Stability of plane Poiseuille flow of a Bingham fluid through a deformable neo-Hookean channel

We study the linear stability of plane Poiseuille flow of a Bingham fluid through a deformable neo-Hookean channel. The analysis reveals the existence of a finite-wave instability in the creeping-flow limit, which is absent for Newtonian fluid flow in the same geometry. Indeed, the instability in plane Poiseuille flow of a Bingham fluid closely resembles the finite-wave instability exhibited by plane Couette flow of a Newtonian fluid past a deformable solid. The motion of the unyielded region near the channel centerline effectively results in a Couette-like drag flow of the fluid in the yielded region, albeit with a parabolic base flow velocity profile. More importantly, the boundary conditions at the interface between yielded and unyielded zones is found to be crucial for the existence of the finite-wave instability. This is in stark contrast to Newtonian channel flow where the conditions at the channel centerline are dictated by the symmetry or antisymmetry of the velocity perturbations. The continuation of the predicted instability in the creeping-flow limit is also shown to determine the stability of the flow of a Bingham fluid at higher Reynolds number. In addition to treating the plug as a rigid solid, we also consider the plug to be an elastic solid and show that the elastic nature of the plug does not have any effect on the predicted finite-wave instability. We further argue, using the elastic plug scenario, that some of the regularized models can introduce model-dependent instabilities which are specific to the parameters used in the models.

Figure 1

Schematic of the flow geometry studied in the present work. Here $y_0{R}^{*}$ and $\tilde{H}{R}^{*}$ are, respectively, the dimensional plug and solid thicknesses. At $y^{*}=R^{*}+\tilde{H}{R}^{*}$ and $-R^{*}-\tilde{H}{R}^{*}$, the deformable solid is perfectly bonded to a rigid solid.

Figure 2

Variation of $c_i$ of the most unstable eigenvalue with $k$ for $T=0$ and $\text{Re}=0$. Here $y_0=0$ defines a Newtonian fluid (varicose mode). The presence of (a) finite-wave instability for $H=10$ and $\mathrm{\Gamma }=0.5$ ($k<1$) and (b) short-wave instability for $H=0.1$ and $\mathrm{\Gamma }=2$ ($k>5$) is shown for pressure-driven flow of a Bingham fluid in the creeping-flow limit.

Figure 3

Neutral stability curve in $k\text{−}\mathrm{\Gamma }$ space for $\text{Re}=0$ and $H=10$. The figure shows the existence of the finite-wave instability for the Bingham fluid absent for the Newtonian fluid.

Figure 4

(a) Reduction of the Bingham fluid problem to the plane Poiseuille flow with fictitious boundary conditions ${\tilde{v}}_x={\tilde{v}}_y=0$ at the channel centerline as $y_0\to 0$. (b) Plane Poiseuille flow of a Newtonian fluid through a channel with both deformable walls with boundary conditions for varicose and sinuous modes at the channel centerline for $y_0=0$. (c) Plane Poiseuille flow of a Newtonian fluid with a deformable wall with no-slip and no-penetration boundary conditions at the upper plate.

Figure 5

Variation of $c_i$ with $k$ of the most unstable mode at $\text{Re}=0$, $\mathrm{\Gamma }=0.5$, and $H=10$. In the limit of $y_0\to 0$, the finite-wave instability predicted here approaches the results for stability of plane Poiseuille flow of a Newtonian fluid with fictitious no-slip boundary conditions ${\tilde{v}}_x=0$ and ${\tilde{v}}_y=0$ at the channel centerline.

Figure 6

Variation of ${\mathrm{\Gamma }}_c$ and $k_c$ with $H$ for $\text{Re}=0$. The finite-wave and short-wave instabilities are the critical modes of instability in $H>1$ and $H<1$ regions, respectively.

Figure 7

(a) Variation of ${\mathrm{\Gamma }}_c$ with $y_0$. (b) Collapse of the data for $H=10$ and 15 shows the existence of the scaling ${\mathrm{\Gamma }}_c\sim 1/H$ for the finite-wave instability (${\mathrm{\Gamma }}_{c}H$ vs $y_0$).

