© 2016 The Korean Society of Rheology and Springer 281 Korea-Australia Rheology Journal, 28(4), 281-300 (November 2016) DOI: 10.1007/s13367-016-0030-7 www.springer.com/13367 pISSN 1226-119X eISSN 2093-7660 Simultaneous pulsatile flow and oscillating wall of a non-Newtonian liquid E.E. Herrera-Valencia 1 , M.L. Sánchez-Villavicencio 2 , F. Calderas 3, * , M. Pérez-Camacho 1 and L. Medina-Torres 4 1 Carrera de Ingeniería Química, Facultad de Estudios Superiores Zaragoza, UNAM, Ciudad de México 09230, México 2 Posgrado en Biología Experimental, Departamento de Ciencias de la Salud, UAM-I, Ciudad de México C.P. 09340, México 3 CIATEC A. C. Omega 201 Industrial Delta, León, Guanajuato C.P. 37545, México 4 Departamento de Ingeniería Química, Facultad de Química, UNAM, Ciudad de México C.P. 04510, México (Received February 12, 2016; final revision received August 21, 2016; accepted September 6, 2016) In this work, analytical predictions of the rectilinear flow of a non-Newtonian liquid are given. The fluid is subjected to a combined flow: A pulsatile time-dependent pressure gradient and a random longitudinal vibration at the wall acting simultaneously. The fluctuating component of the combined pressure gradient and oscillating flow is assumed to be of small amplitude and can be adequately represented by a weakly stochastic process, for which a quasi-static perturbation solution scheme is suggested, in terms of a small parameter. This flow is analyzed with the Tanner constitutive equation model with the viscosity function represented by the Ellis model. According to the coupled Tanner-Ellis model, the flow enhancement can be separated in two contributions (pulsatile and oscillating mechanisms) and the power requirement is always positive and can be interpreted as the sum of a pulsatile, oscillating, and the coupled systems respectively. Both expressions depend on the amplitude of the oscillations, the perturbation parameter, the exponent of the Ellis model (associated to the shear thinning or thickening mechanisms), and the Reynolds and Deborah numbers. At small wall stress values, the flow enhancement is dominated by the axial wall oscillations whereas at high wall stress values, the system is governed by the pulsating noise perturbation. The flow transition is obtained for a critical shear stress which is a function of the Reynolds number, dimensionless frequency and the ratio of the two amplitudes associated with the pulsating and oscillating perturbations. In addition, the flow enhancement is compared with analytical and numerical predictions of the Reiner-Phil- lipoff and Carreau models. Finally, the flow enhancement and power requirement are predicted using bio- logical rheometric data of blood with low cholesterol content. Keywords: pulsatile flow, oscillating wall, stochastic noise, constitutive equations, perturbation solution, analytical solutions 1. Introduction The analysis of the axial and transversal oscillations and pulsating pressure gradient flow of Newtonian and non- Newtonian fluids in a pipe has attracted ample interest due to several applications, among them, in bio-fluids mechanics (blood), flexoelectric membranes, natural and synthetic fibers (spider silk, Kevlar), porous media, enhanced oil recovery operations, polymer science (extrusion with oscillating dies), rheometry and others (Abou-Dakka et al., 2012; Herrera et al., 2009; Herrera et al., 2010; Rey and Herrera-Valencia, 2010; 2012; Rey et al., 2011). In addition, the use of pulsation has also attracted interest in connection with heat, turbulent heat, mass transfers, and coating processes (Herrera et al., 2009; Herrera et al., 2010). It is important to note that the combined flow proposed here has been previously studied only for an isolated perturbation, i.e. either pulsating flow or vibratile flow, i.e. the system (liquid) experiments a pulsating pre- ssure gradient and the wall is fixed and there is no axial or transversal perturbation (Herrera et al., 2009; Herrera et al., 2010). Two of the most interesting effects in pulsating and oscillating non-Newtonian fluids flow are the flow enhancement I(%) and power requirement E(%) which can be calculated though the volumetric flow Q 0 and aver- age time volumetric flow respectively: , (1) . (2) Equations (1) and (2) are a measure of the oscillating and pressure gradient signal noises in the volumetric flow. Several studies have demonstrated that the shear-thinning behavior is the mechanism behind flow enhancement and this enhancement is proportional to the square of the rel- ative amplitude of the oscillating pressure gradient and its magnitude depends on the shape of the viscosity function. Other important factor is the wave-form (triangular, sinu- soidal or square) which has a strong effect on flow Qt () 〈 〉 I % ( ) = 100 Qt () 〈 〉 Q 0 – Q 0 ------------------------- E % ( ) = 100 Qt ( )∇pt () 〈 〉 Q 0 ∇p 0 – Q 0 ∇p 0 ------------------------------------------------- *Corresponding author; E-mail: almotasim@hotmail.com