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The solution of computational fluid dynamics problems is one of the mostcomputationally hard tasks, especially in the case of complex geometries andturbulent flow regimes. We propose to use Tensor Train (TT) methods, whichpossess logarithmic complexity in problem size and have great similarities withquantum algorithms in the structure of data representation. We develop theTensor train Finite Element Method -- TetraFEM -- and the explicit numericalscheme for the solution of the incompressible Navier-Stokes equation via TensorTrains. We test this approach on the simulation of liquids mixing in a T-shapemixer, which, to our knowledge, was done for the first time using tensormethods in such non-trivial geometries. As expected, we achieve exponentialcompression in memory of all FEM matrices and demonstrate an exponentialspeed-up compared to the conventional FEM implementation on dense meshes. Inaddition, we discuss the possibility of extending this method to a quantumcomputer to solve more complex problems. This paper is based on work weconducted for Evonik Industries AG.

Numerical solution of the incompressible Navier-Stokes equations for chemical mixers via quantum-inspired Tensor Train Finite Element Method