2.1. Physical system

A chemically isotropic active spherical particle of radius $a$ is suspended in an unbounded fluid with solute concentration $C$ and subjected to an imposed uniaxial extensional flow as shown in figure 1. The far-field velocity $\boldsymbol {u}_{\infty }$ is the velocity field associated with the imposed uniaxial extensional flow. For extensional flow, the velocity field is expressed as $\boldsymbol {u}_{\infty }=\boldsymbol {\Gamma }\cdot \boldsymbol {x}$ , where $\boldsymbol {x}$ is the position vector, and $\boldsymbol {\Gamma }$ represents the rate-of-strain tensor. This imposed extensional flow is symmetric in nature, with the corresponding velocity scale represented by $Ga$ , where $G$ is the strain rate of the extensional flow. The rate-of-strain tensor for uniaxial extensional flow is of the form

(2.1) \begin{equation} \boldsymbol {\Gamma }=\frac {G}{2} \left[\begin{array}{c@{\quad}c@{\quad}c} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 2\end{array} \right]. \end{equation}

The interfacial activity at the particle surface, represented by $\mathcal {A}$ , results in a formation of excess solute cloud near the particle surface such that the difference between local concentration and far-field solute concentration $C_\infty$ is of the order $\mathcal {C}=|\mathcal {A}|\, a/\mathcal {D}$ , where $\mathcal {D}$ represents the solute diffusivity. This interfacial gradient of solute concentration coupled with mobility results in the formation of diffusiophoretic slip, such that the velocity scale corresponding to the slip is expressed as $\mathcal {U}=|\mathcal {A M}|/\mathcal {D}$ , where $\mathcal {M}$ is the mobility parameter (Michelin et al. Reference Michelin, Lauga and Bartolo2013; Saha et al. Reference Saha, Yariv and Schnitzer2021). Thus the corresponding Péclet number quantifying the solute advection relative to diffusion is expressed as $Pe=\mathcal {U}a/\mathcal {D} = |\mathcal {A M}|\,a/\mathcal {D}^2$ .

For analytical simplification, all equations have been normalised. Consequently, the remainder of the paper employs dimensionless variables unless explicitly stated otherwise. By normalising length variables by $a$ , solute concentration by $\mathcal {C}$ , velocity by $\mathcal {U}$ , and time by $a/\mathcal {U}$ , the non-dimensional form of the advection–diffusion equation, governing the excess solute concentration distribution, is expressed as

(2.2) \begin{equation} Pe \left ( \frac {\partial C}{\partial t} + \boldsymbol {u}\cdot \boldsymbol {\nabla }C \right ) = \nabla ^2 C. \end{equation}

Figure 1.

An active particle with diffusiophoretic slip denoted by $C_+ \gt C_-$ encountering a uniaxial extensional flow $\boldsymbol {u}_{\infty }$ , where $C$ denotes the solute concentration. The asymmetric solute cloud around the particle surface due to surface activity, diffusiophoresis and advection results in self-propulsion of the particle.

The advection term in this equation illustrates the nonlinear coupling between the solute concentration and the velocity field. The isotropic surface flux condition is represented as

(2.3) \begin{equation} \left . \frac {\partial C}{\partial r} \right \vert _{r=1} = -A, \end{equation}

where $A=\mathcal {A}/|\mathcal {A}|=\pm 1$ is dimensionless activity. The far-field attenuation condition is represented as

(2.4) \begin{equation} C|_{r\to \infty }\to 0. \end{equation}

The fluid flow is governed by the continuity and Navier–Stokes equations as

(2.5) \begin{equation} \boldsymbol {\nabla \cdot u}=0, \quad Re \left (\frac {\partial \boldsymbol {u}}{\partial t} + \boldsymbol {u}\boldsymbol{\cdot} \boldsymbol{\nabla} \boldsymbol {u} \right ) = - \boldsymbol {\nabla }p + \nabla ^2\boldsymbol {u}, \end{equation}

where the pressure is non-dimensionalised by $\mu \mathcal {U}/a$ , and $Re=\rho a \mathcal {U}/\mu$ is the Reynolds number, with $\rho$ being the fluid density, and $\mu$ being the fluid viscosity.

