Automatic apparatus, based upon a nickel-tube resonator, for measuring the complex shear modulus of liquids in the kHz range

Notations /3, Prest a outer radius of the tube A apparatus constant t b distance to screen (in App. B) Tliq Bn 3 dB-bandwith B l o n g longitudinal magnetization of RL the tube gin, Vout C O ~ ]~Ni~ONi l velocity of propX agation of torsional waves in a XL tube without losses y C, C~, C2, C3 constants z deff thickness of "immobilized Z liquid layer" Z L f frequency a fn resonance frequency /~ F = H / ( C I ) G = G' + iG" complex shear modulus F H transfer function i 1/-1 Af I moment of inertia per unit ea,, eG,, length ( Ip polar moment per unit length k 3(2n + 1) r/, r/s K parameter indicating devia0 tions from "ideal" resonance £ tube length co L inductance of transmitting coil co~ m ratio of inner and outer radii of the tube Mel torsion moment caused by subscripts elastic deformation of the tube Mext torsion moment exerted on the 0 tube by the liquid n n integer Ni N number of turns in receiving coil superscripts