Figure 8

Variation of ${\mathrm{\Gamma }}_c$ with $y_0$ for constant $\tilde{H}=(1-y_0)H$ at $\text{Re}=0$. The figure shows scaling ${\mathrm{\Gamma }}_c\sim (1-y_0)$.

Figure 9

Eigenfunction plots for the streamwise and normal components of the velocity for the neutrally stable eigenvalue $c=0.950578$ obtained at $\text{Re}=0$, $H=15$, $y_0=0.01$, $k=0.0835$, and $\mathrm{\Gamma }=0.17141$.

Figure 10

Variation of ${\mathrm{\Gamma }}_c$ with Re at $H=0.1$ for the short-wave mode. Increasing Re has a destabilizing effect on the short-wave instability.

Figure 11

Spectra showing the existence of upstream modes for Bingham fluid at (a) $y_0=0.01$ and (b) $y_0=0.5$. The other parameters utilized are $\text{Re}=50$, $k=0.5$, and $H=5$.

Figure 12

Spectra showing the destabilization of downstream modes for Bingham fluid at $y_0=0.01$, $\text{Re}=50$, $k=1$, and $H=5$.

Figure 13

Variation of critical parameters with Re at $H=5$ and $y_0=0.01$ for (a) inviscid and (b) wall modes. The finite-wave mode is the most unstable inviscid mode while the first wall mode is the most unstable wall mode.

Figure 14

Eigenfunction plots for the streamwise and normal components of the velocity for the neutrally stable eigenvalue $c\sim 0.953744+0i$ obtained at $\text{Re}=1000$, $H=5$, $y_0=0.01$, $k=1.392$, and $\mathrm{\Gamma }=0.0009635$.

Figure 15

Eigenfunction plots for the streamwise and normal components of the velocity for the eigenvalue $c\sim -0.201800+0i$ obtained at $\text{Re}=1000$, $H=5$, $y_0=0.01$, $k=0.494$, and $\mathrm{\Gamma }=0.08767$. The maximum variation of the eigenfunctions near the deformable wall indicates the wall mode nature of the upstream mode.

Figure 16

Variation of critical parameters (${\mathrm{\Gamma }}_c$ and $k_c$) with Re at $H=5$ and $y_0=0.5$ for inviscid and wall modes. The inviscid and wall modes are, respectively, the critical modes of instability for $\text{Re}<10$ and $\text{Re}>10$.

Figure 17

Variation of ${\mathrm{\Gamma }}_c$ with Re at $H=15$ for (a) finite-wave and (b) first wall modes. The finite-wave mode is the most unstable inviscid mode while first wall mode is the most unstable wall mode.

Figure 18

Spectra at $\text{Re}={10}^4$, $k=1$, $H=5$, and $y_0=0.01$ showing the destabilization of the inviscid modes of the Bingham fluid which manifests as finite-wave instability. (a) Destabilization of a downstream fluid-solid coupled mode which gives rise to the finite-wave mode with an increase in $\mathrm{\Gamma }$. The eigenvalues are shown for ${10}^{-5}<\mathrm{\Gamma }<{10}^{-4}$ such that for each step $\mathrm{\Gamma }$ increases in steps of ${10}^{-5}$. At $\mathrm{\Gamma }={10}^{-5}$ the mode is stable (blue dot at $c\sim 2.75-0.0017i$); it moves towards the $c_i=0$ line with increasing $\mathrm{\Gamma }$ and becomes unstable at $\mathrm{\Gamma }={10}^{-4}$. (b) The invariable nature of the rigid spectra is shown with respect to variation in the deformability of the wall.

Figure 19

Variation of critical Re with $\mathrm{\Sigma }$ for $H=5$. For (a) $y_0=0.01$, the finite-wave mode is the critical mode, while for (b) $y_0=0.5$, the absence of the finite-wave mode makes short-wave, first inviscid, and wall modes critical modes of instability for different Re ranges.