Fluctuating flow with mass transfer induced by a rotating disk electrode with a superimposed time-periodic modulation

Abstract

Fluctuating flow with mass transfer induced by a rotating disk electrode with a time-periodic angular velocity Ω0[1+α cos(pτ)]b is analyzed by solving numerically the time-dependent von Kármán self-similar form of the Navier–Stokes and convective diffusion equations when the electrochemical reaction at the disk electrode is mass transfer limited. Results are presented for the full frequency range for several values of the waveform parameter b and α<1. At the intermediate frequency p=1.24 (with α=0.9, b=1), the time-periodic flow undergoes a transition from a quasisteady von Kármán flow to a double boundary layer structure. The transition, which is characterized by an abrupt shift from phase lead (p<1.24) to phase lag (p>1.24) in the far-field angular velocity, is discussed in terms of a viscous wave confined to the Stokes layer by axial inflow. The numerical results for b=1 are compared to the low- and high-frequency asymptotic theories of Chawla and Verma [Proc. R. Soc. London Ser. A 386, 163 (1983)], and it is shown that terms of O(p−5/2) neglected by Chawla and Verma are required in the high-frequency expansion to describe the flow in the far field. Electrochemical measurements of the fluctuating mass transfer limited flux induced by the modulating disk electrode are in excellent agreement with the numerical results. The low-frequency calculations show that a judicious choice of b results in the cancellation of all harmonics other than the fundamental near the disk surface for certain field variables. The implications for electrochemical studies are discussed.

Full-spectrum nonlinear response of a sinusoidally modulated rotating disk electrode

“…This situation results in an axisymmetric von Kármán similarity flow that is governed by the nondimensional equations 4,11,12,14 …”

“…12,13 However, experimental corroboration of the predicted nonlinear electrochemical response has been limited to a few modulation frequencies. [12][13][14] The only fullspectrum nonlinear frequency response experiments for a speed-modulated RDE focused solely on the time-averaged ͑mean͒ limiting current, not oscillating components; experiments confirmed that 0 Ϸ 2⍀ 0 was a key resonant frequency for the steady streaming flow generated by the time-averaged Reynolds stresses. 15 To the best of our knowledge, nobody has investigated the full-spectrum frequency response of electrochemical higher harmonics.…”

“…The nonlinear oscillating flow field at a RDE has been evaluated theoretically using low and high frequency asymptotic expansions as well as transient numerical methods ͑finite differences and finite elements͒. [11][12][13][14] The nonlinear electrochemical response has also been computed using finite difference and finite element numerical methods. 12,13 However, experimental corroboration of the predicted nonlinear electrochemical response has been limited to a few modulation frequencies.…”

“…[11][12][13][14] The nonlinear electrochemical response has also been computed using finite difference and finite element numerical methods. 12,13 However, experimental corroboration of the predicted nonlinear electrochemical response has been limited to a few modulation frequencies. [12][13][14] The only fullspectrum nonlinear frequency response experiments for a speed-modulated RDE focused solely on the time-averaged ͑mean͒ limiting current, not oscillating components; experiments confirmed that 0 Ϸ 2⍀ 0 was a key resonant frequency for the steady streaming flow generated by the time-averaged Reynolds stresses.…”

On the Bödewadt–Rosenblat flow

“…Chawla and Verma [5] undertook an exhaustive study of this physical situation and derived specific expressions for quantities of physical interest under various asymptotic conditions. Shwartz, Rehg, Stroeve and Higgins [6] analyzed, numerically and experimentally, the fluctuating flow with mass transfer induced by a rotating disk electrode with a superposed time-periodic angular velocity. Their efforts at comparing their numerical results, with the low and high frequency asymptotic theories of Chawla and Verma [5], were rather fruitless because they overlooked some of the terms of higher order in the solution of Chawla and Verma.…”