X,T), using (13), the zerothorder solution can be written as U0=e(XT). This quasi-equilibrium solution has neglected transients relative to the moving boundary (to zeroth-order). The solution predicts that the dimensionless temperature dies off very quickly a short distance from the moving boundary. This is expected since we are supplying a large amount of energy to the moving boundary It is also possible to obtain the time dependent solution using the method of Laplace transforms, see Andrews and McLone (1976) and Bedding (1994). This solution is U0(,)=21e[1erf(4)]+21[1erf(4+)]. Note that the limit corresponds to the quasi-equilibrium solution

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