At the particle surface, the diffusiophoretic slip is characterised as

(2.6) \begin{equation} \boldsymbol {u}|_{r=1} =M\,\boldsymbol {\nabla }_sC, \end{equation}

where $\boldsymbol {\nabla }_s = (\boldsymbol {I}- \boldsymbol{nn})\cdot \boldsymbol {\nabla }$ represents surface gradient operator, and $M=\mathcal {M}/|\mathcal {M}| = \pm 1$ is the dimensionless mobility. This surface slip condition couples the flow field with the solute concentration. With the reference frame moving with the particle, the flow field as $r\to \infty$ could be represented as

(2.7) \begin{equation} \boldsymbol {u_{\infty }}\to \frac {\epsilon }{2} \left[\begin{array}{c@{\quad}c@{\quad}c} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 2 \end{array} \right] \cdot \boldsymbol {x} - \boldsymbol {U}, \end{equation}

where $\boldsymbol {U}$ represents the particle velocity. The dimensionless extension rate is given by $\epsilon = Ga/\mathcal {U}$ , which represents the ratio of the velocity scales due to imposed uniaxial extensional flow and diffusiophoresis. Thus $\epsilon$ is a measure of the strength of imposed extensional flow. The particle velocity is governed by the Newton’s second law of motion:

(2.8) \begin{equation} \frac {{\rm d}\boldsymbol {U}}{{\rm d}t} = \int _{A_p} \left ( \boldsymbol {\sigma \cdot n} \right )\,{\rm d}A, \end{equation}

where $\boldsymbol {\sigma }=-p\boldsymbol {I}+ \boldsymbol{\nabla} \boldsymbol {u}+(\boldsymbol{\nabla} \boldsymbol {u})^{\rm T}$ is the hydrodynamic stress tensor, $A_p$ is the particle surface, and $\boldsymbol {n}$ is the outward unit normal on $A_p$ . The right-hand side of (2.8) represents the total hydrodynamic force on the particle. At the steady state, the total hydrodynamic force vanishes, and the particle translates in a force-free manner.

The motion of particles in a suspension alters the overall suspension rheology due to the disturbance flow generated by individual particles that interact with external flows, impacting the rheological characteristics of the suspension. The stresslet accounts for the stress due to individual particles to the bulk fluid stress. For dilute suspension with low volume fraction such that the particle–particle hydrodynamic interaction is absent (or negligible), the effective suspension rheology is dependent directly on the stresslet and the volume fraction. In the present study, the stresslet $\boldsymbol {S}$ is determined using the following expression derived by Batchelor (Reference Batchelor1970) for a particle:

(2.9) \begin{equation} \boldsymbol {S}=\int _{A_p} \left [\tfrac {1}{2}\left \{\left (\boldsymbol {\sigma \cdot n}\right )\boldsymbol {x}+\left (\left (\boldsymbol {\sigma \cdot n} \right )\boldsymbol {x}\right )^{\rm T}-\tfrac {2}{3}\boldsymbol {I}\left (\boldsymbol {\sigma \cdot n}\right )\boldsymbol {\cdot} \boldsymbol {x}\right \}-\left (\boldsymbol {un}+\left (\boldsymbol {un}\right )^{\rm T}\right )\right ] {\rm d}A. \end{equation}

Before discussing the asymptotic and numerical approach for solving the nonlinear particle dynamics, it is important to note that the coupled problem involving an active particle and imposed extensional flow will now depend on $Pe$ as well as $\epsilon$ . This coupled and nonlinear nature of the mathematical model cannot be directly solved analytically for arbitrary $Pe$ and $\epsilon$ .

2.2. Asymptotic solution

We obtain the analytical solution based on the following key assumptions. (i) The flow and chemical transport are axisymmetric, and the particle stays very close to the stagnation point of the imposed extensional flow. (ii) The fluid inertia is negligible compared to the viscous effects. (iii) Steady state is assumed. (iv) There is small $\epsilon$ linearisation about the stationary base state and away from any bifurcations (i.e. $Pe \lt 4$ ).

The asymptotic steady-state solution is obtained for motion of the active particle in the limit of negligible fluid inertia, $Re \ll 1$ . In this limit, the governing equation for fluid flow (2.5) and solute concentration distribution (2.2) can be rewritten as

(2.10) \begin{equation} \boldsymbol {\nabla \cdot u}=0, \quad - \boldsymbol {\nabla }p + \nabla ^2\boldsymbol {u}= \boldsymbol {0}, \end{equation}

(2.11) \begin{equation} Pe\, \boldsymbol {u}\cdot \boldsymbol {\nabla }C = \nabla ^2 C. \end{equation}

In the limit $\epsilon \ll 1$ , the asymptotic expansion for any dependent variable (e.g. solute concentration $C$ ) is of the form

(2.12) \begin{equation} C=C^{(0)} + \epsilon C^{(1)} + \epsilon ^2 C^{(2)} + \dotsb . \end{equation}

This is such that the leading order or the base state corresponds to a trivial solution with no flow and a spherically symmetric source distribution for solute concentration around a stationary particle, represented as

(2.13) \begin{equation} \boldsymbol {{u}}^{(0)}=\boldsymbol {0},\hspace {0.5cm}U^{(0)}=0,\hspace {0.5cm}C^{(0)}=\frac {A}{r}. \end{equation}

Considering the axial symmetry of fluid flow and chemical transport, we have chosen an axisymmetric spherical coordinate system $(r,\theta )$ . For an asymptotically small value of $\epsilon$ , the perturbation expansion represents the $O(\epsilon )$ deviation from the base state of spherical symmetry. Considering the axisymmetric nature of the flow field, the fluid flow is expressed through stream function $\psi ^{(1)}$ , such that the radial and tangential velocity components can be expressed as

(2.14) \begin{equation} u_{r}^{(1)}=-\frac {1}{r^2}\frac {\partial \psi ^{(1)}}{\partial \eta } \quad\text {and} \quad u_{\theta }^{(1)}=-\frac {1}{r\sqrt {1-\eta ^2}}\frac {\partial \psi ^{(1)}}{\partial r}, \end{equation}

where $\eta =\cos \theta$ . The Stokes equation (2.5) can thus be expressed as (Leal Reference Leal2007)

(2.15) \begin{equation} E^{4}\psi ^{(1)} = 0, \end{equation}

where $ E^{2} \equiv \frac{\partial^2 }{\partial r^2} - \frac{(1-\eta^2)}{r^2}\frac{\partial^2}{\partial \eta^2}$ . Using the stream function leads to satisfaction of the continuity equation automatically, while the solution of the Stokes equation (2.15) yields a generalised expression of the stream function for a particle in an axisymmetric flow field, where the stream function can be expressed as (Leal Reference Leal2007)

(2.16) \begin{equation} \psi ^{(1)}=\sum _{n=1}^{\infty }\left [\mathtt {A_{_n}}r^{n+3}+\mathtt {B_{_n}}r^{n+1}+\mathtt {E_{_n}}r^{2-n}+\mathtt {F_{_n}}r^{-n}\right ]Q_{_n}(\eta ), \end{equation}

where $Q_{_n}(\eta )$ is the Gegenbauer polynomial. The $O(\epsilon )$ far-field flow velocity from (2.7) becomes

(2.17) \begin{equation} \boldsymbol {u}_{\infty }^{(1)}\to \frac {r}{2}\left [\left (3\eta ^2-1-\frac {2 U^{(1)}}{r} \eta \right )\boldsymbol {e}_r\,\, - \left (3\eta \sqrt {1-\eta ^2}- \frac {2 U^{(1)}}{r} \sqrt {1-\eta ^2}\right )\boldsymbol {e}_\theta \right ], \end{equation}

where $\boldsymbol {e}_r$ and $\boldsymbol {e}_\theta$ are unit vectors in spherical coordinates, with $U^{(1)}$ being the $O(\epsilon )$ particle velocity. Applying the far-field condition on the stream function, the values of the constants can be determined as

(2.18) \begin{equation} \begin{aligned} &\mathtt {A_{_n}}=0 \text { for }n\geqslant 1,\\ &\mathtt {B_{_n}}=0 \text { for }n \geqslant 2,\\ &\mathtt {B}_{1}=U^{(1)} \text { and } \mathtt {B}_{2}=-1. \end{aligned} \end{equation}

Thus the stream function is of the form

(2.19) \begin{equation} \psi ^{(1)}= U^{(1)} r^2\, Q_{_1}(\eta ) -r^3\, Q_{_2}(\eta ) + \sum _{n=1}^{\infty }\left [\mathtt {E_{_n}}r^{2-n}+\mathtt {F_{_n}}r^{-n}\right ]Q_{_n}(\eta ). \end{equation}

The far-field stream function includes $Q_{_1}(\eta )$ and $Q_{_2}(\eta )$ , therefore by considering $n=1$ and $n=2$ , the respective stream function is

(2.20) \begin{equation} \psi ^{(1)}= \left ( U^{(1)} r^2 + \frac {\mathtt {F_{_1}}}{r} +\mathtt {E_{_1}}r \right )Q_{_1}(\eta ) + \left (-r^3+\mathtt {E_{_2}}+\frac {\mathtt {F_{_2}}}{r^{2}}\right )Q_{_2}(\eta ). \end{equation}

By incorporating a no-penetration boundary condition at the particle surface, i.e. $u_{r}\big |_{r=1}=0$ , we get $\mathtt {E{_1}}+\mathtt {F{_1}}+U^{(1)}=0$ and $\mathtt {E{_2}}+\mathtt {F{_2}}=1$ . Substituting $\mathtt {E{_2}}$ in terms of $\mathtt {F{_2}}$ , the stream function takes the form

(2.21) \begin{equation} \psi ^{(1)} = \left [ {U^{(1)}} r^2 + \frac {\mathtt {F_{_1}}}{r} +\mathtt {E_{_1}}r \right ]Q_{_1}(\eta ) + \left [-r^3+1+\mathtt {F_{_2}}\left (\frac {1}{r^{2}}-1\right )\right ]Q_{_2}(\eta ). \end{equation}

The $O(1)$ and $O(\epsilon )$ solute transport equations are respectively of the form

(2.22) \begin{align} \nabla ^2 C^{(0)} =0 \quad\text{for }O(1), \end{align}

(2.23) \begin{align} \nabla ^2 C^{(1)}= Pe\,\boldsymbol {u}^{(1)}\cdot \boldsymbol {\nabla }C^{(0)} \quad\text{for }O(\epsilon ), \end{align}

such that ${ ({\partial C^{(0)}}/{\partial r}})\big |_{r=1}=-A\,\,$ , ${({\partial C^{(1)}}/{\partial r}})|_{r=1}=0$ and $C^{(0)}|_{r\to \infty }=C^{(1)}|_{r\to \infty }=0$ . On applying the boundary conditions, the final expression for the $O(\epsilon )$ solute transport equation is of the form

(2.24) \begin{equation} \nabla ^2 C^{(1)} = Pe\,\,A\left [ \left \{ \frac {U^{(1)}}{ r^2}+\frac {\mathtt {F_{_1}}}{r^5}+\frac {\mathtt {E_{_1}}}{r^3}\right \}P_{_1}(\eta ) - \left \{\frac {1}{r}-\frac {1}{r^4}-\mathtt {F_{_2}}\left (\frac {1}{r^6}-\frac {1}{r^4}\right )\right \}P_{_2}(\eta ) \right ], \end{equation}

where $P_{n}(\eta )$ is the Legendre polynomial of degree $n$ . By solving this equation, the solute concentration is found to be of the form

(2.25) \begin{equation} \begin{aligned} C^{(1)}&=-Pe\, A\left [\frac {U^{(1)}}{2}-\frac {\mathtt {F_{_1}}}{4r^3}+\frac {\mathtt {E_{_1}}}{2r}+\frac {1}{r^2}\left (\frac {3 \mathtt {F_{_1}}}{8}-\frac {\mathtt {E_{_1}}}{4}\right ) \right ]P_{_1}(\eta )\\ &\quad {}-Pe\, A\left [-\frac {r}{4}-\frac {\mathtt {F_{_2}}}{6r^4}+\frac {1}{r^2}\left (-\frac {\mathtt {F_{_2}}}{4}+\frac {1}{4}\right )+\frac {1}{r^3}\left (-\frac {1}{4}+\frac {7\mathtt {F_{_2}}}{18}\right ) \right ]P_{_2}(\eta ). \end{aligned} \end{equation}

The diffusiophoretic slip at the particle surface corresponds to $u_{\theta }\big |_{r=1}=-M \sqrt {1-\eta ^2} ({\partial C^{(1)}}/{\partial \eta })\big |_{r=1}$ . After substituting the expressions of solute concentration in the expression obtained from the slip condition , i.e.

(2.26) \begin{equation} \frac {\mathtt {F_{_1}}-2U^{(1)}-\mathtt {E_{_1}}}{2} + \frac {(3 +2\mathtt {F_{_2}})\eta }{2}={M}\left.\frac {\partial C^{(1)}}{\partial \eta }\right |_{r=1}, \end{equation}

one can determine the value of the unknown constants $\mathtt {E_{_1}}$ and $\mathtt {F_{_2}}$ as

(2.27) \begin{equation} \mathtt {E_{_1}}={U^{(1)}}\left (\frac {12 -3AM\:Pe}{AM\:Pe-8}\right ), \end{equation}

(2.28) \begin{equation} \mathtt {F_{_2}}=\frac {9 \left (-2 +AM\:Pe\right )}{12- AM\:Pe} = -\frac {3}{2} + 3 C_2 M. \end{equation}

For the force-free condition, $\mathtt {E_{_1}}=0$ , which implies that the particle velocity is $U^{(1)}=0$ up to $O(\epsilon )$ . Similar results have been obtained in the studies carried out by Michelin et al. (Reference Michelin, Lauga and Bartolo2013) and Saha et al. (Reference Saha, Yariv and Schnitzer2021). Also, it is important to note that away from bifurcation, the particle is motionless even in the presence of extensional flow due to symmetry of the flow. Here, $C_2$ is the coefficient of the second-degree Legendre polynomial when the solute concentration distribution on the particle surface is expressed in terms of Legendre polynomials as

(2.29) \begin{equation} C|_{r=1}=\sum _{n=0}^{\infty } C_{n}\, P_{n}(\eta ). \end{equation}

For large $r$ , the above solution for $O(\epsilon )$ solute concentration does not satisfy the far-field condition. One can use matched asymptotic expansion to obtain the far-field correction, which has been done for a passive phoretic particle in linear flow at low $Pe$ by Yariv & Kaynan (Reference Yariv and Kaynan2017). Since the solute concentration at the particle surface is independent of the far-field solution, as shown by Saha et al. (Reference Saha, Yariv and Schnitzer2021) and Yariv & Kaynan (Reference Yariv and Kaynan2017), the far-field solution has not been evaluated in the present study. However, the far-field solution can be obtained by applying the matched asymptotic solution, as followed by Saha et al. (Reference Saha, Yariv and Schnitzer2021).

For an axisymmetric particle self-propelling in direction $\boldsymbol {e}$ ( $z$ -direction), the stresslet tensor ( $\boldsymbol {S}$ ) can be expressed in terms of a scalar stresslet intensity ( $S$ ) as $\boldsymbol {S}=S(\boldsymbol {ee}-({1}/{3}){\boldsymbol {I}})$ . The velocity field and the stress tensor are determined directly from the stream function, with an additional determination of fluid pressure through the Stokes equation to determine the diagonal elements of the stress tensor. This yields the stresslet intensity $S$ up to $O(\epsilon )$ to be of the form

(2.30) \begin{equation} S = \left (10\pi - 12 \pi C_2 M\right ) \epsilon = 40\pi \left [1 -\frac {9}{12 - AM\,\,Pe}\right ]\epsilon . \end{equation}

Stresslet intensity $S$ indicates the effective viscosity of the suspension. A positive value of the stresslet represents the case of active particles showcasing puller-type motion, thereby increasing the effective viscosity, whereas a negative value of the stresslet, on the other hand, represents the active particle behaving as a pusher, thereby decreasing the effective viscosity of the active suspension (Saintillan Reference Saintillan2010, Reference Saintillan2018; Lauga & Michelin Reference Lauga and Michelin2016). A higher magnitude of $S$ , irrespective of its sign, represents the impact of a particle in influencing the suspension rheology. It is important to note that the self-propulsion and stresslet depend on the sign of $AM$ . Negative $AM$ represents the scenario of no self-propulsion (Michelin et al. Reference Michelin, Lauga and Bartolo2013), thus the suspension also behaves as a suspension of passive particles. The chemical transport for negative $AM$ gives rise to a diffusiophoretic slip which resists the imposed extensional flow and thereby enhances the stresslet as represented in (2.30). On the other hand, positive $AM$ represents a state of self-propulsion beyond the threshold $Pe=4$ (Michelin et al. Reference Michelin, Lauga and Bartolo2013). The chemical transport for positive $AM$ gives rise to a diffusiophoretic slip that augments the imposed extensional flow and thereby reduces the stresslet as represented in (2.30). It is important to note that for positive $AM$ , due to the first-order dipolar instability resulting in the particle self-propulsion, the assumed motionless isotropic base state is not valid for all values of $Pe$ (Schnitzer Reference Schnitzer2023), thereby limiting the theoretical expression derived in (2.30) to be applicable for $Pe\leqslant 4$ only.

2.3. Numerical solution

In the case of an active particle in imposed uniaxial extensional flow, the asymptotic solution is applicable only when the imposed extensional flow is weak in comparison to the diffusiophoretic slip, i.e. $\epsilon$ is small. Also, for positive $AM$ , due to the dipolar instability at $Pe =4$ , the perturbation solution (2.30) is limited to $Pe\leqslant 4$ only. In addition to this, the particle sets into motion due to instability of dipolar perturbation, and further increase in $Pe$ results in instability of other types of perturbations (such as quadrupolar) (Hu et al. Reference Hu, Lin, Rafai and Misbah2022; Kailasham & Khair Reference Kailasham and Khair2022). To address these limitations, we carry out numerical analysis over a wide range of $\epsilon$ and $Pe$ .

The particle velocity and stresslet intensity are obtained through numerical simulations of the governing equations (2.5) and (2.2) for fluid flow and solute concentration subject to boundary conditions (2.6), (2.7) and (2.3), (2.4), respectively, using the finite element-based simulation framework COMSOL Multiphysics (Zhu & Zhu Reference Zhu and Zhu2023; Shreekrishna, Mandal & Das Reference Shreekrishna, Mandal and Das2024). For simulations, an axisymmetric problem is solved with a spherical particle being at the stagnation point of the imposed extensional flow with unit radius in an axisymmetric computational domain (with non-dimensional size $50\times 100$ ). To inculcate the effects of increasing $Pe$ , along with the possibility of the presence of an initial asymmetry in the concentration distribution, the initial concentration field is expressed following Kailasham & Khair (Reference Kailasham and Khair2022) as

(2.31) \begin{equation} C |_{(r=1,t=0)} = \frac {A}{r} - \delta _{per}\left (\frac {\cos \theta }{r^2}\right ), \end{equation}

with $|\delta _{per}| \lt 1$ , such that initial asymmetry is inculcated with an axisymmetric concentration distribution. We chose the default triangular mesh element setting with linear elements for both the velocity components and the pressure field (P1 + P1). The element size is finer near the particle–fluid interface as compared to the fluid domain where the maximum element sizes 0.008 and 0.5 were used for meshing the particle surface and the fluid domain, respectively. The mesh sizes are selected based upon a convergence test performed with coarse mesh, fine mesh and very fine mesh, where the respective maximum element sizes of the fluid domain and the particle surface were 1, 0.05 (coarse mesh), 0.5, 0.008 (fine mesh), and 0.2, 0.002 (very fine mesh). The domain size $50 \times 100$ is selected post domain independence tests performed on three fluid domains of sizes $30\times60$ , $50\times100$ and $60\times120$ . The Navier–Stokes equation along with the continuity equation is solved in the computation model by using the ‘laminar flow’ interface. This is coupled with the ‘stabilised convection–diffusion’ equation to solve the solute concentration distribution with the application of a slip velocity at the particle surface. The Reynolds number for the fluid flow in the domain was kept at $Re=0.01$ . A time-dependent BDF solver with relative tolerance of $10^{-6}$ is used alongside a direct PARDISO solver.

The comparison between theoretical and numerical simulation is represented in figure 2. The theoretical and numerical simulations exhibit strong similarity for all values of $Pe$ when $AM =-1$ . However, for $AM = 1$ , i.e. the case of self-propulsion, it can be observed that the theoretical and the numerical simulations are in close resemblance for $Pe \sim 4$ , beyond which the theoretical results diverge due to instability, resulting in self-propulsion at $Pe=4$ . When $AM =-1$ , the stresslet is observed to remain positive for all values of $Pe \gt 0$ . This adheres to the fact that the effective viscosity of passive particle suspension is more than the background fluid viscosity. On the contrary, the stresslet changes sign after a particular value of $Pe$ when $AM \gt 0$ . In addition to this, the numerical simulations compare very well with the existing results for the case of active particles in a quiescent medium, as represented in figure 3. Thus numerical simulations are carried out to analyse active particle propulsion and the associated stresslet in the imposed uniaxial extensional flow condition.

Figure 2.

Theoretically and numerically obtained stresslet for a particle subjected to a weak uniaxial extensional flow with $\epsilon =0.01$ and positive and negative values of $AM$ .

Figure 3.

Motion of an active particle in the absence of any imposed background flow. The present numerical simulation is validated with results in the literature. Particle velocity is validated from Kailasham & Khair (Reference Kailasham and Khair2022) and Hu et al. (Reference Hu, Lin, Rafai and Misbah2022), and the stresslet is validated from Michelin et al. (Reference Michelin, Lauga and Bartolo2013). Plots show the variation of particle velocity and the induced stresslet with Péclet number for an active particle in the absence of any imposed flow. The surface plot shows qualitatively the symmetry breaking in the solute concentration around the particle surface.