(to be done ON PAPER )

1. State precisely themathematicalproblem describing axially symmetric heat conduction in a hollow cylinder Ω: Rin ≤ r ≤ Rout, 0 ≤ z ≤ Z, with imposed temperature boundary conditions on all boundaries, starting at a uniform temperatureu. 2. Set up a 2-dimensional (r,z) mesh for the axially symmetric heat transfer problem above, by subdividing [Rin, Rout] into_{init}Mrsubintervals and [0,Z] intoMzsubintervals. Determine (formulas for) the location of nodes: r_{i}, z_{j}, of faces: r_{i-½}, z_{j-½}, areas of radial faces: Ar_{i-½,j}, areas of axial faces: Az_{i,j-½}, and control volumes: ΔV_{ij}. Make sure you specify the range of indices for each. 3. Derive the discrete heat balance for axially symmetric heat conduction in a (hollow) cylinder using the above mesh, for any (implicitness parameter) θ in [0,1]. [This is the conservation law applied on each control volume, in terms of U and fluxes, where fluxes=...., areas=...., volume=.... Do NOT write code, only formulas needed for update.] 4. Outline the explicit Finite Volumeas it will be coded up, do NOT plug these into the expressionalgorithm(formulas, NOT code) for the problem with convective BC at r=Rin, imposed flux at r=Rout, and insulated top and bottom (at z=0, z=Z), starting at uniform initial temperature u_{init}. For simplicity, assume constant properties and mesh uniform in each direction. 5. Find the CFL condition and the stable Δt_{expl}for the explicit scheme (for internal nodes only). [For this you DO need to plug in formulas for fluxes, areas, vol, to see the coefficients]