Pulsated Herschel-Bulkley flows in two-dimensional channels: A model for mucus clearance devices

Pressure oscillations applied to human airways can help patients to evacuate bronchial mucus, a highly non-Newtonian gel. To explore the fluid mechanics aspects of these therapies, we perform numerical simulations of pulsated non-Newtonian fluids in two-dimensional channels. The fluid rheology is modeled with the Herschel-Bulkley law, reproducing two essential nonlinear mechanical properties of the mucus, namely, the yield-stress and shear-thinning/thickening properties. The flow dynamics is simulated using the lattice-Boltzmann method over large ranges of the three main nondimensional parameters, i.e., the pulsation rate or Womersley number $\alpha$, the flow index $n$ quantifying the shear-thinning/thickening effect, and the Bingham number controlling the yield stress. The ratio between the fluctuating and average parts of the oscillatory forcing is examined through three typical cases: a purely oscillating flow, a weakly oscillating flow, and a strongly oscillating flow. For each configuration, specific sets of parameters are found to have a drastic effect on the dynamics of mucus plugs, which suggests new therapeutic strategies for patients suffering from bronchial obstructions.

Figure 1

Scheme of the computational domain.

Figure 2

Analytical solution for the Herchel-Bulkley flow driven by a constant body force, $\text{Bn/Re}=3/10$, $n=1$, and ${\mu }_r=2500$; see Eq. (9).

Figure 3

Analytical solution of the Newtonian oscillatory flow (Womersley flow) in a channel, for ${\xi }_1=0$, ${\xi }_2=1$, $\alpha =5$, $\text{Re}=0.1$ at $t=n\pi /8$ for $n=[0:15]$; see Eq. (11).

Figure 4

Impact of the viscosity ratio on the velocity profile for $\text{Re}=0.1$, $\text{Bn/Re}=0.15$, $n=0.7$, $△$: ${\mu }_r=500$, $•$: ${\mu }_r=1500$, black line: ${\mu }_r=2500$, and red line: ${\mu }_r=3500$.

Figure 5

Velocity profiles (a) $\text{Re}=10$, $n=1$ and different Bn, purple: $\text{Bn}=1$, green: $\text{Bn}=2$, blue: $\text{Bn}=3$, yellow: $\text{Bn}=4$; (b) $\text{Re}=10$, $\text{Bn}=3$, purple: $n=0.7$, blue: $n=1$, green: $n=1.3$. The lines represent the analytical solutions, and the symbols the numerical ones.

Figure 6

Newtonian oscillatory flow for (a) $\text{Re}=10$, $\alpha =1$ and (b) $\text{Re}=10$, $\alpha =7$: numerical (symbols) and analytical (solid lines) velocity profiles at times $t=n\pi /8$ and $n=\{0,...,15\}$.

Figure 7

Velocity profile for $n2\pi /10$, for $n$ being an integer number between 0 and 9 (a) $\text{Re}=0.1$, $\text{Bn/Re}=0.05$, $\alpha =1$, and $n=1.3$; (b) $\text{Re}=0.1$, $\text{Bn/Re}=0.05$, $\alpha =7$, and $n=1.3$.

Figure 8

Space-time diagram of the velocity field during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$ and $n=1.3$. The velocity magnitude is indicated by the color bar. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel.

Figure 9

Space-time diagrams of dynamic viscosity during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$ and $n=1.3$. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel. The blue part represents the “solid” behavior of the fluid, while the white part represents the liquid part. The red line corresponds to the position of the interface $y_{i0}\left(t\right)$.

Figure 10

Maximal flow rate in lattice units (l.u.) for each pair of ($\alpha$, $\text{Bn/Re}$) and (a) $n=1.3$ and (b) $n=0.7$. Dotted line: flow rate for Poiseuille flow.

Figure 11

Phase lag (in degrees) between the driving force and the flow rate for each pair of ($\alpha ,\text{Bn/Re}$) and $n=1.3$.

