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Exact solution of a restricted Euler equation for the velocity gradient tensor. (English) Zbl 0754.76004
The velocity gradient tensor satisfies a nonlinear evolution equation of the form \((dA_{ij}/dt)+A_{ik}A_{kj}- (1/3)(A_{mn}A_{nm})\delta_{ij}=H_{ij}\), where \(A_{ij}=\partial u_ i/\partial x_ j\) and the tensor \(H_{ij}\) contains terms involving the action of cross derivatives of the pressure field and viscous diffusion of the velocity gradient. The homogeneous case (\(H_{ij}=0\)) considered previously by P. Vielliefosse [J. Phys. 43, No. 6, 837-842 (1982); Physica A 125, 150-162 (1984; Zbl 0599.76040)] is revisited here and examined in the context of an exact solution. First the equations are simplified to a linear, second-order system \((d^ 2A_{ij}/dt^ 2)+(2/3)Q(t)A_{ij}=0\), where \(Q(t)\) is expressed in terms of Jacobian elliptic functions. The exact solution in analytical form is then presented providing a detailed description of the relationship between initial conditions and the evolution of the velocity gradient tensor and associated strain and rotation tensors.

MSC:
76A02 Foundations of fluid mechanics
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References:
[1] DOI: 10.1051/jphys:01982004306083700
[2] DOI: 10.1016/0378-4371(84)90008-6 · Zbl 0599.76040
[3] DOI: 10.1063/1.866513
[4] DOI: 10.1017/S0022112091001957 · Zbl 0721.76036
[5] DOI: 10.1063/1.857938
[6] DOI: 10.1063/1.857824 · Zbl 0696.76070
[7] DOI: 10.1063/1.857773 · Zbl 0697.76071
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