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Sketch of the prolate swimmer and the coordinate system used; (r,z) denote cylindrical coordinates and (R,θ) the spherical coordinates with φ the azimuthal angle. The results presented below assume axisymmetric flow.
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In several biologically relevant situations, cell locomotion occurs in polymeric fluids with Weissenberg number larger than 1. Here we present results of three-dimensional numerical simulations for the steady locomotion of a self-propelled body in a model polymeric (Giesekus) fluid at low Reynolds number. Locomotion is driven by steady tangential d...
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... e is the orientation vector of the squirmer, B n is the nth mode of the surface squirming velocity [51], P n is the nth Legendre polynomial, R is the position vector, and R = |R|. (See Fig. 1 for a sketch of the notations.) In a Newtonian fluid, the swimming speed of the squirmer is 2B 1 /3 [51] and thus only dictated by the first mode. In previous studies, it is commonly assumed B n = 0 for n > 2 [55,56]. Consequently, the tangential velocity on the sphere in the comoving frame is expressed as u θ (θ ) = B 1 sin θ + (B 2 ...
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... decays as ∼1/r 3 , whereas the decay is only ∼ 1/r 2 for pusher and puller-type cells (β SW = 0). We numerically estimate the radial decay of the velocity for locomotion in a polymeric fluid by fitting a power law from about r ≈ D to the end of the computational domain. The values of the exponent γ obtained with this procedure are reported in Fig. 10 as a function of the Weissenberg number. The flow, which decays as ∼1/r 3 in the Newtonian case, always decays faster in the polymeric case. We observe that the variation of the decay rate with We is not monotonic and that for the two largest values of the viscosity ratio (β = 0.6,0.3) a maximum is reached near We = 1, which coincides ...
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... the elongation of the polymers in the fluid. We plot Tr(C) in Figs. 11 and 12 for the forced motion of the sphere and for swimming with different polymer relaxation times. In Fig. 11 we compare polymer stretching for forced motion and free swimming at the same speed in the case where We = ...
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... the elongation of the polymers in the fluid. We plot Tr(C) in Figs. 11 and 12 for the forced motion of the sphere and for swimming with different polymer relaxation times. In Fig. 11 we compare polymer stretching for forced motion and free swimming at the same speed in the case where We = ...
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... and β = 0.3. The region around the body where stretching is evident is much larger in the case of forced motion. The spatial decay of stretching is more rapid on the side of the swimmer while the largest elongation is observed in the wake right behind the organism. In Fig. 12 we show the variation of stretching at different values of We. As expected a larger Weissenberg number leads to a larger region of elongated polymers and correlates with a more pronounced elastic ...
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... increase of the magnitude of the polymer elongation is further quantified in Fig. 13 where the maximum of Tr(C) inside our computational domain is displayed as a function of We for the different values of β considered. The relationship between elongation and relaxation time is found FIG. 12. Polymeric stretching field: trace of the polymer con- formation tensor, Tr(C). Comparison of polymer stretching between We = 1 ...
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... between elongation and relaxation time is found FIG. 12. Polymeric stretching field: trace of the polymer con- formation tensor, Tr(C). Comparison of polymer stretching between We = 1 and We = 9 for β = 0.3. to be approximately linear, with a slope s dependent on the viscosity ratio. The dependence of the slope with β is shown in the inset in Fig. 13, and a power law s ≈ β −1/4 provides an appropriate fit to our numerical ...
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... with the location where these maxima are attained (spherical coordinates with R the distance from the center of the swimmer and θ in degrees measured from the front of the swimmer). Three values of the Weissenberg number are considered with β = 0.3. The maximum elongation is in axial stretching, C zz , and occurs just behind the body (see also Fig. 12). The maximum of radial stretching C rr is observed on the swimmer, just off the symmetry line at the front stagnation, while the peak of the shear C rz is characterized by negative values and is observed on the back of the body with values of θ slightly increasing with polymer elasticity; this component will be responsible for the ...
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... components of polymer stretch- ing (bold numbers) and corresponding location where these maxima are attained for different values of We and in the case β = 0.3. The position is reported in spherical polar coordinate, with θ in degrees and R nondimensionalized by the sphere diameter, while the polymeric stresses are in cylindrical coordinate (see Fig. 1). nonzero. Its amplitude is in fact comparable to that of the radial stretching C rr and it attains its maximum value in front of the ...
