Hysteresis in viscoelastic flow instability of confined cylinders

Viscoelastic flow through porous media is relevant in many industrial and biological applications including enhanced oil recovery and biofluids' transport inside the body, where the presence of large polymeric stresses in the porous media leads to viscoelastic instability. In the present study, we numerically investigate viscoelastic instability-induced flow states in the channels consisting of (i) a single cylinder and (ii) two streamwise located cylinders. We unravel the presence of a hysteresis for pulsatile viscoelastic flows. For both geometries, the quantitative value of flow asymmetry around the cylinder resulting from the viscoelastic instability forms a closed hysteresis loop. We investigate the effects of fluid rheological properties and periodic inlet flow rates on the hysteresis loop.

Figure 1

Schematic of the geometries used for simulations: (a) a single cylinder located inside a channel and (b) a pair of streamwise located cylinders inside a channel. $U_{\mathrm{in}}\left(t\right)$ represents time-dependent inlet flow velocity. Channel length, channel width, and cylinder diameter have been denoted by $l, w$, and $d$, respectively. The separation between cylinders for the channel consisting of a pair of cylinders is $l_s=2d$. The values of different geometrical parameters are $l=4$ mm, $w=0.4$ mm, and $d=160 \mu \mathrm{m}$, respectively. The channel length is much larger than the width ($l=10w$) and cylinder diameter ($l=25d$) to avoid any entrance or exit effect (Appendix pp1-s1). Black solid lines represent solid boundaries, whereas green and red lines indicate inlet and outlet, respectively.

Figure 2

(a) Velocity field for the viscoelastic flow around a confined cylinder at (i) $\mathrm{Wi}=1.25$ and (ii) $\mathrm{Wi}=2.5$. (b) Flow asymmetry, $I$, for the viscoelastic flow around a confined cylinder at different Wi. The symbol and the error bars represent the mean value and the standard deviation, respectively. The standard deviation is very small as the flow remains almost steady (Appendix pp1-s3). The fluid rheological parameters are $\beta =0.05$ and $L^2=1000$.

Figure 3

(a) The profile of time-dependent oscillatory $\mathrm{Wi}$ used in the simulation. ${\mathrm{Wi}}_{\min }, {\mathrm{Wi}}_{\max }$, and $T$ are the minimum value, maximum value and time period of Wi. The value of flow asymmetry at different $\mathrm{Wi}$ for (b) $T=10$, (c) $T=20$, and (d) $T=40$. Black dashed lines show the startup regime for the hysteresis, whereas solid lines represent different hysteresis loops. The values of other parameters are $\beta =0.05, L^2=1000, {\mathrm{Wi}}_{\min }=0.62$, and ${\mathrm{Wi}}_{\max }=3.12$.

Figure 4

[(a)–(c)] Flow field and [(d)–(f)] the trace of polymeric stress field at $\mathrm{Wi}=1.25(<{\mathrm{Wi}}_{cr})$ for the different branches of the hysteresis loop shown in Fig. 3. The flow states shown in [(a) and (d)], [(b) and (e)], and [(c) and (f)] correspond to the locations indicated by filled circle ($•$), filled diamond ($⧫$), and filled star ($★$) in Fig. 3, respectively. Other parameters are $\beta =0.05, L^2=1000, T=20, {\mathrm{Wi}}_{\min }=0.62$, and ${\mathrm{Wi}}_{\max }=3.12$.

Figure 5

The area of hysteresis loop ($A_{\mathrm{loop}}$) and the minimum value of flow asymmetry in the hysteresis loop ($I_{\min }$) for different values of (a) oscillation time period (T), (b) minimum $\mathrm{Wi}$ (${\mathrm{Wi}}_{\min }$), and (c) maximum $\mathrm{Wi}$ (${\mathrm{Wi}}_{\max }$). The values of other parameters are ${\mathrm{Wi}}_{\min }=0.62$ and ${\mathrm{Wi}}_{\max }=3.12$ for (a), $T=20$ and ${\mathrm{Wi}}_{\max }=3.12$ for (b), and ${\mathrm{Wi}}_{\min }=0.62$ and $T=20$ for (c). The rheological parameters are $\beta =0.05$ and $L^2=1000$.

Figure 6

The area of hysteresis loop ($A_{\mathrm{loop}}$) and the minimum value of flow asymmetry in the hysteresis loop ($I_{\min }$) for different values of (a) viscosity ratio ($\beta$) at $L^2=1000$ and (b) polymeric chains' extensibility ($L^2$) at $\beta =0.05$. Other parameters are $T=20, {\mathrm{Wi}}_{\min }=0.62$, and ${\mathrm{Wi}}_{\max }=3.12$.

