The Effect of Vibration on Flow Rate of Non-Newtonian Fluid

Acoustic stimulation is a promising method for increasing drainage of non-Newtonian fluids through porous structures in various applications. In this study, a mathematical model is developed for unsteady flow of a bi-viscous incompressible fluid in a circular straight channel. Longitudinal vibrations are superimposed on the flow driven by changing pressure gradients along the channel. Simulations are carried out for a range of relevant dimensionless parameters. Effects of vibration amplitude, frequency, and fluid viscosity ratio on the enhancement of mean flow rate are discussed. Introduction Flows of non-Newtonian fluids in narrow channels and porous structures are encountered in the vast majority of chemical, biomedical and process industries. One of the practical problems encountered with high-viscosity non-Newtonian fluids is augmenting the mean flow rate in a channel at a given mean pressure gradient. It is known that by using sound or vibrations it is possible to increase the time-averaged flow of shear-thinning and viscoplastic fluids. In this study, an unsteady flow of a nonNewtonian incompressible fluid is analyzed in a circular cross section of a channel under the presence of an oscillating pressure gradient. Model Formulation We propose a model which effectively approximates shear-thinning, shear-thickening, and Bingham fluids by using a linear functions approximations. The governing equation for the fluid flow is the momentum equation, we consider an axisymmetric flow in a tube with a constant radius and assume parallel flow. The constitutive law relating the stress and strain rate of the fluid is τ = { μ1 ∂v ∂r if τ < τ1 μ2 ∂v ∂r + τ1 if τ1 ≤ τ < τ2 μ1 and μ2 – dynamic viscosities and τ1 and τ2 – yield stresses. Figure 1: Dependence of shear stress on shear rate for different types of fluids. Figure 2: Linear functions approximation of dependence of shear stress on shear rate. By assuming constant pressure gradient ∂p ∂z = G we obtain ρ ∂v ∂t = −G + μn r ∂v ∂r + μn ∂2v ∂r2 + τn r To model the effects of unsteady pressure gradient on the fluid flow inside the tube, longitudinal vibrations are applied to the channel wall parallel to the direction of the fluid flow. The displacement of the wall is given by w = beiωt b – the displacement amplitude and ω – the angular frequency of the vibration. Define the relative velocity of the fluid with respect to the wall movement: U = (v− ẇ) The governing equation for the flow takes the form ρ ∂U ∂t = ρbω2eiωt−G + μn r ( ∂U ∂r ) + μn ( ∂2U ∂r2 ) + τn r U(R, t) = 0. Instantaneous volume flow rate through the tube