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... results for swimming speed, power, and efficiency as a function of the Weissenberg number and the viscosity ratio show similar trends as those discussed earlier for spherical squirming and will not be repeated. As example of flow, we show in Fig. 14 the flow streamlines and polymer elongation for a prolate swimmer of aspect ratio AR = 4. Large values of Tr(C) are observed in a thin region around the body and in the wake, similarly to the spherical swimmer. The thickness of this stretching boundary layer, as well as the length of the wake, is found to decrease for an elongated ...
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... for a prolate swimmer of aspect ratio AR = 4. Large values of Tr(C) are observed in a thin region around the body and in the wake, similarly to the spherical swimmer. The thickness of this stretching boundary layer, as well as the length of the wake, is found to decrease for an elongated swimmer. Comparing the polymer stretching reported in Fig. 11 and in Fig. 14, we note also that the maximum of Tr(C) is more than twice as big in the case of a spherical swimmer. In addition, for the prolate swimmer the velocity displays a weak overshoot just behind the body and, more interestingly, the streamlines are seen to converge toward the center of the body (z = 0) and then depart further ...
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... swimmer of aspect ratio AR = 4. Large values of Tr(C) are observed in a thin region around the body and in the wake, similarly to the spherical swimmer. The thickness of this stretching boundary layer, as well as the length of the wake, is found to decrease for an elongated swimmer. Comparing the polymer stretching reported in Fig. 11 and in Fig. 14, we note also that the maximum of Tr(C) is more than twice as big in the case of a spherical swimmer. In addition, for the prolate swimmer the velocity displays a weak overshoot just behind the body and, more interestingly, the streamlines are seen to converge toward the center of the body (z = 0) and then depart further downstream. ...
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... Fig. 15 we show the variation of the swimming speed with the prolate aspect ratio. We plot the results in the Newtonian case (black squares) as well as the polymeric case with We = 7 and β = 0.3 (red circles). The swimming speed is normalized with the swimming velocity of the spherical Newtonian squirmer and is seen to decrease with the aspect ...
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... swimming power, normalized by that of the sphere in the Newtonian fluid with the same total viscosity, is shown in Fig. 16 and also decreases with the aspect ratio of the body. The relative reduction in consumed power is increasing with decreasing aspect ...
Citations
... P 1 1 (cos θ) = − sin θ, P 1 2 (cos θ) = −3 cos θ sin θ. The squirmer model has been instrumental in examining various aspects of swimming at zero Reynolds number, including hydrodynamic interactions (Llopis & Pagonabarraga 2010), locomotion in viscoelastic fluids (Zhu et al. 2011; and nutrient uptake (Magar, Goto & Pedley 2003;Magar & Pedley 2005). In that regime, the flow field and squirmer swimming velocity induced by the swimming-gait modal expansion (1.1) can be obtained by superposing those motions induced by each mode separately. ...
... Previous studies of squirmers at non-zero Re have focused on squirmers whose swimming gait involves only the first two, dipolar and quadrupolar modes in (1.1). Wang & Ardekani (2012) developed an asymptotic expansion through O(Re) for the swimming speed of a two-mode squirmer at small Re, which Khair & Chisholm (2014) later extended to O(Re 2 ). (In these works, Re is defined based upon the magnitude of the dipolar squirming mode.) ...
... It is intuitive that fore-aft asymmetric squirmers that are non-motile at Re = 0 (i.e. squirmers with B 1 = 0 but B n / = 0 for at least one odd, non-unity value of n), generally become motile for Re > 0, though with their swimming speed vanishing as Re 0; this is readily demonstrable by adapting the small-Re analyses of Wang & Ardekani (2012) and Khair & Chisholm (2014). Here, we ask whether inertia can also enable fore-aft symmetric squirmers to swim via nonlinear symmetry breaking (figure 1). ...
The squirmer is a popular model to analyse the fluid mechanics of a self-propelled object, such as a microorganism. We demonstrate that some fore-aft symmetric squirmers can spontaneously self-propel above a critical Reynolds number. Specifically, we numerically study the effects of inertia on spherical squirmers characterised by an axially and fore-aft symmetric 'quadrupolar' distribution of surface-slip velocity; under creeping-flow conditions, such squirmers generate a pure stresslet flow, the stresslet sign classifying the squirmer as either a 'pusher' or 'puller'. Assuming axial symmetry, and over the examined range of the Reynolds number Re (defined based upon the magnitude of the quadrupolar squirming), we find that spontaneous symmetry breaking occurs in the puller case above Re ≈ 14.3, with steady swimming emerging from that threshold consistently with a supercritical pitchfork bifurcation and with the swimming speed growing monotonically with Re.
