1. T=1 T=0 At an internal node Heat Balance gives Grid advection Note sign And volume continuity gives Deforming 2-D grid for One-Phase Stefan Problem—Compare with Filling

2. We will Write this as Coefficients from one face i=1 2 3 Uses Central difference with Grid vel at face mid point u grid velocity at node3 = 2/12 for face bolded The addition of this term is explained below

3. i=1 2 3 On noting that

4. Sum over all faces in support gives And that heat balance can be written as Where the contribution from a face are Now note that by Volume continuity

5. Heat Balance Equation Can be written as Where the contribution from a face are So we have volume balance (1) (2) Put (1) in (2) to end up with the final internal node heat balance

6. On moving boundary in bo T =0 from boundary condition Can use this to update node position on boundary Use Boundary condition To arrive at

7. xx yy i (in direction Of average normal To surfaces that meet at p --exact direction not critical) And with the setting Then splitting boundary into 2 segments

8. xx yy i So for a given nodal h field the equation provides an explicit means of finding the A and y displacements at node I that will recover the volume balance NOTE end points on boundary are constrained to move in x-direction or y-direction. At End point 1 set c= 10 -6 and at End point2 set c= 10 6 1 2

9. Steps in Solution for Filling problem 1.Set up grid assuming a small initial radius r = 1.1 say, set  say  2. Calculate and store coefficients 3. Set up boundaries 4. solve steady state problem Fixed =1 NO Fixed =0 NO 5. Calculate update boundary displacements In a time step 15-20 iterations of the following 6. Calculate x- displacements globally through solving Use Current Coefficients with boundary settings Fixed =  x i NO Fixed =  7. Do same for y-displacements 8. Update node point locations (under relax) 9. Repeat to end of iterations 10. At end of time step calcs Set 11. Go to next time step 9A—calculate nodal velocities

10. Last HomeWork—All projects from now on Due Tue April 21 T=1 T=0 Solve this problem on deforming mesh (use time step and initial mesh as in notes) Solve for S t = 0,.5, 1, 10 Compare with fixed grid enthalpy results—Notes given next week