By Kleppner .D And Kolenkow R.

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We present below the final form of the energy equation for constant p case (cf. [34]). 65) ^T = T^{V | ( ^ - •^f x) ) ++ -^[V | ( ^ - •^f ,. ). i = iTix + JTiy + kTiz, i = x,y, z. 67) Here, jplVI^ is the kinetic energy and for simplicity we take the body force to be F = — grad ij). 8) are possible but before dealing with them, we give some definitions. , have been used earlier without defining or explaining them for the reason that greater concern was shown for the mathematical formulation. Some thermodynamical aspects of Stefan problems will be discussed in this section.

The latent heat released during the solidification of this volume element is pl^w\PQ\. The rate at which latent heat is released during solidification of this volume element is PQ lim p/l——lAti; = plV • fix Aw. 16) S(t+At) nx Fig. 5. Geometry in the derivation of the Stefan condition Here, V is the velocity of the interface. On using the energy conservation principle in the cylinder of volume Aw\PQ\ which requires calculating the difference in the rates at which heat flows into the cylinder across the left face and leaves through the face on the right and adding to it the rate at which heat is released and cancelling Aa;, we obtain on I an where fii = —n and ni is the outward normal to the base of the cylinder at P.

94) u{x) > ip{x) =^ u"{x) = 0. 93) implies that the string does not penetrate the obstacle. 94) and the tightness of the elastic string implies that the portions Pi A and BP2 are straight lines and so u"{x) = 0. e.. 96) where [/] stands for the jump in the quantity under consideration at the given point. 95) is equivalent to the following condition [u{x)-^{x)]u"{x) = 0. 97) is valid. 97) is satisfied. 95). 97). 2)- Consider the following minimization problem which is concerned with the minimization of the energy of the above string.