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The Quarterly Journal of Mechanics and Applied Mathematics 1999 52(4):623-644; doi:10.1093/qjmam/52.4.623
© 1999 by Oxford University Press
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Behaviour of a non-local reactive convective problem modelling ohmic heating of foods

AA LaceyA1, DE TzanetisA2 and PM VlamosA2

( A1 Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK A2 Department of Mathematics, National Technical University of Athens, Zografou Campus 157 80, Greece )

We consider the non-local problem,

ut + ux = {lambda}f(u) / ({int}01f(u)dx)2, 0 < x <1,

which models the temperature when an electric current flows through a moving material with negligible thermal conductivity. The potential difference across the material is fixed but the electrical resistivity f(u) varies with temperature. It is found that, for f decreasing with {int}0{infty} f(s)ds < {infty}, blow-up occurs if {lambda} is too large for a steady state to exist or if the initial condition is too big. If f is increasing with {int}0{infty} ds/f(s) < {infty} blow-up is also possible. If f is increasing with {int}0{infty} ds/f(s) = {infty} or decreasing with {int}0{infty} f(s) s = {infty} the solution is global. Some special cases with particular forms of f are discussed to illustrate what the solution can do.


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