Numerical investigation of non-Newtonian fluids in annular ducts with finite aspect ratio using lattice Boltzmann method

In this work the instability of the Taylor-Couette flow for Newtonian and non-Newtonian fluids (dilatant and pseudoplastic fluids) is investigated for cases of finite aspect ratios. The study is conducted numerically using the lattice Boltzmann method (LBM). In many industrial applications, the apparatuses and installations drift away from the idealized case of an annulus of infinite length, and thus the end caps effect can no longer be ignored. The inner cylinder is rotating while the outer one and the end walls are maintained at rest. The lattice two-dimensional nine-velocity (D2Q9) Boltzmann model developed from the Bhatnagar-Gross-Krook approximation is used to obtain the flow field for fluids obeying the power-law model. The combined effects of the Reynolds number, the radius ratio, and the power-law index $n$ on the flow characteristics are analyzed for an annular space of finite aspect ratio. Two flow modes are obtained: a primary Couette flow (CF) mode and a secondary Taylor vortex flow (TVF) mode. The flow structures so obtained are different from one mode to another. The critical Reynolds number Re${}_c$ for the passage from the primary to the secondary mode exhibits the lowest value for the pseudoplastic fluids and the highest value for the dilatant fluids. The findings are useful for studies of the swirling flow of non-Newtonians fluids in axisymmetric geometries using LBM. The flow changes from the CF to TVF and its structure switches from the two-cells to four-cells regime for both Newtonian and dilatant fluids. Contrariwise for pseudoplastic fluids, the flow exhibits 2-4-2 structure passing from two-cells to four cells and switches again to the two-cells configuration. Furthermore, the critical Reynolds number presents a monotonic increase with the power-law index $n$ of the non-Newtonian fluid, and as the radius ratio grows, the transition flow regimes tend to appear for higher critical Reynolds numbers.

Figure 1

Comparison of the azimuthal velocity with analytical solution for η $=$ 0.5, Γ $=$ 3.8, and $n=1$.

Figure 2

Evolution of the friction factor against the Reynolds number for Γ $=$ 14.3 and η $=$ 0.5.

Figure 3

Comparison of streamlines (left), pressure (middle), and azimuthal vorticity (right) fields against results of Huang et al. [21] for $n=1$, Re $=$ 150, η $=$ 0.5, Γ $=$ 3.8.

Figure 4

Variations of the azimuthal velocity profiles for different fluids types for η $=$ 0.5 and Γ $=$ 3.8.

Figure 5

Evolution of the friction factor versus the radius ratio η for different $n$, Re $=$ 100, η $=$ 0.5, and Γ $=$ 3.8).

Figure 6

Contours of streamlines (left) and vorticity (right) fields for $n=1$, Γ $=$ 3.8, and η $=$ 0.5. (a) Re $=$ 40, (b) Re $=$ 60, (c) Re $=$ 65, (d) Re $=$ 68.4, (e) Re $=$ 85, and (f) Re $=$ 100.

Figure 7

Velocity profiles variations for $n=1$ and different Re. (a) The modulus velocity along the vertical direction at $r=d/2$, (b) radial velocity along the radial direction at $z$/$d=1.9$, (c) axial velocity along the radial direction at $r=d/2$, and (d) radial velocity along the axial direction at $r=d/2$.

Figure 8

Velocity profiles for different values of $n$ and Re $=$ 100. (a) Radial velocity along $z$/$d$, and (b) axial velocity along $z$/$d$.

Figure 9

Flow patterns streamlines (left), azimuthal vorticity (middle), and azimuthal velocity (right) for the pseudoplastic fluids ($n=0.5$).

Figure 10

Flow patterns streamlines (left), azimuthal vorticity (middle), and azimuthal velocity (right) for the dilatant fluids ($n=1.5$).

Figure 11

Evolution of the friction factor versus Re for η $=$ 0.5 and Γ $=$ 3.8. (a) $n=0.5$, (b) $n=1$, and (c) $n=1.5$.

Figure 12

Critical transition Reynolds number against $n$ for different η.

Figure 13

Map of flow structures for power-law fluids for η $=$ 0.5 and Γ $=$ 3.8.

Figure 14

Critical steady Reynolds number Re${}_c$ versus aspect ratio Γ for the transition between two- and four-cell flows for η $=$ 0.5.

Figure 15

(a), (b) Computed contours of the stream function lines of Newtonian fluid for different Γ at large radius ratio (η $=$ 0.5).

Figure 16

(a)–(c) Computed contours of the stream function lines (left) and the azimuthal velocity (right) of Newtonian fluid for different Γ at large radius ratio (η $=$ 0.5).