3.1. Effect of Carreau number, viscosity ratio and power-law index

We begin by examining the influence of shear thinning on the lateral migration of a neutrally buoyant spherical particle in plane Couette flow, with a fixed confinement ratio of

kappa equals 0.2

$\kappa = 0.2$

(which is maintained throughout this section unless otherwise specified). The extent of shear thinning is characterised by the Carreau number (

upper C u

$Cu$

), which represents the relative importance of the shear rate to the intrinsic time scale of the fluid; larger values of

upper C u

$Cu$

correspond to stronger shear-thinning effects. The viscosity ratio (

beta

$\beta$

) quantifies the contrast between the infinite-shear and zero-shear viscosities, thereby controlling the strength of viscosity variation between regions of high and low shear rate. The power-law index (

n

$n$

) governs the degree of shear thinning in the intermediate shear-rate regime, with smaller values of

n

$n$

corresponding to more pronounced shear-thinning behaviour. We first examine the effect of the Carreau number by analysing the bifurcation behaviour of the equilibrium particle position. Figure 3(a) presents the regime diagram in the

left parenthesis italic Re Subscript c Baseline comma upper C u right parenthesis

$({\textit{Re}}_c, Cu)$

parameter space for

beta equals 0.1

$\beta = 0.1$

and

n equals 0.25

$n = 0.25$

. The solid demarcating line represents the critical condition separating two distinct migration regimes. For a fixed Carreau number, this boundary corresponds to a critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

; conversely, for a fixed Reynolds number, it defines a critical Carreau number

upper C u Subscript c r

$Cu_{{cr}}$

. Below this boundary (i.e.

italic Re Subscript c Baseline less than italic Re Subscript c r

${\textit{Re}}_c \lt {\textit{Re}}_{{cr}}$

for fixed

upper C u

$Cu$

, or

upper C u less than upper C u Subscript c r

$Cu \lt Cu_{{cr}}$

for fixed

italic Re Subscript c

${\textit{Re}}_c$

), the particle ultimately migrates to the channel centreline irrespective of its initial release position. In contrast, beyond the critical condition (

italic Re Subscript c Baseline greater than italic Re Subscript c r

${\textit{Re}}_c \gt {\textit{Re}}_{{cr}}$

or

upper C u greater than upper C u Subscript c r

$Cu \gt Cu_{{cr}}$

), the centreline becomes unstable and the particle migrates towards an off-centre equilibrium position. The final equilibrium location depends on the initial release position. If the particle is released between the centreline and the lower half of the channel, it migrates to a stable off-centre position within that region. Similarly, if released between the centreline and the upper wall, it approaches a symmetric off-centre equilibrium in the upper half of the channel. Thus, the bifurcation is supercritical in nature: once the critical threshold (

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

for fixed

upper C u

$Cu$

or

upper C u Subscript c r

$Cu_{{cr}}$

for fixed

italic Re Subscript c

${\textit{Re}}_c$

) is exceeded, the centreline loses stability and symmetric off-centre equilibria emerge. Further insight into the nature of the bifurcation is obtained by examining the dependence of the equilibrium position on the Carreau number. Figure 3(b) shows the equilibrium position

upper Z Subscript italic eq Baseline divided by upper H

$Z_{{\textit{eq}}}/H$

as a function of

upper C u

$Cu$

for a fixed channel Reynolds number

italic Re Subscript c Baseline equals 100

${\textit{Re}}_c = 100$

, with

beta equals 0.1

$\beta = 0.1$

and

n equals 0.25

$n = 0.25$

. This analysis is aimed at characterising the scaling behaviour of particle migration in a shear-thinning fluid near the onset of instability. For sufficiently small

upper C u

$Cu$

, the steady equilibrium remains at the channel centreline (

upper Z Subscript italic eq Baseline divided by upper H equals 0.5

$Z_{{\textit{eq}}}/H = 0.5$

), indicating stability of the symmetric state. Beyond a threshold value of

upper C u

$Cu$

, the centreline loses stability and the particle migrates to an off-centre equilibrium position (

upper Z Subscript italic eq Baseline divided by upper H not equals 0.5

$Z_{{\textit{eq}}}/H \neq 0.5$

), with the direction determined by the initial release location. Furthermore, increasing

upper C u

$Cu$

drives the particle progressively farther from the centreline and closer to the wall, reflecting the strengthening influence of shear-thinning effects. The value of

upper C u

$Cu$

at which the transition from centreline to off-centre equilibrium occurs is defined as the critical Carreau number,

upper C u Subscript c r

$Cu_{{cr}}$

. For the present parameter set, the transition is observed between

upper C u equals 1

$Cu = 1$

and

upper C u equals 2

$Cu = 2$

, yielding an estimate

upper C u Subscript c r Baseline almost equals 1.2

$Cu_{{cr}} \approx 1.2$

. This estimate is obtained from simulations performed with refined parameter sampling in this interval (using increments of Carreau number

normal upper Delta upper C u equals 0.1

$\Delta Cu = 0.1$

) near the bifurcation point. In general,

upper C u Subscript c r

$Cu_{{cr}}$

depends on

italic Re Subscript c

${\textit{Re}}_c$

,

n

$n$

,

beta

$\beta$

and

kappa

$\kappa$

; for example, figure 3(a) shows that

upper C u Subscript c r

$Cu_{{cr}}$

varies with the channel Reynolds number

italic Re Subscript c

${\textit{Re}}_c$

. In the present discussion, however, we restrict attention to this representative case. For

upper C u less than upper C u Subscript c r

$Cu \lt Cu_{{cr}}$

, the particle migrates to the centreline irrespective of its initial position, whereas for

upper C u greater than upper C u Subscript c r

$Cu \gt Cu_{{cr}}$

, symmetric off-centre equilibria emerge and the final state depends on the initial condition. The inset of figure 3(b) highlights the near-critical behaviour using the reduced control parameter

delta equals left parenthesis upper C u minus upper C u Subscript c r Baseline right parenthesis divided by upper C u Subscript c r

