Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow

To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s^–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation a radius of the lump - A geometrical constant - b inner radius of the sample holder - c constant - C₁, C₂ apparatus constants - D damping of the pendulum - e_x, e_y, e_z Cartesian basis - e_r, e, e_z orthonormal cylindrical basis - E geometrical constant - E_t, ⁰E_t, _t relative strain tensor - f function of shear rate - F_t relative deformation tensor - G(t) memory function - G^* complex shear modulus - G Re(G^*) - G Im(G^*) - h distance between plates - H^* transfer function - , functional - i imaginary unit: i²= – 1 - I moment of inertia - J_exc excitation current - J₀ amplitude of J_exc - k^* = k – ik complex wave number - K torsional constant - K fourth order tensor - l length of the lump - L mutual inductance - M_dr driving torque - M_liq torque exerted by the liquid - ⁰M_liq, _liq steady state and dynamic part of Mliq - n power of the shear rate - p isotropic pressure - Q quality factor - r radial position - R,R₀, R_c Re(Z^*, Z₀^*, Z_c^*) - s time - t, t time - T temperature - T, ⁰T, stress tensor - u velocity - U lock-in output - ₀ velocity - V_det detector output voltage - V_sig, V_cr signal and cross-talk part of V_det - x Cartesian coordinate - X , X₀, X_c Im(Z^*, Z₀^*, Z_c^*) - y Cartesian coordinate - z Cartesian coordinate, axial position