Chapter 5, Problem 5.139P

Two very long (in the direction normal to the page) bars having the prescribed initial temperature distributions are to be soldered together. At time $t=0,$ the $m=3$ face of the copper (pure) bar contacts the m 4 of the steel (AISI 1010) bar. The solder and flux act as an interfacial layer of negligible thickness and effective contact resistance $R_{t,c}^{"}=2\times {10}^{-5}{\text{m}}^{\text{2}}\cdot \text{K/W}\text{.}$

1. Derive the explicit, finite-difference equation in terms of Fo and $B{i}_c=\Delta x/k{R}_{t,c}^{"}$ for $T_{4,2}$ and determine the corresponding stability criterion.
2. Using $Fo=0.01,$ determine $T_{4,2}$ one time step after contact is made. What is $\Delta t?$ Is the stability criterion satisfied?