... The ratio of the first two modes is denoted by β = B 2 /B 1 , with β > 0 denoting pullers and β < 0 pushers. We keep only the first two squirming modes to analyze the effects of density gradients on swimming, as is common practice in other analysis including the swimming under confinement [38,39], in complex fluids [40][41][42][43][44][45], and at finite inertia [46][47][48][49]. ...
Organisms often swim through density stratified fluids. In this Letter, we investigate the dynamics of small active particles swimming in density gradients and report theoretical evidence of taxis as a result of density stratification ($\textit{densitaxis}$). Specifically, we calculate the effect of density stratification on the dynamics of a force-free spherical squirmer and show that density stratification induces reorientation that tends to align swimming either parallel or normal to the gradient depending on the swimming gait. In particular, particles that propel by generating thrust in the front (pullers) rotate to swim parallel to gradients and hence display (positive or negative) densitaxis, while particles that propel by generating thrust in the back (pushers) rotate to swim normal to the gradients. This work could be useful to understand the motion of marine organisms in ocean, or be leveraged to sort or organize a suspension of active particles by modulating density gradients.
... The propulsion of microorganisms in unbounded fluids and media has attracted a tremendous amount of interest since the pioneering works of Taylor (1951) and Lighthill (1952). Studies have since extended these seminal works to investigate the propulsion of microorganisms in unbounded non-Newtonian and heterogeneous media (Yu, Lauga & Hosoi 2006;Lauga 2007;Leshansky 2009;Zhu et al. 2011;Pak et al. 2012;Datt et al. 2015;Chisholm et al. 2016;Lauga 2016;Gómez et al. 2017;Nganguia & Pak 2018). However, the use of carriers to enhance targeted drug delivery (Lee & Yeo 2015;Wu et al. 2020) or the development of nanotechnologies to manipulate cells in confined spaces (Raveshi et al. 2021) have led to increasing interest in the motion of microorganisms enclosed in various interfaces (Mirbagheri & Fu 2016;Daddi-Moussa-Ider, Lowen & Gekle 2018;Hoell et al. 2019;Nganguia et al. 2020b). ...
The development of novel drug delivery systems, which are revolutionizing modern medicine, is benefiting from studies on microorganisms’ swimming. In this paper we consider a model microorganism (a squirmer) enclosed in a viscous droplet to investigate the effects of medium heterogeneity or geometry on the propulsion speed of the caged squirmer. We first consider the squirmer and droplet to be spherical (no shape effects) and derive exact solutions for the equations governing the problem. For a squirmer with purely tangential surface velocity, the squirmer is always able to move inside the droplet (even when the latter ceases to move as a result of large fluid resistance of the heterogeneous medium). Adding radial modes to the surface velocity, we establish a new condition for the existence of a co-swimming speed (where squirmer and droplet move at the same speed). Next, to probe the effects of geometry on propulsion, we consider the squirmer and droplet to be in Newtonian fluids. For a squirmer with purely tangential surface velocity, numerical simulations reveal a strong dependence of the squirmer's speed on shapes, the size of the droplet and the viscosity contrast. We found that the squirmer speed is largest when the droplet size and squirmer's eccentricity are small, and the viscosity contrast is large. For co-swimming, our results reveal a complex, non-trivial interplay between the various factors that combine to yield the squirmer's propulsion speed. Taken together, our study provides several considerations for the efficient design of future drug delivery systems.
... 14 Studies have since extended these seminal works to investigate the propulsion of microorganisms in non-Newtonian and heterogeneous media. [15][16][17][18][19][20][21][22][23] However, applications including targeted drug delivery, 12,24 or the development of nanotechnologies to manipulate cells, 25 have led to increasing interest in the motion of microorganisms in confined spaces. [25][26][27][28][29][30][31][32][33][34] Theoretical studies of such systems assumed physical interfaces with various properties. ...