Figure 12

Space-time diagrams of the velocity during one period of oscillation for each pair of $\alpha$, $\text{Bn/Re}$ and $n=1.3$.

Figure 13

Space-time diagram of the dynamic viscosity during one period of oscillation for each pair of $\alpha$, $\text{Bn/Re}$ and $n=1.3$. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis is the position inside the channel. The blue part represents the “solid” behavior of the fluid, while the white part represents the liquid part.

Figure 14

Mean flow rate for each pair of ($\alpha$, $\text{Bn/Re}$) and (a) $n=1.3$ and (b) $n=0.7$. Dotted line: flow rate for Poiseuille flow

Figure 15

Space-time diagram of the velocity during one period of oscillation for each pair of $\alpha$, $\text{Bn/Re}$ and $n=1.3$ (high-amplitude, low mean flow). For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis is the position inside the channel.

Figure 16

Space-time diagram of the dynamic viscosity during one period of oscillation for each pair of $\alpha$, $\text{Bn/Re}$ and $n=1.3$ (high-amplitude, low mean flow). For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis is the position inside the channel. The blue part represents the “solid” behavior of the fluid, while the white part represents the liquid part.

Figure 17

(a) Mean flow rate for each pair of ($\alpha$, $\text{Bn/Re}$) and $n=1.3$. (b) Mean flow rate for each pair of ($\alpha$, $\text{Bn/Re}$) and $n=0.7$. Dotted line: flow rate for Poiseuille flow.

Figure 18

Phase lag between force and flow rate response for each pair of ($\alpha$, $\text{Bn/Re}$) and $n=1.3$.

Figure 19

(a) Flow rate over one period for $n=1.3$. (b) Energy of harmonic frequency modes for various pairs ($\alpha ,\text{Bn/Re}$) and $n=1.3$.

Figure 20

Space-frequency diagrams of the spectral energy on the velocity for each pair of $\alpha ,\text{Bn/Re}$ for $n=1.3$. The $x$ axis corresponds to the dimensionless frequency, and the $y$ axis is the position inside the channel.

Figure 21

Space-time diagram of the velocity field during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$ and $n=0.7$. The velocity magnitude is indicated by the color bar. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel.

Figure 22

Space-time diagrams of dynamic viscosity during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$ and $n=0.7$. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel. The blue part represents the “solid” behavior of the fluid, while the white part represents the liquid part.

Figure 23

Phase lag (in degrees) between the driving force and the flow rate for each pair of ($\alpha$, $\text{Bn/Re}$) and $n=0.7$.

Figure 24

Space-time diagram of the velocity field during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$, and $n=0.7$. The velocity magnitude is indicated by the color bar. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel.

Figure 25

Space-time diagrams of dynamic viscosity during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$, and $n=0.7$. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel. The blue part represents the “solid” behavior of the fluid, while the white part represents the liquid part.

Figure 26

Space-time diagram of the velocity field during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$ and $n=0.7$. The velocity magnitude is indicated by the color bar. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel.

Figure 27

Space-time diagrams of dynamic viscosity during one period for each pair of dimensionless numbers $\alpha$, $\text{Bn/Re}$ and $n=0.7$. For each subplot, the $x$ axis corresponds to the temporal evolution, and the $y$ axis gives the position inside the channel. The blue part represents the “solid” behavior of the fluid, while the white part represents the liquid part.

Figure 28

Phase lag between force and flow rate response for each pair of ($\alpha$, $\text{Bn/Re}$) and $n=0.7$.

Figure 29

(a) Flow rate over one period for $n=0.7$. (b) Energy of harmonic frequency modes for various pairs ($\alpha ,\text{Bn/Re}$) and $n=0.7$.

Figure 30

Space-time diagrams of the spectral energy on the velocity for each pair of $\alpha ,\text{Bn/Re}$ for $n=0.7$. The $x$ axis corresponds to the dimensionless frequency, and the $y$ axis is the position inside the channel.