We apply a quantum hydrodynamics of the surface of ~(3)He-~(4)He solutions to account for the in-plane transport of the two-dimensional (2D) spin-aligned atomic hydrogen (H(arrow down)) adsorbed on superfluid helium film. We discuss how the surface flow of 2D H(arrow down) may be traced in experiment thus allowing to study the interaction of the 2D hydrogen with ripplons and surface-bound ~(3)He as well as to observe the superfluidity of the 2D Bose gas of H(arrow down). As an example we consider the formation of the ESR spectrum of the 2D H(arrow down) and find that in spatially non-uniform case the surface flow contributes significantly to the conditions of the ESR spectrum instability observable at high microwave power. We also analyze the conditions at which the surface flow of the 2D hydrogen plays an important role in thermal compression experiments.
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