Figure 7

(a) Flow asymmetry ($I$) around the front cylinder in the geometry having two streamwise located cylinders in a channel at $T=40$. (b) The area occupied by eddies ($A_{\mathrm{eddy}}$) in the region between the cylinders at $T=40$. $A_{\mathrm{eddy}}$ has been normalized with $l_{s}d$. The portions of the hysteresis loop corresponding to the eddy free symmetric (type-1) and asymmetric (type-3) flow states have been indicated by magenta and blue colors, respectively. The green and red portions of the loop represent symmetric and asymmetric flow states with eddies between the cylinders, respectively. The values of other parameters are $\beta =0.05, L^2=1000, {\mathrm{Wi}}_{\min }=0.62$, and ${\mathrm{Wi}}_{\max }=3.12$.

Figure 8

[(a)–(d)] Flow field and [(e)–(h)] the trace of polymeric stress field corresponding to the different portions of the hysteresis loop shown in Fig. 7. The flow states shown in [(a) and (e)], [(b) and (f)], [(c) and (g)], and [(d) and (h)] correspond to the locations indicated by filled circle ($•$), filled diamond ($⧫$), filled star ($★$), and filled triangle ($▴$) in Fig. 7 and represent flow states type-1, type-2a, type-2b, and type-3, respectively. The values of other parameters are $\beta =0.05, L^2=1000, T=40, {\mathrm{Wi}}_{\min }=0.62$, and ${\mathrm{Wi}}_{\max }=3.12$.

Figure 9

(a) Flow asymmetry ($I$) around the front cylinder in the geometry consisting of two streamwise located cylinders at $T=10$. (b) The area occupied by eddies ($A_{\mathrm{eddy}}$) in the region between the cylinders at $T=10$. $A_{\mathrm{eddy}}$ has been normalized with $l_{s}d$. The values of other parameters are $\beta =0.05, L^2=1000, {\mathrm{Wi}}_{\min }=0.62$, and ${\mathrm{Wi}}_{\max }=3.12$.

Figure 10

[(a)–(c)] Flow asymmetry ($I$) around the front cylinder and [(d)–(f)] the area occupied by eddies ($A_{\mathrm{eddy}}$) in the region between the cylinders for [(a) and (d)] ${\mathrm{Wi}}_{\min }=0.62,{\mathrm{Wi}}_{\max }=3.12$; [(b) and (e)] ${\mathrm{Wi}}_{\min }=0.31,{\mathrm{Wi}}_{\max }=3.12$; and [(c) and (f)] ${\mathrm{Wi}}_{\min }=0.62,{\mathrm{Wi}}_{\max }=2.62$. The values of other parameters are $\beta =0.05, L^2=1000$, and $T=20$. $A_{\mathrm{eddy}}$ has been normalized with $l_{s}d$.

Figure 11

[(a) and (b)] Flow asymmetry ($I$) around the front cylinder and [(c) and (d)] the area occupied by eddies ($A_{\mathrm{eddy}}$) in the region between the cylinders for [(a) and (c)] $\beta =0.2$ and $L^2=1000$ and [(b) and (d)] $\beta =0.05$ and $L^2=400$. The values of other parameters are ${\mathrm{Wi}}_{\min }=0.62, {\mathrm{Wi}}_{\max }=3.12$, and $T=20$. $A_{\mathrm{eddy}}$ has been normalized with $l_{s}d$.

Figure 12

(a) The velocity profile at different locations along the length of the channel ($l=25d$) at $\mathrm{Wi}=0.62$ for the channel having a single cylinder. (b) Flow asymmetry ($I$) at different locations along the length of channel at $\mathrm{Wi}=3.12$ for the channels having length $l=25d$ and $l=50d$. The cylinder (the front cylinder in the channel having two cylinders) is located at $x=0$. The entrance and the exit of the channel are at $x=-9.4d$ and $x=15.6d$ for $l=25d$, and $x=-21.9d$ and $x=28.1d$ for $l=50d$.

Figure 13

Cartoons showing coarse numerical meshes ($320\times 32$) close to the cylinders for the channels having (a) a single cylinder and (b) two cylinders. The simulations have been performed using a finer mesh ($2560\times 256$).

Figure 14

Normalized pressure drop across the channel having a single cylinder at (a) $\mathrm{Wi}=0.62$ and (b) $\mathrm{Wi}=3.12$.

Figure 15

Time-dependent flow asymmetry around the confined cylinder at $\mathrm{Wi}=3.12$. Other parameters are $\beta =0.05$ and $L^2=1000$. The standard deviation of the fluctuation of $I$ once the asymmetric flow state becomes fully developed is $\approx 0.6\%$ of the mean value.

Figure 16

Flow asymmetry around a confined cylinder for a quasistatic variation of Wi (flow rate) at $\beta =0.05$ and $L^2=1000$. $\mathrm{\Delta }{t}_{\mathrm{up}}=7.7$ and $\mathrm{\Delta }{t}_{\mathrm{down}}=12.2$ are the lag time of the stress to respond to the increment and decrement of Wi, respectively. The time step of the quasistatic variation of Wi is $\mathrm{\Delta }{t}_{\mathrm{step}}=20$.