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A multi-fluid model is used to study the dynamics of a dusty plasma sheath consists of electrons, ions, and two species of charged dust particles, i.e., nano-size and micron-size particles. It is found that, when the sheath is dominated by the nano-size dust grains, spatially periodic fluctuations are developed in the profiles of the sheath potential, and the number density and velocity of the plasma and dust particles. Due to inertial effects, the fluctuations in the parameters of the micron-size grains are much lower than those of the other parameters. The competition between the electric and ion drag forces plays the primary role in development of the fluctuations. The spatial period of the fluctuations is approximately a few Debye lengths and their amplitude depends on the plasma and dust parameters. The fluctuations are reduced by the increase in the radius, mass density, and Mach number of the nano-size particles, as well as the density and Mach number of the ions. But, they are enhanced by the increase in the plasma number density and the electron temperature. The sheath thickness demonstrates a non-monotonic behavior against variation of the nanoparticle parameters, i.e., it first decreases quickly, shows a minimum, and then increases. However, the sheath width always decreases with the plasma number density and ion Mach number, while grows linearly with the electron temperature.