Thixotropic pumping in a cylindrical pipe

We consider the flow of a thixotropic fluid in a uniform cylindrical pipe, driven by an oscillating pressure gradient or a body force. For a variety of rheological models, solutions can be obtained by integrating ordinary rather than partial differential equations: We illustrate this approach for the thixoviscoplastic Houška model and the thixoviscous simplified Moore–Mewis–Wagner model. We present asymptotic results in the limits of small and large Deborah numbers and numerical results for intermediate Deborah numbers. Under asymmetrical “sawtooth” forcing, thixotropy leads to the net transport of fluid along the pipe, even when there would be no net transport of the corresponding generalized-Newtonian fluid. We propose the name “thixotropic pumping” for this transport mechanism.

Figure 1

A Houška fluid with ${\tau }_{\mathrm{y}0}=0.1$, ${\tau }_{\mathrm{y}1}=0.1$, ${\eta }_{\mathrm{H}1}=1$, $\kappa =0.1$ and (a) $\mathcal{D}=0.001$, (b) $\mathcal{D}=0.03$, (c) $\mathcal{D}=0.1$, (d) $\mathcal{D}=0.3$, (e) $\mathcal{D}=1$, under sinusoidal forcing: the variation with time of $\lambda$ at $r=r_{\mathrm{c}}+(1-r_{\mathrm{c}})n/6$, $n=0$ to 6. (Note that for these parameter values, $r_{\mathrm{c}}=0.4$.) Solid lines are numerical results for $0<\mathcal{D}<\infty$; dashed lines in (a) and (e) correspond to the asymptotic results in the limits (a) $\mathcal{D}=0$ and (e) $\mathcal{D}\to \infty$.

Figure 2

A Houška fluid with ${\tau }_{\mathrm{y}0}=0.1$, ${\tau }_{\mathrm{y}1}=0.1$, ${\eta }_{\mathrm{H}1}=1$, $\kappa =0.1$, under sinusoidal forcing: profiles of (a)–(c) the structure $\lambda$ and (d)–(f) the velocity $w$ at $t=n\pi /16$ for $n=0$ to 8, for (a), (d) $\mathcal{D}=0$; (b), (e) $\mathcal{D}=0.1$; (c), (f) $\mathcal{D}\to \infty$. (Note that for these parameter values, $r_{\mathrm{c}}=0.4$.) The black squares mark the boundary of the plug region, $r=r_{\mathrm{y}}$. In (d) the velocity $w=0$ for $t=0$, $t=\pi /16$ and $t=\pi /8$; in (e), (f) the velocity $w=0$ for $t=0$ and $t=\pi /16$.

Figure 3

A Houška fluid with ${\tau }_{\mathrm{y}0}=0.1$, ${\tau }_{\mathrm{y}1}=0.1$, ${\eta }_{\mathrm{H}1}=1$, $\kappa =0.1$, under sinusoidal forcing. (a)–(c) The flux $Q\left(t\right)$ over a period, for (a) $\mathcal{D}=0.03$; (b) $\mathcal{D}=0.1$; (c) $\mathcal{D}=0.3$. (d) The maximum flux $Q_{max}$. (e) The total volume of fluid transported during the forward phase, $V_{\pi }$. The dashed lines correspond to the asymptotic results in the limits $\mathcal{D}=0$ (heavy dashed) and $\mathcal{D}\to \infty$ (light dashed).

Figure 4

A Houška fluid with ${\tau }_{\mathrm{y}0}=0.1$, ${\tau }_{\mathrm{y}1}=0.1$, ${\eta }_{\mathrm{H}1}=1$, $\kappa =0.1$, under sawtooth forcing. (a)–(c) The flux $Q\left(t\right)$, for $T_0=n\pi /8$, $n=0$ to 4 and (a) $\mathcal{D}=0$; (b) $\mathcal{D}=0.1$; (c) $\mathcal{D}\to \infty$. (d) The total volume of fluid transported during the forward phase, $V_{\pi }$, for $T_0=n\pi /8$, $n=0$ to 4. (e) The total volume of fluid transported over a full period, $V_{2\pi }$, for $T_0=n\pi /8$, $n=0$ to 3 (for $T_0=\pi /2$, $V_{2\pi }=0$). Arrows show the direction of increasing $T_0$.

Figure 5

A Houška fluid with (a), (b) ${\tau }_{\mathrm{y}0}=0.1$, ${\tau }_{\mathrm{y}1}=0$, ${\eta }_{\mathrm{H}1}=1$, $\kappa =0.1$ (yield stress independent of $\lambda$); (c), (d) ${\tau }_{\mathrm{y}0}=0.1$, ${\tau }_{\mathrm{y}1}=0.1$, ${\eta }_{\mathrm{H}1}=0$, $\kappa =0.1$ (viscosity independent of $\lambda$), under sawtooth forcing. (a), (c) The net flux during the forward phase, $V_{\pi }$, for $T_0=n\pi /8$, $n=0$ to 4. (b), (d) The total volume of fluid transported $V_{2\pi }$ over a full period, for $T_0=n\pi /8$, $n=0$ to 3 (for $T_0=\pi /2$, $V_{2\pi }=0$). Arrows show the direction of increasing $T_0$.

Figure 6

A simplified MMW fluid with $b=1$, $\kappa =1$ and (a), (c), (e) $a=1$ and $c=0.5$ (shear-thinning, $n=0.5$); (b), (d), (f) $a=0.5$ and $c=1$ (shear-thickening; $n=1.5$), under sinusoidal forcing. The flux $Q\left(t\right)$ is shown for (a), (b) $\mathcal{D}=0.01$; (c), (d) $\mathcal{D}=0.1$; (e), (f) $\mathcal{D}=1$. The dashed lines correspond to the asymptotic results in the limits $\mathcal{D}=0$ (heavy dashed) and $\mathcal{D}\to \infty$ (light dashed).

Figure 7

A simplified MMW fluid with (a), (b) $a=1$, $b=1$, $c=0.5$, $\kappa =1$ (shear-thinning); (c), (d) $a=0.5$, $b=1$, $c=1$, $\kappa =1$ (shear-thickening), under sawtooth forcing. (a), (c) The net flux during the forward phase, $V_{\pi }$, for $T_0=n\pi /8$, $n=0$ to 4. (b), (d) The total volume of fluid transported over a full period, $V_{2\pi }$, for $T_0=n\pi /8$, $n=0$ to 3 (for $T_0=\pi /2$, $V_{2\pi }=0$). Arrows show the direction of increasing $T_0$.