Extensions of the Ferry shear wave model for active linear and nonlinear microrheology

Abstract

The classical oscillatory shear wave model of Ferry et al. [J. Polym. Sci. 2:593-611, (1947)] is extended for active linear and nonlinear microrheology. In the Ferry protocol, oscillation and attenuation lengths of the shear wave measured from strobe photographs determine storage and loss moduli at each frequency of plate oscillation. The microliter volumes typical in biology require modifications of experimental method and theory. Microbead tracking replaces strobe photographs. Reflection from the top boundary yields counterpropagating modes which are modeled here for linear and nonlinear viscoelastic constitutive laws. Furthermore, bulk imposed strain is easily controlled, and we explore the onset of normal stress generation and shear thinning using nonlinear viscoelastic models. For this paper, we present the theory, exact linear and nonlinear solutions where possible, and simulation tools more generally. We then illustrate errors in inverse characterization by application of the Ferry formulas, due to both suppression of wave reflection and nonlinearity, even if there were no experimental error. This shear wave method presents an active and nonlinear analog of the two-point microrheology of Crocker et al. [Phys. Rev. Lett. 85: 888 -891 (2000)]. Nonlocal (spatially extended) deformations and stresses are propagated through a small volume sample, on wavelengths long relative to bead size. The setup is ideal for exploration of nonlinear threshold behavior.

Purely pulsating flow of a viscoelastic fluid in a pipe revisited: The limit of large Womersley number

“…(22) in Ref. [15]) is qualitatively similar to our (35), but with a dependence of the thickness and the waviness of the viscoelastic boundary layer on the Deborah number quantitatively different from that given here for an oscillating pressure gradient in terms of the functions qðcÞ and sðcÞ.…”

“…c ¼ 1), where shear waves are effectively underdamped and thus extend to the whole fluid domain. This behavior has been reported in several viscoelastic fluid flows induced by an oscillating plate (e.g., [15,14,16]; see Discussion below).…”

“…Curiously, the structure of this solution is qualitatively similar to that for the Maxwell fluid flow between two parallel plates oscillating synchronously [15,14], but with quantitatively different dependence of the reciprocal length scale with the Deborah number. This connection between the shear waves generated by an oscillating plate and the present pulsating pipe flow generated by an oscillating pressure gradient is discussed further below.…”

“…It is interesting to note that this waviness is similar to that found in underdamped (Ferry) shear waves generated by an oscillating plate (see, e.g., Refs. [15,14,16]). In fact, the structure of the solution for the velocity above the oscillating plate in the case of a Maxwell fluid (e.g., Eq.…”

“…Since the blood is known to behave as a non-Newtonian fluid [4], analytical solutions for several non-Newtonian fluid models have also been obtained later for this idealized pipe flow, as well as for some other more general pulsating pipe flows, not only in relation to blood flow, but to some other physiological and industrial processes, many of them in relation to the flow enhancement effect [5][6][7][8][9][10][11][12][13][14]. Some of these works, in addition to other ones for the viscoelastic fluid flow generated by an oscillating plate [15,16], are cited below in connection to the solutions described in the present work. Here we consider the flow in a pipe due to a purely periodic pressure gradient of a fluid modelled by a simple Maxwell model in order to analyse the interesting pulsating behavior of a viscoelastic fluid in the limit of large Womersley numbers, of interest for the blood flow in arteries.…”