A squirmer enclosed in a droplet represents a minimal model for some drug delivery systems. In the case of a spherical squirmer swimming with a spherical cage in a Newtonian fluid [Reigh et al., “Swimming with a cage: Low-Reynolds-number locomotion inside a droplet,” Soft Matter 13, 3161 (2017)], it was found that the squirmer and droplet always propelled in the same direction albeit at different speeds. We expand the model to include particles' shape and medium's heterogeneity, two biologically relevant features. Our results reveal a novel behavior: a configuration that consists of a spherical squirmer and a spheroidal droplet in highly heterogeneous media yields a backward motion of the droplet.
... The velocity of the flow in front of the cargo-puller model is observed to decay more rapidly than that for the cargo-pusher model at Re = 25, as shown in figures 5(a) and 5(b). This result proves that the cargo-puller model swims faster than the cargo-pusher model, where a more rapid decay of the velocity in front of the body leads to a faster swimmer (Zhu et al. 2011). ...
... This pattern is different from that of an individual squirmer because the carried cargoes change the structure of the flow field for these assemblies. However, this result indicates that a more rapid decay leads to larger efficiency (Zhu et al. 2011;Ouyang et al. 2022). The pusher-cargo assemblies with different-shaped cargoes maintain the same decay as |u| ≈ O(r −4 ) at the first stage (r/a < 2.5); by further increasing r/a, the velocity decay for the pusher-cargo model 2 is faster than that for the pusher-cargo model 1. ...
We numerically investigate the hydrodynamics of a spherical swimmer carrying a rigid cargo in a Newtonian fluid. This swimmer model, a ‘squirmer’, which is self-propelled by generating tangential surface waves, is simulated by a direct-forcing fictitious domain method (DF-FDM). We consider the effects of swimming Reynolds numbers ( Re ) (based on the radius and the swimming speed of the squirmers), the assembly models (related to the cargo shapes, the relative distances ( d s ) and positions between the squirmer and the cargo) on the assembly's locomotion. We find that the ‘pusher-cargo’ (pusher behind the cargo) model swims significantly faster than the remaining three models at the finite Re adopted in this study; the term ‘pusher’ indicates that the object is propelled from the rear, as opposed to ‘puller’, from the front. Both the ‘pusher-cargo’ and ‘cargo-pusher’ (pusher in front of the cargo) assemblies with an oblate cargo swim faster than the corresponding assemblies with a spherical or prolate cargo. In addition, the pusher-cargo model is significantly more efficient than the other models, and a larger d s yields a smaller carrying hydrodynamic efficiency η for the pusher-cargo model, but a greater η for the cargo-pusher model. We also illustrate the assembly swimming stability, finding that the ‘puller-cargo’ (puller behind the cargo) model is stable more than the ‘cargo-puller’ (puller in front of the cargo) model, and the assembly with a larger d s yields more unstable swimming.
... 3 This model has been successfully used in considering the self-propelled organisms' nutrient uptake, 5,35 their two-body hydrodynamic interactions, [36][37][38] collective behaviors, 39,40 and hydrodynamics in non-Newtonian fluids. [30][31][32]41 The primary motivation of this study is to elucidate how the shear-dependent rheology, fluid inertia, and the relative position between the propeller and the cargoes affect the assembly's hydrodynamics. Meanwhile, we expect to find potentially the most efficient cargo-carrying assembly. ...
This paper simulates the locomotion of a micro-swimmer towing cargo through a shear-dependent non-Newtonian fluid. We investigate the effect of the shear-dependent rheology (refers to the power-law index n), swimming Reynolds numbers ( Re), and the relative position (refers to the distance d s and the concerning angle θ) between the swimmer and the cargoes on the assemblies' locomotion. For a swimmer towing a cargo, we find that a cargo-puller, cargo-pusher, or pusher-cargo (three typical towing models) swims faster in the shear-thickening fluids than in the shear-thinning fluids at Re ≤ 1. Moreover, the pusher-cargo swims significantly faster than the counterpart puller-cargo at Re ≤ 1. For a swimmer towing two cargoes, we find that the maximum negative swimming speeds can be achieved at θ = 30° and 150°, corresponding to two typical regular-triangle structures assembled by the squirmer and the cargoes. Interestingly, some regular-triangle assemblies (puller with θ = 30° and pusher with θ = 150°) can maintain a swimming opposite to their initial orientation. In addition, we obtain a relation of energy expenditure P ∼ Re ⁿ ⁻¹ ; it is also found that the assembly swimming in the shear-thinning fluids is more efficient than in the shear-thickening ones. Our results provide specified guidance in the designing of cargo-carrying micro-swimming devices.
