Abstract
An internal energy function of the mass density, the volumetric entropy and their gradients at n-order generates the representation of multi-gradient fluids. Thanks to Hamilton’s principle, we obtain a thermodynamical form of the equation of motion which generalizes the case of perfect compressible fluids. First integrals of flows are extended cases of perfect compressible fluids. The equation of motion and the equation of energy are written for dissipative cases, and are compatible with the second law of thermodynamics.
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Notes
Due to the fact that \(\varepsilon ,_{\rho ,_{x_{j_1}} \ldots ,_{x_{j_p}}} d\rho ,_{x_{j_1}} \ldots ,_{x_{j_p}}= d\rho ,_{x_{j_1}} \ldots ,_{x_{j_p}}\varepsilon ,_{\,\rho ,_{x_{j_1}} \ldots ,_{x_{j_p}}}\) and \(\varepsilon ,_{\eta ,_{x_{j_1}} \ldots ,_{x_{j_p}}} d\eta ,_{x_{j_1}} \ldots ,_{x_{j_p}}= d\eta ,_{x_{j_1}} \ldots ,_{x_{j_p}}\varepsilon ,_{\,\eta ,_{x_{j_1}} \ldots ,_{x_{j_p}}}\), we indifferently permute the position of the two terms in the summation.
For example, when \(A=\left[ \begin{array}{ccc} a_{11}, &{} a_{12}, &{} a_{13} \\ a_{21}, &{} a_{22}, &{} a_{23} \\ a_{31}, &{} a_{32}, &{} a_{33} \end{array} \right] ,\) then
\(\mathrm {div}A=\left[ a_{11},_{x_{1}}+a_{21},_{x_{2}}+a_{31},_{x_{3}},\, a_{12},_{x_{1}}+a_{22},_{x_{2}}+ a_{32},_{x_{3}},\, a_{13},_{x_{1}}+a_{23},_{x_{2}}+a_{33},_{x_{3}} \right] ,\ \mathrm {and}\)
\(\mathrm {div}_{2}\,A={a_{11}}_{,x_{1},x_{1}}+{a_{21}}_{,x_{2},x_{1}}+{a_{31}} _{,x_{3},x_{1}}+{a_{12}}_{,x_{1},x_{2}}+{a_{22}}_{,x_{2},x_{2}}+{a_{32}} _{,x_{3},x_{2}}+{a_{13}}_{,x_{1},x_{3}}+{a_{23}}_{,x_{2},x_{3}}+{a_{33}} _{,x_{3},x_{3}}.\)
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The results contained in the present paper have been partially presented in Wascom 2017.
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The paper is dedicated to Professor Tommaso Ruggeri.
The work was supported by National Group of Mathematical Physics GNFM-INdAM (Italy).
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Gouin, H. Multi-gradient fluids. Ricerche mat 68, 195–209 (2019). https://doi.org/10.1007/s11587-018-0397-5
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DOI: https://doi.org/10.1007/s11587-018-0397-5