Peristaltic transport of a compressible viscous liquid through a tapered pore

The axisymmetric peristaltic flow of a viscous compressible liquid through a tapered pore has been studied. This peristaltic flow is actually a result of the influence of ultrasonic radiation on the flow of a liquid through a porous medium which deforms the walls of the pores in the shape of travelling transversal waves exactly like peristaltic pumping [R.F. Ganiev, L.E. Ukrainskii, K.V. Frolov, Wave mechanism for the acceleration of a liquid flowing in capillaries and porous media, Sov. Phys. Dokl. 34 (1989) 519-521]. In this article, the wave amplitude is related to the power output of an acoustic source, while the wave speed is expressed in terms of the shear modulus of the porous medium and the dimensionless radius of the pore (variable radius/radius at inlet). A perturbation technique has been employed to analyze the problem where the amplitude ratio (wave amplitude/pore radius at inlet) is chosen as a parameter. In the second-order approximation a net flow induced by the travelling wave is found. The net flow rate is calculated for various values of the compressibility of the liquid, the Reynolds number and wavelength parameter. The calculations disclose that the compressibility of the liquid has a strong influence on the net flow induced. Furthermore, by a comparison with the flow induced by the pressure gradient in an oil reservoir, the net flow induced by a travelling wave cannot be neglected, although it is a second-order effect.