Consider a bath containing a fluid of specific heat cf and mass density ρf that surrounds the end x = L of a one-dimensional rod. Suppose that the bath is rapidly stirred in a manner such that the bath temperature is approximately uniform throughout, equaling the temperature at x = L, u(L, t). Assume that the bath is thermally insulated except at its perfect thermal contact with the rod, where the bath may be heated or cooled by the rod. Determine an equation for the temperature in the bath. (This will be a boundary condition at the end x = L.) (Hint: See Exercise 1.3.2.)
Reference Exercise 1.3.2.
Two one-dimensional rods of different materials joined at x = x0 are said to be in perfect thermal contact if the temperature is continuous at x = x0:
u(x0−, t) = u(x0+, t)
and no heat energy is lost at x = x0 (i.e., the heat energy flowing out of one flows into the other). What mathematical equation represents the latter condition at x = x0? Under what special condition is ∂u/∂x continuous at x = x0?
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