Lubrication theory of drag on a scanning probe in structured water, near a hydrophilic surface

The drag is evaluated, in lubrication theory, on a parabolic-cylindrical tip moving parallel to a flat sample, in water, close enough that the presumed nanometer-thick, structured-water, "interphase" coatings of both it and the sample overlap. Assuming coatings of width, w, and of viscosity, eta, much larger than bulk water's, the friction force is predicted to equal -2pietacRphi(D/2w), where R is the tip's radius of curvature, c is its speed, D is the tip-sample separation, and Phi is a universal function whose value only differs noticeably from ln(2w/D) for D < w.