Cooperation and competition of viscoelastic fluids and elastomeric microtubes subject to pulsatile forcing

We analyze the dynamic behavior of viscoelastic fluids in elastic tubes subject to pulsatile pressure gradients. In particular, we study the dynamic permeability, a response function that relates linearly flow and pressure gradient in frequency domain. We have found resonances that can be associated to the elasticity of the tube and resonances that can be associated to the elasticity of the fluid. There is a rich phenomenology that includes cooperation and competition of both elasticities. Tuning of the system parameters allows for the excitation of the different modes in the system, sometimes giving responses that are larger than the ones of the corresponding systems with elasticity in only one of its elements. This behavior is similar to the one of two coupled oscillators. Our results are relevant for small confining geometries with low Young moduli. For example, for a microtube with a radius of a few hundred microns with the elasticity of polydimethylsiloxane (PDMS), in which a viscoelastic fluid, with the rheological parameters of blood, is driven by a pulsatile pressure gradient, resonance frequencies on the sound range are predicted. In a wide frequency range, the dynamic permeability is much lower than the one of the same fluid flowing in a poly(methyl methacrilate) (PMMA) tube. Our results are potentially useful for tailoring composite laboratory-on-a-chip devices, where introducing materials with different parameters in a device would induce an increase or decrease of the amplitude of the longitudinally averaged flow. We demonstrate that the magnitude of the dynamic permeability for a composite tube, at a given frequency of the driving pressure drop, determines the amplitude of the average flow along the composite tube.

Figure 1

Magnitude of the dynamic permeability as a function of frequency for three systems: viscoelastic fluid in an elastic tube (blue [dark solid] line), Newtonian fluid in an elastic tube (pink [light dashed] line), and viscoelastic fluid in a rigid tube (green [light solid] line). Parameters have been chosen to show the resonances of the three systems on the same figure. For the elastic tube: $E=5\mathrm{Pa}$, ${\rho }_w=5\times {10}^4\text{kg}/{\text{m}}^3$, $\nu =0.5$, and $h=1\mu \mathrm{m}$. For the fluid: $\rho =1050\text{kg}/{\text{m}}^3$, $\mu =4\times {10}^{-3}\text{kg}/\text{m}\text{s}$, $t_r=1\times {10}^{-3}$ s (in the viscoelastic case), and $t_r=0$ s (in the Newtonian case). The radius was chosen as $a=6\times {10}^{-5}$ m. Values of $a^{*}$ and ${\omega }^{*}$, for the parameters used in this figure, are $3.3\times {10}^{-4}$ m and 220 rad/s, respectively.

Figure 2

Magnitude of the dynamic permeability as a function of frequency for different values of the wall density and three different systems: viscoelastic fluid in an elastic tube (blue [dark solid] lines), Newtonian fluid in an elastic tube (pink [light dashed] lines), and viscoelastic fluid in a rigid tube (green [light solid] lines). For the elastic tube, $E=1.0\times {10}^4\mathrm{Pa}$, $\nu =0.5$, and $h=10\mu \mathrm{m}$. For the fluid, $\rho =818\text{kg}/{\text{m}}^3$, $\mu =1\text{kg}/\text{m}\text{s}$, and $t_r=4\times {10}^{-5}$ s (in the viscoelastic case) and $t_r=0$ s (in the Newtonian case). The radius was chosen as $a=1\times {10}^{-4}$ m. For the first three figures, wall densities correspond to realistic materials. For figure (d), the density is too large to be realistic; it was chosen to illustrate how each of the maxima coincide with the ones of elasticity in only one of the system's elements. Values of $a^{*}$ and ${\omega }^{*}$ for these figures are, respectively, (a) $3.3\times {10}^{-4}$ m and 69431 rad/s; (b) $6.7\times {10}^{-4}$ m and 17132 rad/s; (c) $1.1\times {10}^{-3}$ m and 6853 rad/s; and (d) $3.3\times {10}^{-3}$ m and 685 rad/s.

