Capsules Rheology in Carreau-Yasuda Fluids

Abstract

In this paper, a Multi Relaxation Time Lattice Boltzmann scheme is used to describe the evolution of a non-Newtonian fluid. Such method is coupled with an Immersed-Boundary technique for the transport of arbitrarily shaped objects navigating the flow. The no-slip boundary conditions on immersed bodies are imposed through a convenient forcing term accounting for the hydrodynamic force generated by the presence of immersed geometries added to momentum equation. Moreover, such forcing term accounts also for the force induced by the shear-dependent viscosity model characterizing the non-Newtonian behavior of the considered fluid. Firstly, the present model is validated against well-known benchmarks, namely the parabolic velocity profile obtained for the flow within two infinite laminae for five values of the viscosity model exponent, n = 0.25, 0.50, 0.75, 1.0, and 1.5. Then, the flow within a squared lid-driven cavity for Re = 1000 and 5000 (being Re the Reynolds number) is computed as a function of n for a shear-thinning (n < 1) fluid. Indeed, the local decrements in the viscosity field achieved in high-shear zones implies the increment in the local Reynolds number, thus moving the position of near-walls minima towards lateral walls. Moreover, the revolution under shear of neutrally buoyant plain elliptical capsules with different Aspect Ratio (AR = 2 and 3) is analyzed for shear-thinning (n < 1), Newtonian (n = 1), and shear-thickening (n > 1) surrounding fluids. Interestingly, the power law by Huang et al. describing the revolution period of such capsules as a function of the Reynolds number and the existence of a critical value, Rec, after which the tumbling is inhibited in confirmed also for non-Newtonian fluids. Analogously, the equilibrium lateral position yeq of such neutrally buoyant capsules when transported in a plane-Couette flow is studied detailing the variation of yeq as a function of the Reynolds number as well as of the exponent n.

Keywords: dynamic forcing IBM; immersed boundary method (IBM); moving least squares; multi relaxation time (MRT); non-Newtonian rheology; particle margination.

Figure 1

Apparent viscosity as a function of the shear rate for different model parameters. (a) Distribution of the viscosity for a = 1, 2, 3, 4, 5; n = 0.25; ${\nu }_{\infty }/{\nu }_0$ = 0; and $\lambda =0.1$. (b) Distribution of the viscosity for a = 2; n = 0.25, 0.50, 0.75, 1.0, 1.25, 1.5, 1.75, and 2.0; ${\nu }_{\infty }/{\nu }_0$ = 0; and $\lambda =0.1$. (c) Distribution of the viscosity for a = 2; n = 0.25; ${\nu }_{\infty }/{\nu }_0$ = 0, 0.1, 0.2, 0.3, 0.4, and 0.5; and $\lambda =0.1$. (d) Distribution of the viscosity for a = 2; n = 0.25; ${\nu }_{\infty }/{\nu }_0$ = 0; and $\lambda ={10}^{-6},{10}^{-3},{10}^{-2},{10}^{-1}$, and 1.0.

Figure 2

Carreau–Yasuda flow within two parallel laminae at Re = 200. (a) Schematic of the physical problem. (b) distribution of the normal velocity component for different values of n taken at $x=0.5$L; dots represent numerical predictions while lines are for the analytical solutions; (c) mesh refinement study on $u_x\left(y\right)$ profiles obtained with n = 0.75.

Figure 3

Carreau–Yasuda fluid in a square lid-driven cavity. Distribution of the x (left plot) and y (right plot) components of the velocity field taken at x = 0.5 L (left plot) and y = 0.5 L (right plot) for n = 0.2, 0.4, 0.6, 0.8, and 1.0 at Re = 1000 (a) and Re = 5000 (b). Close-ups emphasize the near-wall velocity profiles implying the central vortex displacements obtained lowering n. Dots represent numerical predictions obtained by Napolitano and Pascazio [48] while lines are for present model solutions.

Figure 4

Rigid elliptical particles rotating under shear in a Carreau–Yasuda fluid. (a) schematic of the physical problem with characteristics dimensions and length. (b,c) distribution of the angular velocity for an elliptical particle with aspect ratio 2 for n = 0.5, 1, and 1.5 obtained for Re = 100 (b) and 500 (c).

Figure 5

Contour plot of conserved thermodynamical quantities at Re = 500. Contour plot of the apparent viscosity (a), the y-component of the velocity (b) and out-of-plane vorticity (c) obtained for n = 0.5, 1, and 1.5 at Re = 500.

Figure 6

Revolution period of elliptical particles in a Carreau–Yasuda fluid. Revolution period as a function of the Reynolds number for elliptical particles with aspect ratio 2 (a) and 3 (b) rotating in a a Carreau–Yasuda fluid with n = 0.5, 1.0, and 1.5. Dots are for present model solutions while solid lines represent analytical predictions.

Figure 7

Particles margination in a non-Newtonian Couette flow. (a) schematic of the physical problem with reference lengths; (b) trajectory of a particle navigating a linear laminar flow for Re = 25, 50, and 75 as a function of n; (c) distribution of the angular velocity over the horizontal coordinate for Re = 25, 50, and 75 and n = 1; (d) trajectory of a particle navigating a linear laminar flow for Re = 100, 150, and 200 as a function of n; (e) distribution of the angular velocity over the horizontal coordinate for Re = 100, 150, and 200 and n = 0.2; (f) equilibrium lateral positions as a function of the Reynolds number for n = 0.2, 0.4, 0.6, 0.8, and 1.