|We derive horizontal fluid motions on the solar surface over large areas covering the quiet-Sun magnetic network from local correlation tracking of continuum intensity images (granules), and Doppler velocities from the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO). From these we calculate horizontal divergence, vertical component of vorticity, and kinetic helicity of fluid motions. We study the relationship between fluid divergence and vorticity (kinetic helicity), and that between vorticity (kinetic helicity) and magnetic field. We find that the vorticity (kinetic helicity) around small-scale fields exhibits a hemispherical pattern (in sign) similar to that followed by the magnetic helicity of large-scale active regions (containing sunspots). This is consistent with helioseismological results on the Coriolis force induced rotations of supergranular motions. In addition, we find that stronger magnetic fields (at the small-scales), due to Lorentz forces, tend to suppress the vortical motions around them resulting in rapid decrease of fluid vorticities beyond magnetic flux densities of about 300 G (HMI). We also show that such action of magnetic fields leads to marked changes in the correlations between fluid divergence and vorticity, and speculate on the implications of our results to the diagnosis of local dynamo action if present and to the dynamical evolution of magnetic helicity in the small-scale.