$\delta = (Cu – Cu_{{cr}})/Cu_{{cr}}$

, with the vertical axis plotted as

upper Z Subscript italic eq Baseline divided by upper H minus 0.5

$Z_{{\textit{eq}}}/H – 0.5$

. The observed square-root variation of

delta

$\delta$

with

upper Z Subscript italic eq Baseline divided by upper H minus 0.5

$Z_{{\textit{eq}}}/H – 0.5$

near onset is consistent with the canonical scaling of a supercritical pitchfork bifurcation. Additional insight is obtained by examining the variation of the equilibrium position with the channel Reynolds number. Figure 3(c) shows the steady-state equilibrium position

upper Z Subscript italic eq Baseline divided by upper H

$Z_{{\textit{eq}}}/H$

as a function of

italic Re Subscript c

${\textit{Re}}_c$

for several values of

upper C u

$Cu$

, with

n equals 0.25

$n = 0.25$

and

beta equals 0.1

$\beta = 0.1$

. For each fixed

upper C u

$Cu$

, the particle remains at the channel centreline (

upper Z Subscript italic eq Baseline divided by upper H equals 0.5

$Z_{{\textit{eq}}}/H = 0.5$

) up to a critical Reynolds number, beyond which it migrates to an off-centre equilibrium position, thereby forming a bifurcation diagram in

italic Re Subscript c

${\textit{Re}}_c$

. The Reynolds number at which this transition occurs is defined as the critical Reynolds number,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

. For example, in the Newtonian limit (

upper C u equals 0

$Cu = 0$

) and for the present confinement ratio

kappa equals 0.2

$\kappa = 0.2$

, the critical value is approximately

italic Re Subscript c r Baseline almost equals 153

${\textit{Re}}_{{cr}} \approx 153$

. For

italic Re Subscript c Baseline greater than italic Re Subscript c r

${\textit{Re}}_c \gt {\textit{Re}}_{{cr}}$

, the particle migrates away from the centreline and settles at an off-centre equilibrium position (

upper Z Subscript italic eq Baseline divided by upper H not equals 0.5

$Z_{{\textit{eq}}}/H \neq 0.5$

). As

upper C u

$Cu$

increases, this critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

decreases monotonically, indicating that shear-thinning effects reduce the inertial threshold required to destabilise the centreline equilibrium. Consistent with the trends observed earlier, for a fixed

italic Re Subscript c

${\textit{Re}}_c$

, larger values of

upper C u

$Cu$

also shift the off-centre equilibrium progressively closer to the channel wall.

Figure 3.

(a) Regime diagram in the

left parenthesis italic Re Subscript c Baseline comma upper C u right parenthesis

$({\textit{Re}}_c,\,Cu)$

parameter space for

kappa equals 0.2

$\kappa = 0.2$

,

beta equals 0.1

$\beta = 0.1$

and

n equals 0.25

$n = 0.25$

. The solid line represents the critical Reynolds number,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

, separating two distinct migration regimes. For

italic Re Subscript c

${\textit{Re}}_c$

values to the right of the boundary, the particle migrates to an off-centre equilibrium position, whereas for

italic Re Subscript c

${\textit{Re}}_c$

values to the left of the boundary, the particle returns to the channel centreline. (b) Normalised equilibrium position of the particle,

upper Z Subscript italic eq Baseline divided by upper H

$Z_{{\textit{eq}}}/H$

, as a function of the Carreau number,

upper C u

$Cu$

. The results exhibit a supercritical pitchfork bifurcation, in which the particle migrates from the channel centreline to off-centre equilibrium positions as

upper C u

$Cu$

exceeds the critical value,

upper C u Subscript c r

$Cu_{{cr}}$

. Note that this

upper C u Subscript c r

$Cu_{{cr}}$

is a function of

italic Re Subscript c

${\textit{Re}}_c$

. Here, the bifurcation shown corresponds to

italic Re Subscript c Baseline equals 100

${\textit{Re}}_c = 100$

,

kappa equals 0.2

$\kappa = 0.2$

,

beta equals 0.1

$\beta = 0.1$

and

n equals 0.25

$n = 0.25$

. Inset: variation of

upper Z Subscript italic eq Baseline divided by upper H minus 0.5

$Z_{{\textit{eq}}}/H – 0.5$

with

delta equals left parenthesis upper C u minus upper C u Subscript c r Baseline right parenthesis divided by upper C u Subscript c r

$\delta = (Cu – Cu_{{cr}})/Cu_{{cr}}$

near the bifurcation point, showing square-root scaling close to onset. (c) Normalised equilibrium position across the channel as a function of the channel Reynolds number

italic Re Subscript c

${\textit{Re}}_c$

for different Carreau numbers, with

beta equals 0.1

$\beta = 0.1$

,

kappa equals 0.2

$\kappa = 0.2$

and

n equals 0.25

$n = 0.25$

.