... The slip velocity v sq is commonly used for spherical swimmers (squirmers) in the Stokes flow 28,29 . Elliptical squirmers were studied in Newtonian 30-34 as well as viscoelastic fluids 35,36 . However, there is a lack of studies of their motion in liquid crystals except for passive ellipsoidal/spheroidal particles [37][38][39] . ...
... We use simplified Lighthill-Blake squirmer boundary condition 30,40 with only two modes on the elliptical surface of the squirmer (c.f. 35 ). We expect, based on experimental results 4,27 , that the most striking effects happen due to the anchoring of the nematic director on the boundary and the generic type of the swimmer (that is, pusher/puller). ...
Swimming bacteria successfully colonize complex non-Newtonian environments exemplified by viscoelastic media and liquid crystals. While there is a significant body of research on microswimmer motility in viscoelastic liquids, the motion in anisotropic fluids still lacks clarity. This paper studies how individual microswimmers (e.g., bacteria) interact in a mucus-like environment modeled by a visco-elastic liquid crystal. We have found that an individual swimmer moves faster along the same track after the direction reversal, in faithful agreement with the experiment. This behavior is attributed to the formation of the transient tunnel due to the visco-elastic medium memory. We observed that the aft swimmer has a higher velocity for two swimmers traveling along the same track and catches up with the leading swimmer. Swimmers moving in a parallel course attract each other and then travel at a close distance. A pair of swimmers launched at different angles form a "train”: after some transient, the following swimmers repeat the path of the "leader”. Our results shed light on bacteria penetration in mucus and colonization of heterogeneous liquid environments.
... With increasing Re (increasing the inertial effect), the velocity decay with r becomes faster (from approximate O(r −4 ) to O(r −8 ) for the pusher (β = −1) and from approximate O(r −2.5 ) to O(r −3 ) for the puller (β = 1)). Combined with the swimming speeds as in figure 4(a), this result is in agreement with the conclusion that a more rapid decay leads to a larger efficiency (Zhu et al. 2011). For the cases of the squirmer dumbbells at Re = 25, it is seen in figure 13(b) that the velocity decay with |β| = 1 is identical to that of an individual squirmer. ...
We study the hydrodynamics of a spherical and dumbbell-shaped microswimmer in a tube. Combined with a squirmer model generating tangential surface waves for self-propulsion, a direct-forcing fictitious domain method is employed to simulate the swimming of the microswimmers. We perform the simulations by considering the variations of the swimming Reynolds numbers ( Re ), the blockage ratios ( κ ) and the relative distances ( d s ) between the squirmers of the dumbbell. The results show that the squirmer dumbbell weakens the inertia effects of the fluid more than an individual squirmer. The constrained tube can speed up an inertial pusher (propelled from the rear) and an inertia pusher dumbbell; a greater distance d s results in a slower speed of an inertial pusher dumbbell but a faster speed of an inertial puller (propelled from the front) dumbbell. We also illustrate the swimming stability of a puller (stable) and pusher (unstable) swimming in the tube at Re = 0. At a finite Re , we find that the inertia and the tube constraint competitively affect the swimming stability of the squirmers and squirmer dumbbells. The puller and puller dumbbells swimming in the tube become unstable with increasing Re , whereas an unstable–stable–unstable evolution is found for the pusher and pusher dumbbells. With increasing κ , the puller and puller dumbbells become stable while the pusher and pusher dumbbells become unstable. In addition, we find that a greater d s yields a higher hydrodynamic efficiency η of the inertial squirmer dumbbell.
... The relationship between hydrodynamic efficiency η of = 0.2 is shown in Figure 10. η of the pusher and puller increa the pusher has a larger η than the puller under the same Re tionship between power-law exponent γ(u *~( r*) -γ as shown in ing Figure 11 and Figure 10, we can see that fast velocity de which is the same as the previous conclusion [13,33]. There ex for the puller at Re = 3.0; this may be attributed to the breakag as shown in Figure 7. ...
... η of the pusher and puller increases with increasing Re, but the pusher has a larger η than the puller under the same Re. Figure 11 shows the relationship between power-law exponent γ(u *~( r*) -γ as shown in Figure 6 and Re. Combining Figures 10 and 11, we can see that fast velocity decay corresponds to large η, which is the same as the previous conclusion [13,33]. There exists a sudden reduction of γ for the puller at Re = 3.0; this may be attributed to the breakage of the upstream tip vortex as shown in Figure 7. ...