Figure 3

Magnitude of the dynamic permeability as a function of frequency for different radii and four different systems: viscoelastic fluid in an elastic tube (blue [dark solid] lines), Newtonian fluid in an elastic tube (pink [light dashed] lines), viscoelastic fluid in a rigid tube (green [light solid] lines), and Newtonian fluid in a rigid tube (black dashed line). For the elastic tube, ${\rho }_w=987\text{kg}/{\text{m}}^3$, $E=1.0\times {10}^4\mathrm{Pa}$, $\nu =0.5$, and $h=10\mu \mathrm{m}$. For the fluid, $\rho =818\text{kg}/{\text{m}}^3$, $\mu =1\text{kg}/\text{m}\text{s}$, $t_r=4\times {10}^{-5}$ s (in the viscoelastic case), and $t_r=0$ s (in the Newtonian case). Values of $a^{*}$ and ${\omega }^{*}$, for the parameters used in this figure, are $3.3\times {10}^{-4}$ m and 69431 rad/s, respectively.

Figure 4

Two resonance frequencies for viscoelastic fluid in an elastic tube, obtained from blue line in Fig. 3. The low resonance frequency, ${\omega }_{\mathrm{res}1}$, associated to the elasticity of the tube, is marked by crosses. The high resonance frequency, ${\omega }_{\mathrm{res}2}$, associated to the viscoelasticity of the fluid, is marked by circles. Red (dark dashed), green (light solid), and black (dark solid) lines, used as reference, indicate the characteristic frequencies of the system, given by Eqs. (16), (17), and (20), respectively.

Figure 5

Effect of different relaxation times in the dynamic permeability. Magnitude of the dynamic permeability as a function of frequency for two different fluid systems with different relaxation times: viscoelastic fluid with $t_r^1$ in an elastic tube (blue [dark] solid line), viscoelastic fluid with $t_r^1$ in a rigid tube (green [light] solid line), viscoelastic fluid with $t_r^2>t_r^1$ in an elastic tube (blue [dark] dashed line), and viscoelastic fluid with $t_r^2>t_r^1$ in a rigid tube (green [light] dashed line). The two values of the relaxation time used were $t_r^1=4\times {10}^{-5}\mathrm{s}$ (solid lines) and $t_r^2=5\times {10}^{-5}\mathrm{s}$ (dashed lines), corresponding to ${\omega }_{t_r}/{\omega }_{\mu }=0.20$ and ${\omega }_{t_r}/{\omega }_{\mu }=0.16$; and $t_r{\omega }^{*}=0.69$ and $t_r{\omega }^{*}=0.86$, respectively. The rest of the parameters are as in Fig. 2.

Figure 6

(a) Orange (light dashed) line is the magnitude of the dynamic permeability of a composite tube of total length of 2 cm in which the first third of the tube has the elastic properties of PDMS and the rest of the tube has the elastic properties of PMMA. As reference, the permeabilities of a tube made solely of PDMS and a tube made solely of PMMA are shown in blue (dark solid) and green (light solid) lines, respectively. (b) Average flow, along the flow direction, as a function of time for the three systems, with the same color coding used in panel (a), when the fluid is driven with an oscillatory pressure drop $\mathrm{\Delta }p\left(t\right)=-\mathrm{\Delta }{p}_0\text{cos}\left({\omega }_{0}t\right)$, with ${\omega }_0=2800$ rad/s and $\mathrm{\Delta }{p}_0=20\mathrm{Pa}$. Fluid parameters were chosen as $\rho =1050\text{kg}/{\text{m}}^3$, $\mu =4\times {10}^{-3}\text{kg}/\text{m}\text{s}$, and $t_r=1\times {10}^{-3}$ s, which gives a Deborah number $\frac{{\omega }_{t_r}}{{\omega }_{\mu }}=2.6$. For the driving frequency, ${\omega }_0$, the Womersley number is $\sqrt{\frac{{\omega }_0}{{\omega }_{\mu }}}=1.1$. Elastic parameters of the first part of the composite tube were chosen as those of PDMS, ${\rho }_w=987\text{kg}/{\text{m}}^3,E=3.6\times {10}^5\text{Pa,}\text{and}\nu =0.5$, and those of the second part of the composite tube were chosen as those of PMMA, ${\rho }_w=1200\text{kg}/{\text{m}}^3,E=3.8\times {10}^9\text{Pa},\text{and}\nu =0.36$. Radius and tube wall thickness were chosen as $a=1\times {10}^{-4}$ m and $h=1\times {10}^{-5}$ m. We have verified that the value of $\kappa a<5\times {10}^{-2}$ for PDMS and $\kappa a<5\times {10}^{-4}$ for PMMA in the frequency range of panel (a), showing that the example used in this proof of concept is within the range of parameters in which our approximation is valid.