Three graphs depict particle migration in non-Newtonian fluids.

Two line graphs depict the variation of critical Reynolds number with Carreau number for different parameters.

Having established the influence of the Carreau number on the critical Reynolds number, we now examine how the remaining rheological parameters such as viscosity ratio

beta

$\beta$

and power-law index

n

$n$

affect the onset of lateral migration. The dependence of

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

on

upper C u

$Cu$

for different viscosity ratios is shown in figure 4(a) for

beta equals 0.9

$\beta = 0.9$

,

0.5

$0.5$

and

0.1

$0.1$

, with

n equals 0.25

$n = 0.25$

. In each case,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

is determined from simulations performed with unit increments of

italic Re Subscript c

${\textit{Re}}_c$

near the transition point. For all viscosity ratios,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

decreases monotonically with increasing

upper C u

$Cu$

. Moreover, the reduction becomes more pronounced as

beta

$\beta$

decreases, corresponding to a larger viscosity contrast between low- and high-shear regions and therefore stronger shear-thinning effects. We next consider the role of the power-law index

n

$n$

. Figure 4(b) shows the variation of

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

with

upper C u

$Cu$

for

n equals 0.25

$n = 0.25$

,

0.5

$0.5$

and

0.75

$0.75$

, at a fixed viscosity ratio

beta equals 0.5

$\beta = 0.5$

. The results indicate that

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

decreases as

n

$n$

decreases, with the sensitivity becoming increasingly significant at lower values of

n

$n$

. Since smaller

n

$n$

corresponds to stronger shear-thinning behaviour, this trend is consistent with the variations observed with

upper C u

$Cu$

and

beta

$\beta$

. Taken together, these results demonstrate that enhanced shear thinning, whether achieved by increasing

upper C u

$Cu$

, decreasing

beta

$\beta$

or reducing

n

$n$

, lowers the critical Reynolds number required to destabilise the centreline equilibrium and promotes particle migration at weaker inertial forcing.

Three line graphs depict the variation of dimensionless lift force on a neutrally buoyant sphere in confined Couette flow.

To further understand the bifurcation behaviour described above, we examine the stability of both the centreline and off-centre equilibrium positions by analysing the lateral force acting on the particle. Specifically, we compute the non-dimensional lift force as a function of the particle’s lateral position across the channel height, as shown in figure 5(a–c). The lift force is evaluated by placing the particle at various

z

$z$

-locations while allowing free translation and rotation in all directions except

z

$z$

, where translation is constrained. The resulting lift force is normalised using

upper F Subscript s Baseline equals rho upper V Subscript w Superscript 2 Baseline a Superscript 4 Baseline divided by upper H squared

$F_s = \rho V_w^2 a^4 / H^2$

(Ho & Leal Reference Ho and Leal1974), allowing direct comparison across different flow conditions and rheological parameters. Figure 5 presents the lift-force profiles for variations in the Carreau number (

upper C u

$Cu$

), viscosity ratio (

beta

$\beta$

) and power-law index (

n

$n$

), with all other parameters held constant. Figure 5(a) illustrates the effect of

upper C u

$Cu$

for

n equals 0.25

$n = 0.25$

,

beta equals 0.1

$\beta = 0.1$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

. For a Newtonian fluid (

upper C u equals 0

$Cu = 0$

), the lift-force profile exhibits a single zero crossing at the channel centreline, with a negative slope, indicating that the centreline is a stable equilibrium. As

upper C u

$Cu$

increases, reflecting stronger shear-thinning effects, two additional off-centre zero crossings emerge. These off-centre crossings have negative slopes, identifying them as stable equilibrium positions, while the slope at the centreline becomes positive, signalling the loss of its stability. This transition indicates that stronger shear thinning promotes off-centre migration, even when the particle initially resides at the centreline. Figure 5(b) explores the role of the viscosity ratio

beta

$\beta$

for

upper C u equals 10

$Cu = 10$

,

n equals 0.25

$n = 0.25$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

. For higher

beta

$\beta$

values (closer to unity), the lift-force profiles resemble those of Newtonian fluids, with a single stable equilibrium at the centreline. Reducing

beta

$\beta$

, which increases the contrast between zero and infinite shear viscosities, destabilises the centreline and promotes the appearance of off-centre stable equilibria. This demonstrates that the distribution of shear-dependent viscosity across the channel can significantly influence particle migration, with stronger viscosity contrasts favouring lateral displacement from the centre. Figure 5(c) highlights the influence of the power-law index

n

$n$

for

upper C u equals 10

$Cu = 10$

,

beta equals 0.5

$\beta = 0.5$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

. As

n

$n$

decreases, corresponding to stronger shear-thinning behaviour, the centreline equilibrium is progressively destabilised, and stable off-centre equilibria emerge. This shows that it is not only the magnitude of shear thinning (via

upper C u

$Cu$

), but that also the intrinsic rheological character of the fluid (via

beta

$\beta$

and

n

$n$

) critically governs the equilibrium positions of the particle. Collectively, these results reveal a consistent and unified trend: increasing the intensity of shear thinning, whether by increasing

upper C u

$Cu$

, decreasing

beta

$\beta$

or reducing

n

$n$

, leads to an earlier onset of lateral migration and the formation of stable off-centre equilibrium positions. By systematically varying one parameter at a time, the lift-force profiles provide a clear and quantitative picture of how shear thinning modulates particle stability, highlighting the intricate interplay between fluid rheology and inertial effects in confined flows. To place these findings in context, it is useful to compare them with Newtonian fluids. Previous studies have shown that the lift force decreases with increasing Reynolds number, a phenomenon attributed to inertial screening (Fox, Schneider & Khair Reference Fox, Schneider and Khair2021). As Reynolds number increases, the disturbance flow induced by the particle becomes increasingly confined to its vicinity, thereby reducing hydrodynamic interactions with the channel walls and leading to a weaker lift force. In shear-thinning fluids at fixed

italic Re Subscript c

${\textit{Re}}_c$

, we observe a qualitatively similar reduction in lift force on increasing

upper C u

$Cu$

, decreasing

beta

$\beta$

or reducing

n

$n$

(see figure 5
a–c), although the underlying mechanism is different. Here, the shear-thinning rheology reduces the viscosity in regions of high shear rate near the particle surface, thereby diminishing local viscous stresses. This rheological effect confines the disturbance flow more closely to the particle, effectively screening particle–wall interactions which in turn reduces the overall lift force. These insights emphasise that even subtle changes in the fluid’s shear-rate-dependent properties can markedly alter the behaviour of suspended particles, offering a predictive framework for understanding particle migration in complex fluids.

Heat maps showing dimensionless viscosity contours in plane Couette flow for different Carreau numbers.

Next, to examine the reduction in the critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

with increasing

upper C u

$Cu$

and decreasing

beta

$\beta$

and

n

$n$

, kinematic viscosity

nu left parenthesis x comma z right parenthesis

$\nu (x,z)$

contours are plotted, where the viscosity is normalised by the zero-shear-rate viscosity

nu 0

$\nu _{0}$

. Figure 6(a–d) presents steady-state viscosity contours for increasing Carreau numbers,

upper C u equals 0.1 comma 1 comma 10

$Cu = 0.1,\, 1,\, 10$

and

100

$100$

, at fixed

beta equals 0.1

$\beta = 0.1$

,

n equals 0.25

$n = 0.25$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

. These contours illustrate how varying

upper C u

$Cu$

affects the local viscosity field around the particle and, consequently, its equilibrium. At low Carreau number (

upper C u equals 0.1

$Cu = 0.1$

, figure 6
a), the fluid exhibits very weak shear thinning, resulting in a nearly uniform viscosity field across the channel. In this regime, viscosity is only slightly reduced in the vicinity of the particle due to localised shear. Consequently, the particle remains near the channel centreline, as the weak and nearly symmetric viscosity gradients generate negligible lift asymmetry. As

upper C u

$Cu$

increases (figure 6
b–d), the shear-thinning response becomes progressively stronger, leading to a pronounced reduction in viscosity near the particle surface, where local velocity gradients are largest. This reduction in viscosity locally around the particle modifies the local stress distribution which eventually destabilises the centreline equilibrium, causing the particle to migrate towards an off-centre equilibrium position, the exact location of which depends on the initial particle position. The destabilisation of the centreline can be rationalised through the Carreau constitutive relation: larger local velocity gradients induce a stronger reduction in viscosity near the particle surface, which decreases the local viscous force. This reduction effectively makes local inertial force stronger than the viscous force, thereby accelerating the onset of inertial migration away from the centreline. At low

upper C u

$Cu$

, where the fluid behaviour is nearly Newtonian, the centreline remains a stable equilibrium. However, as

upper C u

$Cu$

increases, the enhanced shear thinning generates progressively stronger viscosity gradients, which in turn promote off-centre migration. This mechanism is reflected quantitatively in the critical Reynolds number,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

, for lateral migration. For instance, at

beta equals 0.9

$\beta = 0.9$

(figure 4
a), the Newtonian limit (

upper C u right arrow 0

$Cu \to 0$

) yields

italic Re Subscript c r Baseline almost equals 153

${\textit{Re}}_{{cr}} \approx 153$

, whereas at

upper C u equals 10

$Cu = 10$

,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

decreases markedly to

italic Re Subscript c r Baseline almost equals 96

${\textit{Re}}_{{cr}}\approx 96$

. This reduction demonstrates that local shear thinning lowers the effective viscous resistance, facilitating an earlier onset of off-centre equilibrium positions as

upper C u

$Cu$

increases. Overall, the viscosity contours provide a clear visualisation of how shear thinning modulates the local stress field around the particle, thereby governing its migration behaviour in confined flows.

Heat map showing Reynolds number contours around a particle in a fluid flow.

To rationalise this migration using the local Reynolds number distribution, we quantify the spatial variation of

italic Re Subscript script l Baseline left parenthesis bold italic x right parenthesis

${\textit{Re}}_{\ell }(\boldsymbol{x})$

, following Lashgari et al. (Reference Lashgari, Picano, Breugem and Brandt2012) and Patel, Rothstein & Modarres-Sadeghi (Reference Patel, Rothstein and Modarres-Sadeghi2022), in the mid-plane (i.e.

x z

$xz$

-plane) passing through the particle centre. The local Reynolds number is defined as

italic Re Subscript script l Baseline left parenthesis bold italic x right parenthesis equals upper V Subscript w Baseline upper H divided by nu left parenthesis x comma z right parenthesis

${\textit{Re}}_{\ell }(\boldsymbol{x}) = V_w H / \nu (x,z)$

, which provides a local measure of the balance between inertial and viscous stresses and can therefore be interpreted as an effective local Reynolds number. Contours of

italic Re Subscript script l Baseline left parenthesis bold italic x right parenthesis

${\textit{Re}}_{\ell }(\boldsymbol{x})$

at steady state for

upper C u equals 0.1

$Cu = 0.1$

,

1

$1$

,

10

$10$

and

100

$100$

are shown in figure 7(a–d) for

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

,

beta equals 0.1

$\beta = 0.1$

and

n equals 0.25

$n = 0.25$

. As the Carreau number increases from

upper C u equals 0.1

$Cu = 0.1$

to

100

$100$

,

italic Re Subscript script l

${\textit{Re}}_{\ell }$

increases both around the particle surface and in the bulk region far away from the particle. Away from the particle, the distribution of

italic Re Subscript script l

${\textit{Re}}_{\ell }$

is nearly uniform. This uniform far-field value corresponds to the Reynolds number based on the effective base viscosity associated with the imposed background shear rate according to the Carreau relation. The increase in the far-field

italic Re Subscript script l

${\textit{Re}}_{\ell }$

with increasing

upper C u

$Cu$

is due to the reduction in the base viscosity for a fixed background shear rate in the absence of the particle, as predicted by the Carreau rheology. This behaviour is also evident in the viscosity contour plots shown in figure 6(a–d). Closer to the particle surface, however, the distribution of

italic Re Subscript script l Baseline left parenthesis bold italic x right parenthesis

${\textit{Re}}_{\ell }(\boldsymbol{x})$

shows a significant local increase compared with its far-field value for each Carreau number. This local amplification arises from the enhanced shear rates generated around the particle, which further reduce the viscosity in shear-thinning fluids. Consequently, the local viscous stresses decrease, leading to a locally more inertia-dominated region in the vicinity of the particle. Although the global channel Reynolds number

italic Re Subscript c

${\textit{Re}}_c$

is fixed, the particle experiences a locally higher effective Reynolds number over a finite region surrounding it. In comparison with the Newtonian case at the same

italic Re Subscript c

${\textit{Re}}_c$

, the shear-thinning-induced reduction in viscous stress causes the particle to effectively experience a higher Reynolds number flow in its immediate neighbourhood. As a result, the equilibrium position shifts away from the centreline. For example, configurations that remain centred in the Newtonian limit shift to an off-centre equilibrium position at finite

upper C u

$Cu$

, as shown in figure 4(a). Specifically, while the Newtonian case (

upper C u right arrow 0

$Cu \to 0$

) yields

italic Re Subscript c r Baseline almost equals 153

${\textit{Re}}_{{cr}} \approx 153$

, increasing the Carreau number to

upper C u equals 10

$Cu = 10$

reduces the critical Reynolds number to

italic Re Subscript c r Baseline almost equals 96

${\textit{Re}}_{{cr}} \approx 96$

, consistent with the enhanced local inertial effects discussed above. Overall these results indicates that even in the absence of flow curvature, shear thinning alone can generate localised inertia-dominated regions around the particle, destabilising the centreline equilibrium and causing off-centre migration.

Classical studies by Ho & Leal (Reference Ho and Leal1974) and Schonberg & Hinch (Reference Schonberg and Hinch1989) examined inertial migration of particles in shear and Poiseuille flows at small but finite Reynolds numbers using matched asymptotic expansions. In their framework, the flow is divided into distinct regions, one in the immediate vicinity of the particle (called the Stokes region) where viscous and pressure stresses dominate and the Stokes equations are valid, and the other far from the particle (called the Oseen region) where inertial stresses dominate. Consistency of the asymptotic analysis requires an intermediate region in which viscous and inertial stresses are comparable, which enables matching of the inner and outer solutions. In the absence of such a region, the matched asymptotic expansion approach breaks down. Further, Hood et al. (Reference Hood, Lee and Roper2015) revisited this problem numerically and found that, even at moderate channel Reynolds numbers (

italic Re Subscript c Baseline equals 10

${\textit{Re}}_c=10$

), viscous and pressure stresses exceed inertial contributions at all radial distances from the particle, leaving no region of co-dominance and contradicting the predictions of Ho & Leal (Reference Ho and Leal1974) and Schonberg & Hinch (Reference Schonberg and Hinch1989). At higher channel Reynolds numbers (

italic Re Subscript c Baseline equals 50

${\textit{Re}}_c=50$

and

80

$80$

), inertial stresses increase but scale linearly with

italic Re Subscript c

${\textit{Re}}_c$

, acting as a passive correction rather than establishing a new dominant balance. Further, in Newtonian shear flows, particle migration at finite Reynolds numbers has been interpreted in terms of inertial screening (Ho & Leal Reference Ho and Leal1974; Fox et al. Reference Fox, Schneider and Khair2021; Anand & Subramanian Reference Anand and Subramanian2023). Fox et al. (Reference Fox, Schneider and Khair2021) also showed that increasing particle Reynolds number confines the particle-induced velocity disturbance to a thin region near the particle surface, thereby weakening long-range hydrodynamic interactions and particle–wall coupling. Existing studies (Ho & Leal Reference Ho and Leal1974; Anand & Subramanian Reference Anand and Subramanian2023) show that, for ambient shear flow, the characteristic inertial screening length scales as

script l Subscript s Baseline tilde a italic Re Subscript p Superscript negative 1 divided by 2

$\ell _s \sim a {\textit{Re}}_{\!p}^{-1/2}$

, or equivalently in channel scaling,

script l Subscript s Baseline tilde upper H italic Re Subscript c Superscript negative 1 divided by 2

$\ell _s \sim H {\textit{Re}}_c^{-1/2}$

. When

italic Re Subscript c Baseline much less than 1

${\textit{Re}}_c \ll 1$

,

script l Subscript s Baseline much greater than upper H

$\ell _s \gg H$

and the walls lie within the inner Stokes region, with inertia acting as a regular perturbation. When

italic Re Subscript c Baseline equals upper O left parenthesis 1 right parenthesis

${\textit{Re}}_c = O(1)$

or larger,

script l Subscript s Baseline less than or equivalent to upper H

$\ell _s \lesssim H$

and the walls lie in the Oseen region. In the present case, one further insight may be drawn. Since

italic Re Subscript c Baseline greater than upper O left parenthesis 1 right parenthesis

${\textit{Re}}_c \gt O(1)$

, it follows that

script l Subscript s Baseline less than or equivalent to upper H

$\ell _s \lesssim H$

, and the confining walls lie in the Oseen region. Consequently, inertial effects influence the global disturbance field rather than acting solely as a regular perturbation. Furthermore, owing to shear thinning, the viscosity decreases with increasing

upper C u

$Cu$

, leading to a reduction in the effective base viscosity at the imposed shear rate as per Carreau fluid rheology. As a result, even far from the particle, the effective local Reynolds number increases in correspondence with the channel Reynolds number

italic Re Subscript c

${\textit{Re}}_c$

. In addition, the shear rate is significantly elevated in the immediate vicinity of the particle, producing a further local reduction in viscosity and a pronounced enhancement of the local Reynolds number, as demonstrated by the spatial distributions of

italic Re Subscript script l

${\textit{Re}}_{\ell }$

shown in figure 7. Notably, this near-particle amplification of

italic Re Subscript script l

${\textit{Re}}_{\ell }$

exceeds its far-field value. Such a localised enhancement of inertia is consistent with a stronger spatial localisation of the particle-induced disturbance. Consequently, relative to a Newtonian fluid at the same nominal channel Reynolds number, the effective inertial screening length,

script l Subscript s

$\ell _s$

, is further reduced in the shear-thinning case. This reduction in the screening length, together with the associated attenuation of the disturbance field, is reflected in the variation of the non-dimensional lift force across the channel height, as shown in figure 5(a), obtained at a fixed channel Reynolds number

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

with

kappa

$\kappa$

,

n

$n$

and

beta

$\beta$

held constant. As

upper C u

$Cu$

increases from

0

$0$

(Newtonian) to

100

$100$

, the magnitude of the lift force decreases at all fixed positions

z divided by upper H

$z/H$

, leading to progressively flatter lift-force profiles at higher

upper C u

$Cu$

. This reduction in lift is consistent with enhanced inertial screening of the particle-induced velocity disturbance. At larger

upper C u

$Cu$

, the disturbance becomes increasingly confined to the immediate vicinity of the particle surface, thereby weakening long-range hydrodynamic interactions between the particle and the confining wall, particularly wall-mediated interactions responsible for lift are weakened and reducing the net lift force. A similar trend is observed upon decreasing

beta

$\beta$

(figure 5
b) or decreasing the power-law index

n

$n$

(figure 5
c), both of which intensify shear thinning. In each case, stronger shear thinning leads to a further reduction in the lift force, consistent with a greater spatial localisation of the particle-induced disturbance. Overall, our results show that, even in the absence of flow curvature, shear thinning alone is sufficient to destabilise the centreline equilibrium and trigger off-centre inertial migration. This transition can occur at relatively low channel Reynolds numbers, as shear thinning reduces the critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

compared with a Newtonian fluid. The mechanism arises from the formation of localised inertia-dominated regions around the particle, where reduced viscous stresses and increased inertial effects produce a significant local rise in

italic Re Subscript script l

${\textit{Re}}_{\ell }$

. The resulting migration behaviour differs fundamentally from that observed in Newtonian channel flows. Overall these findings highlight the important role of the shear-thinning rheology in modifying the local stress balance and in turn affecting the nature of inertial migration.

Figure 8.

Time-dependent trajectories of a particle released from different initial positions. (a) Effect of the Carreau number at

n equals 0.25

$n = 0.25$

,

kappa equals 0.2

$\kappa = 0.2$

,

beta equals 0.1

$\beta = 0.1$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

. (b) Effect of the viscosity ratio at

upper C u equals 10

$Cu = 10$

,

n equals 0.25

$n = 0.25$

,

kappa equals 0.2

$\kappa = 0.2$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

. (c) Effect of the power-law index at

upper C u equals 10

$Cu = 10$

,

beta equals 0.5

$\beta = 0.5$

,

kappa equals 0.2

$\kappa = 0.2$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

. (d) Influence of the Carreau number,

upper C u

$Cu$

, on the particle migration time. The evolution of the normalised particle position,

upper Z divided by upper H

$Z/H$

, with

upper G t

$Gt$

illustrates the time taken to attain the equilibrium position. Results correspond to

italic Re Subscript c Baseline equals 50

${\textit{Re}}_c = 50$

and

beta equals 0.5

$\beta = 0.5$

,

n equals 0.25

$n = 0.25$

for

kappa equals 0.2

$\kappa = 0.2$

.

Four line graphs showing the time-dependent trajectories of a particle under different conditions.

Figure 8(a–c) presents the lateral migration trajectories of particles released from two different initial positions,

upper Z 0 divided by upper H equals 0.4

$Z_0/H=0.4$

(below the centreline) and

upper Z 0 divided by upper H equals 0.6

$Z_0/H=0.6$

(above the centreline), within the channel. The particles are free to translate and rotate, and their trajectories are followed until they reach steady-state equilibrium positions in the

z

$z$

-direction. In figure 8(a), the Carreau number is varied while fixing the other parameters as

beta equals 0.1

$\beta =0.1$

,

n equals 0.25

$n=0.25$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c=150$

. For

upper C u equals 1

$Cu=1$

, the particle migrates to the centreline (

upper Z divided by upper H equals 0.5

$Z/H=0.5$

), independent of the initial position. As

upper C u

$Cu$

increases, however, two stable equilibria appear symmetrically about the channel centreline; a particle released from

upper Z 0 divided by upper H equals 0.4

$Z_0/H=0.4$

migrates to an off-centre position between the lower wall and the centreline, whereas a particle released from

upper Z 0 divided by upper H equals 0.6

$Z_0/H=0.6$

stabilises symmetrically between the centreline and the upper wall. This demonstrates that sufficiently strong non-Newtonian effects break the uniqueness of the centreline equilibrium, and the final state becomes sensitive to the initial release height. Figure 8(b) illustrates the role of the viscosity ratio

beta

$\beta$

for

upper C u equals 10

$Cu=10$

,

n equals 0.25

$n=0.25$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c=150$

. At larger values of

beta

$\beta$

, the centreline remains the sole equilibrium position, and trajectories are independent of initial lateral positions. Reducing

beta

$\beta$

, however, produces two off-centre equilibria symmetrically about the channel centreline, such that a particle released near the lower wall remains trapped below the centreline while one released near the upper wall stabilises above it. A similar trend is observed when varying the power-law index

n

$n$

(figure 8
c) at fixed

upper C u equals 10

$Cu=10$

,

beta equals 0.5

$\beta =0.5$

and

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c=150$

: for

n equals 1

$n=1$

(Newtonian case), particles invariably migrate to the centreline, but as

n

$n$

decreases, corresponding to stronger shear-thinning behaviour, the centreline equilibrium becomes unstable and stable off-centre equilibria emerge. To further understand the migration dynamics in comparison with the Newtonian case, we consider a parameter set for which the particle migrates to the centreline, namely

italic Re Subscript c Baseline equals 50

${\textit{Re}}_c = 50$

,

beta equals 0.5

$\beta = 0.5$

and

n equals 0.25

$n = 0.25$

, as shown in figure 8(d). It can be observed that as the Carreau number increases from

upper C u equals 0.1

$Cu = 0.1$

to

100

$100$

, the time required for the particle to reach the centreline decreases. In addition to that, we have observed that the streamwise distance travelled by the particle before reaching its equilibrium position is also reduced. This behaviour indicates that in a shear-thinning fluid the particle experiences a lower effective hydrodynamic force than its Newtonian counterpart. The reduction in viscosity with increasing shear rate decreases the overall viscous resistance acting on the particle, allowing it to migrate more rapidly and over a shorter streamwise distance toward the centreline. Taken together, these trajectories provide direct numerical confirmation of the multiple stable equilibrium positions predicted by the lift-force analysis shown in figure 5(a–c). The viscosity contours (figure 6) and the local Reynolds number distributions (figure 7) further clarify the underlying mechanism, demonstrating that shear thinning enhances local inertial effects through viscosity reduction in high-shear regions around the particle. As a consequence, the migration dynamics and equilibrium locations differ fundamentally from the Newtonian case, and particle migration is governed primarily by the coupled influence of fluid rheology and inertia. All the above results correspond to a confinement ratio

kappa equals 0.2

$\kappa = 0.2$

. It is therefore important to examine how these equilibrium states and migration characteristics are modified as the degree of confinement is varied. This aspect is addressed in the following section.

3.2. Effect of confinement ratio

To elucidate the role of geometric confinement on inertial migration, we examine the effect of the confinement ratio

kappa equals a divided by upper H

$\kappa = a/H$

, which quantifies the particle size relative to the channel height. In the present study,

kappa

$\kappa$

is varied between

0.05

$0.05$

and

0.2

$0.2$

, corresponding to weak confinement (

kappa equals 0.05 comma 0.1

$\kappa = 0.05, 0.1$

) and strong confinement (

kappa equals 0.2

$\kappa = 0.2$

). Figure 9(a) shows the normalised stable equilibrium position as a function of the Reynolds number

italic Re Subscript c

${\textit{Re}}_c$

for different confinement ratios at fixed rheological parameters

upper C u equals 10

$Cu = 10$

,

beta equals 0.5

$\beta = 0.5$

and

n equals 0.25

$n = 0.25$

. It is evident that the critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

increases with increasing confinement. In particular, for strong confinement (

kappa equals 0.2

$\kappa = 0.2$

), the onset of off-centre migration occurs at a significantly higher

italic Re Subscript c

${\textit{Re}}_c$

compared with the weakly confined cases (

kappa equals 0.05 comma 0.1

$\kappa = 0.05, 0.1$

), indicating a stabilising influence of the channel walls. The influence of shear-thinning at a fixed confinement is illustrated in figure 9(b), which compares equilibrium positions for two Carreau numbers at

kappa equals 0.1

$\kappa = 0.1$

, with

beta equals 0.5

$\beta = 0.5$

and

n equals 0.25

$n = 0.25$

. At low

italic Re Subscript c

${\textit{Re}}_c$

, the particle remains centred; however, beyond a critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

, a symmetry-breaking bifurcation occurs and stable off-centre equilibrium positions emerge. Importantly, increasing the Carreau number leads to a systematic reduction in

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

, consistent with the trends observed under stronger confinement. This demonstrates that shear thinning accelerates the onset of inertial migration irrespective of the confinement level. Figure 9(c) summarises these trends by showing the variation of the critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

with the Carreau number

upper C u

$Cu$

for confinement ratios

kappa equals 0.05 comma 0.1

$\kappa = 0.05, 0.1$

and

0.2

$0.2$

, at fixed rheological parameters

n equals 0.25

$n = 0.25$

and

beta equals 0.5

$\beta = 0.5$

. In the limit of very small

upper C u

$Cu$

, corresponding to a flow approaching the Newtonian regime,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

decreases with decreasing confinement ratio and approaches closer to

italic Re Subscript c r Baseline almost equals 148

${\textit{Re}}_{{cr}} \approx 148$

specifically when

kappa equals 0.05

$\kappa = 0.05$

. This value is in good agreement with the analytically obtained critical Reynolds number reported by Anand & Subramanian (Reference Anand and Subramanian2023) for a neutrally buoyant particle in a Newtonian fluid. As the Carreau number increases,

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

decreases markedly, indicating an enhanced role of shear thinning in promoting the instability. Furthermore, for a fixed value of

upper C u

$Cu$

, reducing the confinement ratio, equivalently increasing the channel height relative to the particle size, leads to a further reduction in

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

. Overall, the dependence of

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

on

kappa

$\kappa$

reflects a balance between wall-induced hydrodynamic interactions and viscosity-driven asymmetries. Under strong confinement, enhanced particle–wall interactions generate stabilising wall-induced lift forces that delay the onset of bifurcation. In contrast, weak confinement diminishes wall effects, allowing non-Newtonian modifications of the flow field to dominate the lateral force balance and trigger earlier symmetry breaking. It is further observed from figure 9(c) that, at lower Carreau numbers, corresponding to the near-Newtonian limit, the variation of the critical Reynolds number

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

with confinement ratio is relatively weak. However, as

upper C u

$Cu$

increases, a pronounced dependence of

italic Re Subscript c r

${\textit{Re}}_{{cr}}$

on confinement emerges, indicating that confinement effects become significantly amplified in the shear-thinning regime.

Three line graphs depict the relationship between normalized equilibrium position, critical Reynolds number, and Carreau number.

Two line graphs depict variations in normalized particle velocities with the Carreau number.

Next, we look at the translational and angular velocities of the particle at steady state. Figure 10(a, b) shows the magnitude of particle translational velocity

left parenthesis upper U equals StartAbsoluteValue bold italic upper U EndAbsoluteValue right parenthesis

$(U=|\boldsymbol{U}|)$

normalised by the wall velocity

upper V Subscript w

$V_w$

, and the magnitude of angular velocity

left parenthesis upper Omega equals StartAbsoluteValue bold italic upper Omega EndAbsoluteValue right parenthesis

$(\varOmega = |\boldsymbol{\varOmega }|)$

normalised by the imposed shear rate

upper G

$G$

, for varying Carreau numbers as a function of confinement ratio

kappa

$\kappa$

. The results correspond to

italic Re Subscript c Baseline equals 150

${\textit{Re}}_c = 150$

,

beta equals 0.1

$\beta = 0.1$

and

n equals 0.25

$n = 0.25$

. It is observed that both the translational and angular velocities decrease with increasing

upper C u

$Cu$

. Furthermore, as the confinement ratio

kappa

$\kappa$

decreases (i.e. for wider channels), the translational velocity decreases, whereas the rotational velocity increases. This behaviour is linked to the shift in the equilibrium position of the particle. For the present parameter set, all cases except

kappa equals 0.2

$\kappa = 0.2$

at

upper C u equals 0.1

$Cu = 0.1$

migrate toward off-centre equilibrium positions between the bottom wall and the centreline, since the particle is initially released in the lower half of the channel. As

upper C u

$Cu$

increases and

kappa

$\kappa$

decreases, the equilibrium position moves progressively closer to the wall. The reduced local streamwise fluid velocity near the wall leads to a decrease in translational speed, while the enhanced velocity gradients in the wall vicinity promote increased particle rotation.