... The relationship between hydrodynamic efficiency η of the squir = 0.2 is shown in Figure 10. η of the pusher and puller increases with the pusher has a larger η than the puller under the same Re. Figure tionship between power-law exponent γ(u *~( r*) -γ as shown in Figure ing Figure 11 and Figure 10, we can see that fast velocity decay corr which is the same as the previous conclusion [13,33]. There exists a sud for the puller at Re = 3.0; this may be attributed to the breakage of the u as shown in Figure 7. ...
In this paper the propulsion of elliptical objects (called squirmers) by imposed tangential velocity along the surface is studied. For a symmetric velocity distribution (a neutral squirmer), pushers (increased tangential velocity on the downstream side of the ellipse) and pullers (increased tangential velocity on the upstream side of the ellipse), the hydrodynamic characteristics, are simulated numerically using the immersed boundary-lattice Boltzmann method. The accuracy of the numerical scheme and code are validated. The effects of Reynolds number (Re) and squirmer aspect ratio (AR) on the velocity u*, power expenditure P* and hydrodynamic efficiency η of the squirmer are explored. The results show that the change of u* along radial direction r* shows the relation of u*~r*−2 for the neutral squirmer, and u*~r*−1 for the pusher and puller. With the increase of Re, u* of the pusher increases monotonically, but u* of the puller decreases from Re = 0.01 to 0.3, and then increases from Re = 0.3 to 3. The values of P* of the pusher and puller are the same for 0.01 ≤ Re ≤ 0.3; P* of the pusher is larger than that of the puller when Re > 0.3. η of the pusher and puller increases with increasing Re, but the pusher has a larger η than the puller at the same Re. u* and P* decrease with increasing AR, and the pusher and puller have the largest and least u*, respectively. The values of P* of the pusher and puller are almost the same and are much larger than those of the neutral squirmer. With the increase of AR, η increases for the neutral squirmer, but changes non-monotonically for the pusher and puller.
... They find that, with increasing Re, the shear-thinning rheology affects the swimming speeds of the squirmers, in a different manner to the swimming in a shear-thickening fluid. Besides the rheological properties of fluids, the microswimmers' anisotropic shape also affect their swimming behaviors [31][32]. A classical rod-shaped microswimmer model assembled by several squirmers in tandem has attracted considerable attention recently [8,[32][33][34], as it can well reflect the actual geometries of the bacteria or microorganisms. ...
... It is seen that the velocity decay for the pusher (β=-3) in the shear-thinning fluid is faster than in the shear-thickening fluid (|u|≈r -3.2 for n=0.8 and r -1.2 for n=1.2 ) at Re=5. This pattern agrees with the conclusion that a more rapid decay leads to a larger efficiency [31]. Note that an opposite pattern (swim faster but decay slower) is found for the puller (β=3) at Re=5 because the puller induces negative flows in front of the body. ...
We employ an immersed boundary-lattice Boltzmann (IB-LB) scheme to simulate a cylindrical (a classical self-propelled model) and a rod-shaped squirmer swimming in a channel filled with power-law fluids. The power-law index n, the channel blocking ratio κ (squirmer diameter/channel width), and the swimming Reynolds number Re are respectively set at 0.8≤n≤1.2, 0.2≤κ≤0.5 and 0.05≤Re≤5 to investigate the microswimmer’ swimming speed, its power expenditure (P), and its hydrodynamic efficiency (η). The results show that increasing n yields a faster squirmer at a low Re (Re≤0.5). On further increasing Re (Re≥1), a larger n results in a slower pusher (a squirmer propelled from the rear), or a faster puller (a squirmer propelled from the front). Increasing the channel’s width (decreasing κ) can lead to a slower puller or a puller rod squirmer. A definition of puller/pusher will be provided later. It is also found that, with shear-thinning, it is easier to unstabilize a puller than with shear-thickening, when increasing Re. Swimming in a shear-thinning fluid expends more power P than in a shear-thickening fluid, and P is scaled with Re according to P~ Ren-1 (0.05≤Re≤1). In addition, a stronger channel constraint (κ=0.5) yields a higher η for the puller and the weak inertial pusher, whereas a weaker channel constraint (κ=0.2) results in a higher η for the pusher with the increased fluid inertia.
