Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model

The unidirectional oscillating flow of a viscoelastic fluid with the fractional Maxwell model is studied. The flow is produced by a periodic pressure gradient in an infinite straight pipe. Exact solutions are obtained in the time and frequency domains by using Fourier transform. The fractional Maxwell model exhibits resonance phenomena similar to that of the Maxwell model. However, the number of the resonance peaks, the amplitudes of the velocity enhancement and the frequencies where resonance occurs are totally different. The amplitudes of resonance peaks decay rapidly with frequency. In addition, a critical dimensionless radius is found for the fractional Maxwell model. Below the critical radius,the resonance behavior is weakened, but above this value, the enhancement is strengthened.