Material surfaces tend to show roughness that has fractal characteristics, and this affects the nature of contacts between surfaces. In particular, as smaller and smaller scales of roughness are added to a model, the areas in contact break up to form larger numbers of smaller contacts. The total area in contact diminishes towards zero. What this means for the constriction resistance is examined, and using the ‘Cantor Dust’ fractal model, it is shown that, in realistic situations, it tends towards a finite limit. If however, the contact area reduces fast enough as the scale is reduced, an infinite limit is possible. The effect is dependent on the clustering of the contacts determined by the largest scales. In the absence of such clustering, the resistance can tend to zero
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