Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Penetrative internally heated convection in two and three dimensions   Convection of an internally heated fluid, confined between top and bottom
plates of equal temperature, is studied by direct numerical simulation in two
and three dimensions. The unstably stratified upper region drives convection
that penetrates into the stably stratified lower region. The fraction of
produced heat escaping across the bottom plate, which is one half without
convection, initially decreases as convection strengthens. Entering the
turbulent regime, this decrease reverses in two dimensions but continues
monotonically in three dimensions. The mean fluid temperature, which grows
proportionally to the heating rate ($H$) without convection, grows
proportionally to $H^{4/5}$ when convection is strong in both two and three
dimensions. The ratio of the heating rate to the fluid temperature is likened
to the Nusselt number of Rayleigh-B\'enard convection. Simulations are reported
for Prandtl numbers between 0.1 and 10 and for Rayleigh numbers (defined in
terms of the heating rate) up to $5\times10^{10}$.

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The categories of this paper are: hep-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: stat
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: math
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg