3.1. Temperature profile

In figure 2, we describe the effects of certain parameters on the resulting temperature profile due to self-thermophoresis. In the first case (figure 2a–c), the microswimmer director is leaning towards the wall, i.e. the cold portion is facing away from the isothermal wall $(\theta _p=60^{\circ })$. Similarly, in figure 2(d–f), the inclination angle is $120^{\circ }$. The presence of the wall breaks the symmetry of the temperature distribution along the director axis.

Figure 2. Spatial variation of the scaled temperature $(\bar {\mathcal {T}}=T/(T_{max}-T_{min}))$ in the $x$–$z$ plane, both inside the microswimmer and in the surrounding fluid medium, for a fixed wall-swimmer distance of $\delta =0.5$ and a coverage angle of the metal coating $\varphi _{cap}=90^{\circ }$. The orientation angle ($\theta _{p}$) is $60^{\circ }$ in (a–c), while it is $120^{\circ }$ in (d–f). The thermal conductivity contrast has been varied as 0.1, 1 and 10 from left to right panels. The thick orange arc in each figure represents the metal-coated area of the surface. $(a,d)\, \mathcal {K} = 0.1, (b,e)\ \mathcal {K} = 1,\ (c,f)\ \mathcal {K} = 10.$

The surface temperature of the microswimmer $\mathcal {T}_S$ is highly dependent on its distance from the wall, as shown in figures 3(a) and 3(c). In these figures, the variation of the angle $\theta_d$ denotes distance along the surface, measured clockwise from the director axis $\boldsymbol{d}$. Sensing a nearby low-temperature thermal obstacle, the micromotor surface temperature drops when the swimmer approaches the wall, irrespective of its inclination. Besides, the locations of the maximum and minimum temperatures on the particle surface vary with the distance. For both of the inclinations considered, the minimum temperature location shifts more towards the wall as the swimmer enters a wall-adjacent zone. In the former configuration, where the heated portion is closer to the wall, the maximum temperature location is shifted away from the wall as $\delta$ decreases. However, in the latter configuration, the maximum temperature location has only a negligible shift in the same direction. The above-mentioned maxima and minima locations of the surface temperature control the direction of surface slip flow, which, in the present case, is directed from a colder point to a hotter one on the surface.

Figure 3. (a,b) Surface temperature $(\mathcal {T}_S)$ variation along the surface for different wall distances and thermal conductivity ratios, respectively. Here, the angle $\theta _d$ is at the surface of the microswimmer, measured clockwise from the director axis $\boldsymbol {d}$. In (a), we have taken $\mathcal {K}=1$, while in (b) $\delta =1$. Other parameters are $\varphi _{cap} =90 ^{\circ }$ and $\theta _{p} =60 ^{\circ }$. In (c,d), similar variations are shown for a different inclination of $\theta _{p} = 120 ^{\circ }$. In each plot, the central vertical line provides visual assistance to identify the degree of asymmetry around the director axis, $\boldsymbol { d}$.

Comparing the temperature distributions in figures 2(a–c) or 2(d–f) we find that, as the particle material becomes relatively less thermally conductive than that of the surrounding fluid (i.e. a change of $\mathcal {K}$ from 10 to 0.1), the hot-spot on the particle surface becomes progressively more localized owing to a poorer heat loss to the fluid. Such a physical situation can be imagined for metal-coated polymer spheres which bear distinct thermal properties, i.e. perfect conduction and perfect insulation, respectively. This ideal condition holds true only in the limit of $\varkappa _p \ll \varkappa _f$ or $\mathcal {K} \ll 1$. Consequently, such cases result in an enhanced temperature asymmetry around the micromotor. Along similar lines, in the above figures for $\mathcal {K}=0.1$ and 1, the metal-coated and uncoated halves show a vivid contrast in the isotherm contours coming out from the surface of the particle. On the other hand, with $\varkappa _{p}$ much higher than $\varkappa _{f}$, the hot area occupies a greater extent of the particle. However, in the limit of $\mathcal {K} \gg 1$, the particle becomes highly conducting and behaves almost as an isothermal body. This trend is reflected in figure 2(c,f) where the fluid-medium isotherms cut the surface almost parallel to it, and the insulating nature of the uncoated half is merely observed.

Strikingly, although the cap angle is low in figure 2, a high particle conductivity acts behind the trend of diminishing temperature heterogeneity around the particle, a phenomenon that is expected for the case of a very high metallic coverage, i.e. $\varphi _{cap} \to 180^{\circ }$. For a generalized quantitative assessment of this observation, we define a temperature heterogeneity parameter $\mathcal {H}_T$ as

(3.1)\begin{equation} \mathcal{H}_T=\frac{\mathcal{T}_{S,max}-\mathcal{T}_{S,min}}{\mathcal{T}_{S,max} +\mathcal{T}_{S,min}}, \end{equation}

representing a scaled difference between the maximum and minimum swimmer surface temperatures. This quantification is motivated from the deformation parameter frequently employed in drop deformation studies (Taylor Reference Taylor1966; Poddar et al. Reference Poddar, Mandal, Bandopadhyay and Chakraborty2018, Reference Poddar, Mandal, Bandopadhyay and Chakraborty2019b). To untangle the behaviour of the said heterogeneity from that due to non-axisymmetry around the wall normal, we have chosen $\theta _p=180^{\circ }$ in figure 4. In figure 4(a), the effect of high thermal conductivity ratio $\mathcal {K}$ has been captured for different gaps from the wall. Here, a low coating coverage, $\varphi _{cap} =40 ^{\circ }$, has been chosen. On the other hand, figure 4(b) highlights the variations of $\mathcal {H}_T$ with high coating coverages, $\varphi _{cap}$, for different gaps from the wall. This case corresponds to a low conductivity ratio, $\mathcal {K}=0.1$. A comparison of the two figures re-affirms the common trend of decreasing heterogeneity for the two cases: (i) $\{ \mathcal {K} \gg 1 , \text { but low} \ \varphi _{cap} \}$ and (ii) $\{ \mathcal {K} \ll 1, \text { but high} \ \varphi _{cap} \}$. In addition, figure 4 suggests that, with decreasing gap between the microswimmer and the wall, the heterogeneous nature of surface temperature increases due to the tendency of reaching the limit of $\mathcal {T}_{S,min} \to \mathcal {T}_{wall}=0$. However, the impact of increasing $\mathcal {K}$ or $\varphi _{cap}$ on the temperature heterogeneity is asymptotically less pronounced at smaller gaps.

Figure 4. (a) Variation of temperature heterogeneity parameter, $\mathcal {H}_T$, with high $\mathcal {K}$ for different gaps from the wall. Here, a low coating coverage, $\varphi _{cap} =40 ^{\circ }$, has been chosen. (a) Variation of $\mathcal {H}_T$ with high coverage angles, $\varphi _{cap}$, for different gaps from the wall. Here, a low conductivity ratio, $\mathcal {K}=0.1$, has been chosen. In both panels, $\theta _{p}=180^{\circ }$. The wall-swimmer configuration has been shown as an inset to the panel (a).

The effect of thermal conductivity contrast between the particle and fluid is also prominently extended in the fluid domain. For example, in the case of a highly conducting particle, the heated zone in the fluid almost surrounds the full circumference of the particle, as portrayed in figure 2(c,f).

The effects of contrasting thermal conductivities on the surface temperature profile have been demonstrated in figures 3(b) and 3(d) for inclinations $60^{\circ }$ and $120^{\circ }$, respectively. We have chosen typical values of $\mathcal {K}$ for which the particle is more conductive than the fluid ($\mathcal {K}>1$) and vice versa ($\mathcal {K}<1$). With the hot surface facing the wall, the $\mathcal {K}>1$ condition causes a raise in the surface temperature before reaching $\theta \approx 155^ \circ$; thereafter, the temperature decays until $\theta \approx 325^ \circ$ and again raises beyond this location. Locations of these turnover points on the surface as well as the behaviour of temperature with changing $\mathcal {K}$ between these points are modified when the cold portion faces the wall.

3.2. Effect of thermal conductivity contrast in an unbounded flow

Before investigating the coupled interplay between the thermal conductivity contrast and the bounding wall on the microswimmer velocity, we delineate its behaviour in an unbounded domain. In an unbounded flow, the rotational component of the swimmer velocity does not exist since the wall-induced asymmetry in the temperature and flow field are absent in this case. The linear velocity magnitude $|\boldsymbol {V}_{ub}|=\sqrt {V_x^2+V_z^2}$ depends only on the metallic cap coverage, $\varphi _{cap}$, and the thermal conductivity ratio, $\mathcal {K}$, as portrayed in figure 5(a). It is observed that the effect of $\varphi _{cap}$ on $|\boldsymbol {V}_{ub}|$ is symmetric about the $50\,\%$ coverage $(\varphi _{cap}=90^{\circ })$. On the other hand, the effect of increasing $\mathcal {K}$ is to decrease the velocity magnitude due to reduced temperature gradients.

Figure 5. In (a), the unbounded swimming velocity magnitude $|\boldsymbol {V}_{ub}|$ is shown in the $\mathcal {K}-\varphi _{cap}$ plane. In (b,c), the variations in velocity components ($V_z$ and $V_x$, respectively) with the thermal conductivity contrast ($\mathcal {K}$) are shown for both unbounded and near-wall scenarios. The orientation and velocity direction of the particle are shown schematically in each figure.

The wall-induced distortion of the temperature field weakens asymptotically and, beyond a certain height, the thermal boundary condition at the wall is likely to become inconsequential. A simultaneous decay in the hydrodynamic disturbance also takes place, tending towards a situation equivalent to that of an unbounded scenario. We quantify this distance with an effective separation distance, $\delta _{ub}$, at which a velocity component of the microswimmer becomes $99\,\%$ of the unbounded velocity magnitude $|\boldsymbol {V}_{ub}|$. This effective distance is a function of the thermal and configurational parameters, i.e. $\delta _{ub}=\delta _{ub}(\mathcal {K},\theta _p,\varphi _{cap})$. As an example of this functional dependence, we show two reference configurations of the particle motion in figures 5(a) and 5(b), where either of the linear velocity component of the microswimmer exists in an unbounded domain. While in the first instance (figure 5b), the $\delta _{ub}$ for the vertical velocity component reduces from $\delta _{ub}=7.58$ for $\mathcal {K}=0.1$ to $\delta _{ub}=6.89$ for $\mathcal {K}=10$, in the second instance, (figure 5c) the $\delta _{ub}$ for the horizontal velocity component reduces from $\delta _{ub}=1.61$ for $\mathcal {K}=0.1$ to $\delta _{ub}=0.63$ for $\mathcal {K}=10$. The difference in the $\delta _{ub}$ for different velocity components indicates a contrasting nature of the wall effects in different flow directions.

3.3. Combined interplay of bounding wall and thermal conductivity contrast

In both the figures 6(a) and 6(b), the downward translation of the swimmer is intensified due to the presence of the wall, and, subsequently, it reaches an optimum (maximum magnitude) at a certain vertical distance $(\delta _{opt})$. Away from the wall, this motion is retarded. When the coated surface faces the wall $(\theta _p=60^\circ )$, $V_z$ remains negative until a certain distance of $\delta _{cr} \approx 0.3$ from the wall; then it shows a trend of positive $V_z$. Finally, the swimmer gradually reaches a velocity that it would have attained if it were isolated. In the second configuration $(\theta _p=120^\circ )$, where the uncoated surface is nearer to the wall, the swimmer continues to translate downwards irrespective of the wall distance. For both the configurations, the location of $\delta _{opt}$ is shifted far from the wall, as the particle becomes increasingly more conductive (increasing $\mathcal {K}$). Such a consequence implies the coupled interplay between the particle-to-fluid thermal conductivity ratio and the wall distance in influencing the particle velocity.

Figure 6. Variation of translational and rotational velocity components of the microswimmer with distance from the wall $(\delta )$ and various thermal conductivity ratios $(\mathcal {K})$. The heated cap coverage is identically chosen as $\varphi _{cap} =90 ^{\circ }$ for all the cases. In (a,b), $\theta _{p} =60 ^{\circ }$ and $120\,^{\circ }$, respectively. Due to symmetry reasons, the (c) and (d) are applicable for both $\theta _{p}= 60 ^{\circ }$ or $120\,^{\circ }$.

The vertical translational velocity $V_z$ can be calculated from the following reduced version of the general force-free relation (2.13):

(3.2)\begin{equation} V_z={-}F_z^{(Thrust)}/f_z^{T}, \end{equation}

where $f_z^T$ is the resistance factor for translation in the $z$ direction. Thus, the velocity reversal in the $z$ direction can be explained solely from the behaviour of the vertical phoretic thrust, considering that the associated resistance factor, $f_z^T$, does not change its sign. As previously discussed, a variation in the swimmer-wall distance intervenes with the swimmer surface temperature distribution, and thereby its gradient along the surface is also affected. Because of this, the slip flow at the particle surface (2.9) is modified. Simultaneous to the distortions in the temperature field, the hydrodynamic stress distribution around the particle is also disturbed by the confining boundary. Both of these mechanisms work behind an altered thrust force experienced by the microswimmer (2.16).

Following the same figures, the thermal conductivity contrast $(\mathcal {K})$ can neither shift the critical distance for the velocity reversal, nor can it alter the sign of velocity in both the swimmer inclinations considered. However, for a specific wall to swimmer distance $(\delta )$, the swimmer translates slower with escalating values of $\mathcal {K}$. While uncovering the associated physical mechanism, we refer to the attenuation of the asymmetry in temperature around the coated and uncoated faces of the microswimmer with high particle conductivity (see figure 2c,f), causing attenuation in the surface temperature gradient. This occurrence can be quantitatively visualized from the fact that in figure 3(b,d), with increasing $\mathcal {K}$, the maximum temperature falls, but the minimum temperature hikes, despite their corresponding locations on the surface being hardly affected. Hence, a weaker surface temperature gradient (i.e. $\mathcal {T}_{max}-\mathcal {T}_{min}$) is generated and, eventually, the surface flow is weakened. Accordingly, a diminishing surface traction strength on the swimmer surface results. It is to be noted that, in figure 6(b), very close to the wall $( \delta \lesssim 0.05)$, the variation in $V_z$ shows a non-monotonic dependence on the thermal conductivity contrast, $\mathcal {K}$. In the narrow gap region, strong hydrodynamic stresses build up, which are critically coupled with the flow modulations due to variations in the thermal conductivity. Accordingly, the thrust force varies non-monotonically with $\mathcal {K}$, as portrayed in figure 7.

Figure 7. Variation of the vertical thermophoretic thrust $(F_z^{(Thrust)})$ with $\delta$ and for different values of $\mathcal {K}$. The parameters are similar to figure 6(b).

In figure 6(c,d), we portray the variations in wall-parallel translational and rotational velocities with the parameters $\delta$ and $\mathcal {K}$. The particle is slowed down as it approaches the wall, and it moves faster with lessened particle thermal conductivities. The near-wall rotation remains in the clockwise direction throughout, although the magnitude decays with strengthened particle thermal conductivity. While exploring other interesting facets of the velocity components, we have shown the variations in $V_x$ and $\varOmega _y$ in figures 8(a) and 8(b), respectively, for a different coverage angle of $\varphi _{cap} =140^{\circ }$ and inclination, $\theta _{p} =150 ^{\circ }$. As suggested by figure 8(a), with increasing distances from the wall, the horizontal migration velocity of the microswimmer also shows an increasing tendency. However, the swimmer is subsequently retarded beyond a specific separation height. Further, as the particle thermal conductivity reduces, the location of this maximum point shifts far from the wall. We also observe a change in the direction of the particle rotation from counterclockwise to clockwise at a fixed distance from the wall $(\delta \approx 0.11)$.

Figure 8. Variation of wall-parallel translational and rotational velocities of the microswimmer with distance from the wall $(\delta )$ and different thermal conductivity ratios $(\mathcal {K})$. Other parameters are chosen as $\varphi _{cap} =140 ^{\circ }$ and $\theta _{p} =150 ^{\circ }$.

For gaining an insight into the physical origin of such behaviours, we first look into the expressions of $V_x$ and $\varOmega _y$. Unlike the vertical velocity, the parallel rotation and translation velocities can only be obtained by solving a coupled system (2.13). This gives $V_x$ and $\varOmega _y$ in the following forms:

(3.3a,b)\begin{equation} V_x=\dfrac{{F_x^{(Thrust)} f_{y}^R}- C_y^{(Thrust)} f_{x}^R }{f_{x}^Rf_{y}^T-f_{x}^T f_{y}^R} \quad \text{and} \quad \varOmega_y=\frac{C_y^{(Thrust)} f_{x}^T- F_x^{(Thrust)} f_{y}^T}{f_{x}^Rf_{y}^T-f_{x}^Tf_{y}^R}, \end{equation}

where different hydrodynamic resistance factors are defined as $F_{x}^{(Drag,T)}=f_{x}^TV_x,$ $F_{x}^{(Drag,R)}=f_{x}^R\varOmega _y$, $C_{y}^{(Drag,T)}=f_{y}^T V_x$ and $C_{y}^{(Drag,R)}=f_{y}^R\varOmega _y$. It can be inferred from (3.3b) that the actual sign of the rotational velocity is a complex function of the hydrodynamic resistance factors as well as the thrust force and torque. From the previous investigations, it is known that these resistance factors increase in magnitude with a gradual descent of the particle towards the wall (Goldman, Cox & Brenner Reference Goldman, Cox and Brenner1967; Chaoui & Feuillebois Reference Chaoui and Feuillebois2003). Now, since the hydrodynamic resistance term in the denominator remains negative always (see figure 9), it is evident that the sign of $\varOmega _y$ is set by the relative importance of the phoretic torque ($C_y^{(Thrust)} f_{x}^T$) and the phoretic force $(F_x^{(Thrust)} f_{y}^T)$, adjusted by appropriate units. From a physical perspective, these two effects correspond to the counterclockwise torque due to the propulsive effects and the clockwise torque due to its forward movement $(+V_x)$, respectively. The counterclockwise torque is a net effect of the differently directed surface flows based on the locations of peak points in the surface temperature (a diagrammatic representation of a similar scenario has been provided later in figure 16d). The mechanism of the clockwise torque can be understood by considering the case when a lateral force causes a spherical particle to translate without any rotation parallel to a plane wall. The existence of confinement creates high velocity gradients in the lower half of the particle that faces the wall. In turn, the shear stress that the fluid exerts on the particle becomes much stronger in this region than that in the upper half away from the wall. This asymmetry in stress build-up exerts a torque on the particle. For translation along the $+x$ direction, this generated toque always acts in the clockwise direction, and the torque magnitude increases with a raise in $V_x$.

Figure 9. Detailed variation of different terms in (3.3b). The parameters are similar to figure 8(b). The yellow bubble signifies the location where $C_y^{(Thrust)}\times f_{x}^T$ becomes smaller than $F_x^{(Thrust)}\times f_{y}^T$.

Inspection of the temperature profiles at different distances from the wall reveals that the minimum temperature location shifts more towards the wall with a decreasing gap, while the maximum point remains almost unaffected. Subsequently, greater strength of the clockwise surface flow and an ensuing counterclockwise torque result. Also, as the micromotor approaches closer to the wall, it is subjected to the elevated hydrodynamic stresses because of near-wall velocity gradients. Such effects result in the relative dominance of the counterclockwise torque surpassing the opposing clockwise torque. Thus, at the reduced distances from the wall, we have a negative sign of $\varOmega _y$. However, beyond a critical height, the effect of increasing $V_x$ (see figure 8a) becomes dominant and so does the corresponding clockwise torque.

It is exciting to note that the critical gap sizes $(\delta _{cr})$ at which the particle switches its vertical swimming direction (figure 6a) and angular rotation (figure 8b) are independent of the conductivity ratio, $\mathcal {K}$. The physical origin of such behaviour can be perceived by investigating the way the factor $\mathcal {K}$ influences the thermophoretic thrust. Any characteristic shift in thrust force or torque has to be assisted by corresponding modulations in the thermophoretic slip flow around the particle. For a better understanding, we first appeal to figures 3(b) and 3(d) where it has been observed that the locations of the peaks of surface temperature, $\mathcal {T}_{S,max}$ and $\mathcal {T}_{S,min}$, remain unaltered with variations in $\mathcal {K}$. These peak locations are crucial in governing the fluid flow directions at the surface, as indicated by (2.9). As a consequence, the direction of surface flow will also remain unaltered with $\mathcal {K}$ owing to fixed peak temperature locations. This characteristic feature of surface flow is responsible for not altering the condition (i.e. the specific value of $\delta$) required for vanishing phoretic thrust force and torque.

3.4. Existence of fixed points

Here, we report the existence of fixed points in the vertical component of velocity of the self-thermophoretic microswimmer. It has been found that the critical gap size for the reversal of vertical motion, $\delta _{cr}$, depends both on the metallic coating coverage, $\varphi _{cap}$, and the swimmer orientation, $\theta _p$, as depicted in figure 10(a). Conceptually similar fixed points were reported in the literature dealing with self-diffusiophoretic microswimmers near confinements (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015a; Nasouri & Golestanian Reference Nasouri and Golestanian2020).

Figure 10. (a) Phase map demarcating the limits of metallic coating coverage, $\varphi _{cap}$, and the swimmer orientation, $\theta _p$, for the existence of fixed points in the vertical component of velocity, $V_z$. (b) Critical gap size, $\delta _{cr}$, vs. the coating coverage, $\varphi _{cap}$, for different swimmer orientations, $\theta _p$. Data points have been obtained by searching in a grid of 50 equally spaced points (in log scale) for $\delta$ and with a linear increment of $2.5^{\circ }$ for $\varphi _{cap}$. (c) The vertical velocity, $V_z$, vs. gap thickness, $\delta$, for different coating coverages and $\theta _p=50^{\circ }$.

Figure 10(a) suggests that for $\theta _p \ge 90^{\circ }$, the fixed points do not appear, while for $\theta _p < 90^{\circ }$, the fixed points appear even for very low extent of metallic coating. In addition, for low values of $\theta _p$, e.g. for $\theta _p \lesssim 10^{\circ }$, a minimum metallic coverage, $\varphi _{cap} \gtrsim 17.5 ^{\circ }$, is required for the fixed points to exist. However, for greater orientation angles, $10^{\circ } < \theta _p <90^{\circ }$, the fixed points appear even for a minimal coating extent of $\varphi _{cap} \gtrsim 2.5 ^{\circ }$. It is to be noted that the disappearance of fixed points below these minimum coating angles is due to the mathematical artefact that a minimum cutoff gap is required to be set while searching for the fixed points employing a numerical algorithm. The minimum cutoff gap has been presently chosen as $\delta =0.01$, while the same was chosen as $\delta =0.02$ in Uspal et al. (Reference Uspal, Popescu, Dietrich and Tasinkevych2015b). Similarly, the upper limit of $\delta$ during the numerical searching of fixed points has been chosen as $\delta =10$. The reader may refer to the work of Ishimoto & Gaffney (Reference Ishimoto and Gaffney2013) for the practicality behind choosing such a limiting value. This is the reason behind the vanishing fixed points in the common region near $\theta _p \to 90^{\circ }$ and $\varphi _{cap} \gtrsim 160^{\circ }$ (see the upper right corner of the grey half in figure 10a), indicating the probable fixed points beyond the highest gap considered here.

The variation of the critical gap size with the coating coverage has been highlighted in figure 10(b). The figure suggests a non-monotonic dependence of $\delta _{cr}$ on $\varphi _{cap}$, especially when the orientation angle, $\theta _{p},$ is high. In these cases, the critical gap has a minimum value at a certain value of coating coverage for the respective orientation. Any shift from these coverage extents causes an increase in the critical gap magnitude. Figure 10(c) displays the location of fixed points for varying coating coverages on the $V_z$ vs. $\delta$ curves for a specific swimmer orientation, $\theta _p=50^{\circ }$. This behaviour was not reported concerning self-diffusiphoresis (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015a). The said distinguishing non-monotonicity in the present result can be attributed to the contrasting wall-boundary conditions and the consequent variations in flow field as pointed out in the discussion following the later (3.6). Besides, the angle $\theta _{p}$ has an effect of raising the maximum gap at which a fixed point can occur, as highlighted in the inset to figure 10(b).

3.5. Swimming trajectories

The near-wall movement of the microswimmer has been captured in the present study by considering only the deterministic forces acting on the swimmer and neglecting any stochastic contribution due to thermal fluctuation to the translational and rotational diffusion. While deriving the governing differential equations, we have assumed a quasi-steady-state condition and neglected any rotational diffusion associated due to thermal fluctuations. The validity of these assumptions can be justified by considering the fact that the ensuing diffusive transport behaviour due to thermal fluctuations becomes important only when $a > ({k_B T_{ref}}/{\mu \tilde {u}_{drift}} )^{1/2}$ (Golestanian et al. Reference Golestanian, Liverpool and Ajdari2007), where $k_B$ and T _ref are the Boltzmann constant and reference temperature, respectively. With a practical value of the drift velocity as $\tilde {u}_{drift} \sim 10$ $\mathrm {\mu }$m s$^{-1}$ in water, the above conditions suggest that the particle radius (a) has to be greater than a few 100 nm for negligible enhancement in diffusive transport, consistent with the present analysis with the assumption of $a \geq 1$ $\mathrm {\mu }$m. Thus the swimming trajectories can be characterized by considering the following dynamic system:

(3.4)\begin{equation} \dfrac{\textrm{d}x}{\textrm{d}t}=V_x,\quad \dfrac{\textrm{d}z}{\textrm{d}t}=V_z, \quad \dfrac{\textrm{d}\theta_p}{\textrm{d}t}=\varOmega_y. \end{equation}

In the case of a self-diffusiophoretic micromotor near a wall (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016), the near-wall adjustments in the concentration distribution provide a cushioning effect against the steric collision between the swimmer and the wall. Conversely, in the present situation, the swimmer often approaches very close to the wall, leading to an inevitable crashing against the wall, and no further information about its motion can be retrieved beyond this point. Such a contrast arises due to different physical conditions of the concentration and temperature fields at the plane wall in the respective cases. In the case of a comparable self-diffusiophoretic swimmer, the physical circumstance of the impermeability of solutes at the wall demands a zero-flux boundary condition. Here, on the other hand, the wall is maintained isothermal by using a constant temperature bath therefore allowing a finite normal temperature gradient at the wall. The dimensionless heat flux absorbed by the cold isothermal wall from the heated fluid takes the form

(3.5)\begin{equation} q''_w= \left.\frac{\partial \mathcal{T}_{f}}{\partial z}\right|_{z=0}= \left.\frac{1-\cos(\eta)}{\sinh(\xi_0)}\frac{\partial T_f}{\partial \xi} \right|_{\xi=0}, \end{equation}

which upon using (2.7) yields

(3.6)\begin{equation} q''_w=\frac{(1-\cos(\eta))^{3/2}}{2\sinh(\xi_0)} \sum_{m=0}^{\infty}\sum_{n=m}^{\infty} (2 n+1) A_{n,m} P_n^{m}(\cos\eta)\cos(m\phi). \end{equation}

This allows for corresponding adjustments in the temperature distribution keeping the whole wall-adjacent region cool, conforming to the boundary condition of a cold wall. As a consequence, the accumulation of heat in the gap between the particle and wall is nullified. Such a physical scenario is in stark contrast to the comparable self-diffusiophoretic problem where solute molecules are forced to accumulate in the gap, leading to a fluid flow favourable for the near-wall cushioning characteristics. Another wall boundary condition analysed in relation to self-diffusiophoreis is the one having chemiosmotic slip (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2016). An analogous system for a self-thermophoretic swimmer should contain a varying temperature boundary condition at the wall, allowing for a thermo-osmotic slip (Lou et al. Reference Lou, Yu, Liu, Chen and Yang2018). This has been left as an intriguing extension of the present work.

The problem of steric collision has been circumvented by employing an electrostatic-type, short-ranged repulsive force at the plane wall (Spagnolie & Lauga Reference Spagnolie and Lauga2012):

(3.7)\begin{equation} \boldsymbol{F}_{rep}= \dfrac{\alpha_1 \exp{(-\alpha_2 \delta)}}{1-\exp{(-\alpha_2 \delta)}} \hat{\boldsymbol{e}}_{\boldsymbol{z}} . \end{equation}

The parameters $\alpha _1,\alpha _2$ have typical values 200, 100, respectively, so that the swimmer does not approach closer than $\approx$0.01 times the swimmer radius. Such a scenario was previously encountered by others in relation to ‘squirmers’ (Spagnolie & Lauga Reference Spagnolie and Lauga2012; Li & Ardekani Reference Li and Ardekani2014; Poddar, Bandopadhyay & Chakraborty Reference Poddar, Bandopadhyay and Chakraborty2020), as well as self-diffusiophoretic microswimmers (Ibrahim & Liverpool Reference Ibrahim and Liverpool2016), and different forms of repulsive potentials were employed. It is noteworthy that the squirmer models only deal with the hydrodynamics of the microswimmer, and the wall-induced distortion of the scalar field (e.g. temperature or solute) is not captured. Hence, the wall-bounded motion of an auto-thermophoretic microswimmer cannot be predicted by either of the self-diffusiophoretic or the squirmer models.

An alternative solution approach based on the singular perturbation theory can be adopted to alleviate the divergence or extreme slow convergence of the flow field in the near-contact regime, as described by others (O'neill & Stewartson Reference O'neill and Stewartson1967; Yariv & Brenner Reference Yariv and Brenner2003). However, it was shown by O'neill & Stewartson (Reference O'neill and Stewartson1967) that the presently adopted exact approach using infinite series gives accurate results even for sphere-wall distances as small as $\delta =2 \times 10^{-4}$, which is far lower than the approximate minimum distances considered in this study, i.e. $\delta \approx 0.01$ (also known as cutoff distance). In the present case, the infinite series have been truncated to $n_{max}=70$ terms for the lowest distance considered. This has been chosen to ensure that the various series coefficients in the temperature distribution ((2.7)) as well as the complementary flow fields (details in Dean & O'Neill Reference Dean and O'Neill1963, O'Neill Reference O'Neill1964 and Pasol et al. Reference Pasol, Chaoui, Yahiaoui and Feuillebois2005) converge with an error of ${\le } 10^{-6}$ between the series coefficients corresponding to $n_{max}$ and $n_{max}+1$.

Some of the limiting circumstances in the micromotor trajectory have to be treated cautiously. As $\varphi _{cap} \to 180^\circ$, the temperature asymmetry about the director $\boldsymbol {{d}}$ vanishes completely, and motion of the particle is similar to a uniformly heated particle near a wall. In this condition, asymmetric distribution of the driving influences provides an attractive force along the ‘$-z$’ direction, leading to a ‘direct impact’ onto the wall. Virtually, a similar scenario of swimmer trajectory arises if the micromotor is launched with director pointing along $`-z$’ (i.e. $\theta _{p,0}=180^\circ$, where $\theta _{p,0}$ is the initial launching orientation of the micromotor), irrespective of the coverage angle, $\varphi _{cap}$. It is to be noted that a fully covered particle should not, ideally, have a director because the concept of director exists only for an asymmetric particle. However, for consistency in mathematical analysis, that condition has been treated here as a limit when the coating coverage angle becomes increasingly higher, which is mathematically equivalent to $\varphi _{cap} \to 180^{\circ }$. The condition $\theta _{p,0}=0^\circ$ is relatively involved. In that case, consistent with the common intuition, the swimmer escapes along an upward straight line as long as the cap coverage angle ($\varphi _{cap}$) is below a critical value. Beyond this $\varphi _{cap}$, the downward attraction effect due to the wall (discussed later) becomes prominent, and a downward direct impact occurs. The ‘direct impact’ and ‘upward escape’ swimming states have been denoted by black and blue triangles, respectively, in the phase diagrams to follow.

3.5.2. Case-I: From sliding to escape with small heated cap

In figure 11(a), the swimming trajectories of the micromotor have been illustrated with the help of a phase diagram on the $\mathcal {K}-\theta _{p,0}$ plane for a cap coverage angle of $\varphi _{cap}=25^\circ$ and initial launching height of $h_0=2$. For low values of $\mathcal {K}$, until a launching orientation $\theta _{p,0} \lesssim 77.5^\circ$ (the increment of $\theta _{p,0}$ considered here is $2.5^\circ$), the micromotor only escapes away from the wall, never to return. However, when the swimmer is launched from an orientation leaning more towards the wall, the swimmer slides along the wall, keeping a small gap from it and maintaining a fixed angular orientation. The scenario changes as the thermal conductivity ratio $\mathcal {K}$ increases beyond a critical value of $\mathcal {K}_{cr} \approx 1.30$. (The critical values of $\mathcal {K}$ are found with a fine resolution until an accuracy of first two significant digits is obtained.) With the initial orientations more towards the wall, the $\mathcal {K}_{cr}$ value becomes higher, and eventually for $\theta _{p,0} \gtrsim 82.5 ^\circ$, the sliding to escape transition onsets at $\mathcal {K}_{cr} \approx 7.15$.

Figure 11. Case-I: (a) phase map of the microswimmer trajectories on the $\mathcal {K}-\theta _{p,0}$ plane. (b) Comparison of swimming trajectories for different thermal conductivity contrast with $\theta _{p,0}=90^{\circ }$. In panel (i), $\mathcal {K}=10$, and in panel (ii), $\mathcal {K}=5$. In panel (i), the microswimmer escapes from the wall and attains a fixed orientation $\theta _p= 48.7 ^{\circ }$. In panel (ii), it slides along the wall with a fixed velocity of $V_{slide}= 0.018$ with a fixed orientation $\theta _{p,end}= 33.23 ^{\circ }$. For both panels (a) and (b), we have chosen $\varphi _{cap}=25^{\circ }$ and $h_ 0=2$. (c) Variation of sliding velocity $(V_{slide})$ and final orientation angle $(\theta _{p,{end}})$ with $\mathcal {K}<\mathcal {K}_{cr}$ as per panel (b). Panels (d,e) correspond to the variations in $\theta _{p}$ and $\theta _{v}$ for the trajectories in panels (b)(i) and (b)(ii), respectively. Hereafter, the yellow arrows on micromotors indicate the instantaneous director orientations in that trajectory. Black arrows in panel (b) indicate the direction of impending motion, while the corresponding director direction $(\theta _p)$ and trajectory direction $(\theta _v)$ are shown in panels (d) and (e).

Figure 11(b) demonstrates two typical trajectories of a micromotor launched from $\theta _{p,0}=90^\circ$ with $\mathcal {K}$ values below or above the critical one. In both the cases, the micromotor shows an impending motion with $\varOmega _y<0$ and $V_z<0$. Now, similar to our previous discussions in § 3.1, a micromotor with $\mathcal {K}=5$ experiences stronger surface temperature gradient in comparison with its $\mathcal {K}=10$ counterpart. This condition promotes a subsequent raise in the horizontal component of micromotor velocity $(+V_x)$ as well as causing a faster migration in the vertically downward direction $(-V_z)$. Under the competitive driving forces for rotation (refer to § 3.3), once a critical condition is reached, the micromotor experiences a net-zero rotation rate, $\varOmega _y=0$. Here, the hydrodynamic resistance to the downward approach is not sufficient to prevent a steric collision, and the repulsive force $(\boldsymbol {F}_{rep})$ plays its role by contributing to the upward resistive force, leading to a net-zero vertical velocity, i.e. $V_z=0$. In contrast to this, with $\mathcal {K}=10$, a micromotor with a lesser amount of counterclockwise rotation also experiences a reduced hydrodynamic resistance. Still, it continues to rotate while remaining in the wall-adjacent region. Eventually, the angular orientation reaches a point where the particle gains an upward velocity component $(V_z>0)$, and it is ultimately reflected from the wall. As it moves upward, the rotation becomes diminishingly small, thereby letting the particle proceed with a fixed orientation of $48.7^\circ$.

The increasing particle conductivity not only causes a characteristic shift of the trajectories but also results in modulations in the sliding velocity and the fixed tilt angle with which the swimmer traverses (see figure 11c). As the particle-to-fluid thermal conductivity ratio increases towards the critical value, the sliding velocity diminishes, and the director tilts more towards the wall.

The figures 11(d) and 11(e) depict the variations of the angle of the propulsion direction, $\theta _v$, and the director orientation, $\theta _p$, as the particle moves in the trajectories in figure 11(b)(i) and figure 11(b)(ii), respectively. The deviations of the near-wall propulsion directions $(\theta _v)$ from the unbounded case (showing propulsion along $\theta _p$) become clear from these figures.

3.5.3. Case-II: From stopping to escape with small heated cap

Here, we present the case when the micromotor has a small heated cap, and a transition from stationary to escaping trajectories takes place for a critical value of the thermal conductivity ratio $\mathcal {K}_{cr}$. This critical condition can be realized for either of the following conditions: (i) the fluid is more conductive than the particle (demonstrated in figure 12a) or (ii) the particle is more conductive than the fluid (figure 12b). Comparing figures 12(a) and 12(b), we find that the critical thermal conductivity contrast shifts from $\mathcal {K}_{cr}\approx 0.17$ to 3.7, as the heated area increases from $\varphi _{cap}=40^\circ$ to $50^\circ$. Figure 12(c)(ii) depicts that, before the critical condition in the phase diagram is reached $(\mathcal {K}<0.17)$, the micromotor comes very close to the wall while rotating in the counterclockwise direction under strong thermophoretic propulsive torque. Soon it reaches a condition where the director $(\boldsymbol { d})$ points vertically upward, i.e. $\theta _{p,end}=0$. Such an equilibrium orientation lacks any driving force for axial migration (i.e. $V_x=0$) as well as rotation (i.e. $\varOmega _y=0$). At the same time, the downward phoretic thrust force is balanced by the tremendous hydrodynamic resistive force and the repulsive potential of the wall $\boldsymbol {F}_\text {rep}$. Thus, the motion of the swimmer is completely arrested, and it reaches a ‘stationary’ state. However, when the $\mathcal {K}$ value increases beyond $\mathcal {K}_{cr}$ (refer to figure 12c(i)), the counterclockwise rotation is weaker, and it migrates downward at a slower pace. As a result, well before the orientation reaches the $\theta _p=0$ state, the micromotor attains the critical position for the reversal of $-V_z$ to $+V_z$, a phenomenon causing a reflecting trajectory similar to figure 11(b)(i).

Figure 12. Case-II: (a,b) phase map of the microswimmer trajectories on the $\mathcal {K}-\theta _{p,0}$ plane for coverage angles $\varphi _{cap}=40 ^{\circ }$ and $50^{\circ }$, respectively. In both panels, $h_0=2$. (c) Comparison of swimming trajectories for different thermal conductivity contrast with $\theta _{p,0}=100 ^{\circ }$ and $\varphi _{cap}=40 ^{\circ }$. In panel (i), $\mathcal {K}=1$, and in panel (ii), $\mathcal {K}=0.1$. In panel (i) the microswimmer escapes from the wall and attains a fixed orientation $\theta _p= 36.5^{\circ }$. In panel (ii), it becomes stationary at position $(x_{stop},\delta _{stop})=(3.92,0.027)$ and a vertical fixed orientation, i.e. $\theta _p = 0 ^{\circ }$. In panels (d) and (e), we have shown the variations of $\theta _{p}$ and $\theta _{v}$ with $x$ for the trajectories in panels (c)(i) and (ii), respectively.

For a better insight of the trajectories of figure 12(c)(i) and (ii), we have plotted the variations of the corresponding $\theta _p$ and $\theta _v$ with $x$ in figures 12(d) and 12(e), respectively, for those trajectories. Since the particle comes to a standstill in the trajectory of figure 12(c)(ii), the angle $\theta _v$ cannot be defined as the swimmer stops. This point has also been highlighted in figures 12(e).

3.5.4. Case-III: From stopping to escape with a large heated cap

Here, the cap coverage is high $(\varphi _{cap}=145^\circ )$, but the nature of transition is almost similar to Case-II presented above. The only distinguishing factor in the transition is the height above the wall $(\delta _{cr})$ where the sign flip of $V_z$ takes place for $\mathcal {K}>\mathcal {K}_{cr}$. Similar to Case-I, $\mathcal {K}_{cr}$ is a function of the initial orientation $(\theta _{p,0})$. Figure 13(a) shows that, in the domain $0<\theta _{p,0}<12.5^\circ$, we have $\mathcal {K}_{cr} \approx 0.65$, while for $12.5^\circ \lesssim \theta _{p,0}<17.5^\circ$, we have $\mathcal {K}_{cr} \approx 1$. As illustrated in figure 13(b)(i), the micromotor never comes too close to the wall, and the critical height is just below the initial launching height $(h_0)$. This provides a sense of repulsive action from the wall at a much greater vertical distance than that in Case-II. It is to be noted that we have not plotted variations of angles $\theta _p$ and $\theta _v$ in figure 13 because those hardly provide any new information, having already shown the variations of all characteristically different trajectories (escape, slide, stop) for the previous two cases.

Figure 13. Case-III: (a) phase map of the microswimmer trajectories on the $\mathcal {K}-\theta _{p,0}$ plane for coverage angle $\varphi _{cap}=145^{\circ }$. (b) Comparison of swimming trajectories for different thermal conductivity contrast with $\theta _{p,0}=10 ^{\circ },\varphi _{cap}=145^{\circ }$ and $h_0=2$. In panel (i), $\mathcal {K}=1$, and in panel (ii), $\mathcal {K}=0.1$. In panel (i), the microswimmer escapes from the wall and attains a fixed orientation $\theta _p= 7.8 ^{\circ }$. In panel (ii), it becomes stationary at position $(x_{stop},\delta _{stop})=(0.42,0.015)$ and a vertical fixed orientation, i.e. $\theta _{p,stop} = 0 ^{\circ }$.

3.5.5. Influence of initial height, cap coverage and initial orientation

The final swimming feature of the micromotor has been summarized in figure 14(a–c) for all the possible ranges of the heated cap angle $\varphi _{cap}$ and the initial launching angle $\theta _{p,0}$. These figures help elucidate some key differences in the trajectory characteristics with those presented previously in connection with the self-diffusiophoresis phenomenon near a solute-impenetrable plane wall (Uspal et al. Reference Uspal, Popescu, Dietrich and Tasinkevych2015a; Ibrahim & Liverpool Reference Ibrahim and Liverpool2016; Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). In order to readily compare with such a widely studied problem, the geometry of the swimmer-wall system and the swimmer orientation have been defined in an exactly similar manner to those of Mozaffari et al. (Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016). Firstly, the thermophoretic force is of a strikingly distinct nature in imparting a downward motion to the micromotor even when the director initially faces away from the wall (e.g. in figure 14a). Secondly, in the corresponding problems addressed, it was found that the micromotor does not achieve a velocity of $V_z<0$ for any extent of the active area if $\theta _{p,0}<90^\circ$. In sharp contrast to these observations, in the present scenario, we observe that the micromotor can have $V_z<0$ even when $\theta _{p,0}<90^\circ$. An example of this outcome is provided in figure 17 in the Appendix for a typical heated cap coverage of $\varphi _{cap}=90^\circ$ and equal conductivities of the two media $(\mathcal {K}=1)$.Further, the phase maps in figure 14(a) demonstrate that, when the micromotor has a large heated cap, the zone of stopping points on the maps penetrates the region of $\theta _{p,0}<90^\circ$, and even a fully vertical orientation with $\theta _{p,0}=0^\circ$ can result in the case of complete stationary trajectories.

Figure 14. Phase maps categorizing the swimming behaviours of the auto-thermophoretic microswimmer for different values of the coverage angle of the metal cap $(\varphi _{cap})$ and initial launching angle $(\theta _p)$. (a), (b) and (c) correspond to different initial heights as shown. The thermal conductivity ratio has been chosen as $\mathcal {K}=1$. Here the phase maps have been constructed on a grid of $5^\circ$ intervals on both the coordinates. $(a)\, \delta_{0} = 1, (b)\, \delta_{0} = 4,\ (c) \delta_{0} = 6$

As the capped area becomes more extensive, the distortion of the temperature field becomes intrinsically different, which can be visualized by comparing the temperature profiles in figure 15(a,b). This is also reflected in the appearance of two distinct maxima points on the $\mathcal {T}_S \ \text {vs.} \ \theta$ curves for high $\varphi _{cap}$ instead of just one for low cap angle (please refer to figure 15c). Consequently, the pattern of surface slip flow is modified as portrayed schematically in figure 15(a,b). In the former case, the surface slip flow takes place from the colder region (point $\mathcal {T}_{min}$ in figure 15a) to the hotter region (point $\mathcal {T}_{max}$ in figure 15a), and the particle gains an upward velocity component $(V_z>0)$. In contrast to this, for $\varphi _{cap}=160^\circ$, the continuity of flow demands a pair of new circulating rolls due to a reorganization of the surface temperature distribution. As a cumulative effect, the downward surface traction surpasses the upward one, and, subsequently, the motion of the micromotor becomes downward ($V_z<0$). An involved relationship of $V_z$ on both $\varphi _{cap}$ and $\theta _{p}$ has been presented in figure 15(d). While this only provides an insight into the impending downward or upward motion during launching, a sliding to escape or escape to stopping transition of swimming states takes place due to qualitatively similar reasons of alterations in particle rotation $(\varOmega _y)$ and vertical motion $(V_z)$ at a critical transition, as discussed in Case-I of § 3.5.1.

Figure 15. Panels (a,b) show the direction of surface flow for low and high coverage angles $\varphi _{cap}=60^\circ$ and $160^\circ$, respectively. Here, the background colour denotes the scaled temperature field. (c) Variation of surface temperature along the surface of the micromotor for different coverage angles $(\varphi _{cap})$. Here, $\theta _{p} =10 ^ \circ$ has been chosen for (a–c). (d) Variation of the vertical component of the micromotor velocity $(V_z)$ with $\varphi _{cap}$ for different $\theta _{p}$. (e) Variation of surface temperature along the surface of the micromotor for different coverage angles $(\varphi _{cap})$ with $\theta _{p}=120^\circ$. Other parameters are chosen as $\delta =1$ and $\mathcal {K} =1$.

The mechanism by which changing cap angle $(\varphi _{cap})$ influences the swimmer surface flow is more intricate than the corresponding implication due to thermal conductivity contrast $(\mathcal {K})$. We have earlier noted that modulations in $\mathcal {K}$ hold the capacity of changing only the relative magnitudes of the peak surface temperatures $(\mathcal {T}_{max}-\mathcal {T}_{min} )$, but not the peak locations on the swimmer surface (refer to figure 3). Conversely, the cap angle $(\varphi _{cap})$ variation has the potential of shifting the locations of the peak temperatures. Also, it can create or destroy new peak points (see figure 15e for example), thereby regulating the surface flow to a great extent.

Secondly, during the swimming states where the micromotor ends up being motionless, we observe that the director is pointed vertically upward (i.e. $\theta _{p,end} = 0^\circ$) in the final state (refer to figures 12c(ii) and 13b(ii)). Contrarily, in the previously reported studies (Mozaffari et al. Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016), the final orientation of the stationary states was found to be $\theta _{p,end} = 180^\circ$. Also, in the present scenario, the swimmer becomes stationary only after sliding along the wall until it reaches the $\theta _p=0^\circ$ condition, while remaining at a close distance from the wall ($\delta _{stop} \lesssim 0.1$). However, in the self-diffusiophoresis problems, the micromotor reaches ‘stationary’ states with the final separation height being as high as ${\sim}5$ times the particle radius. The difference between the two problem categories also stems from inherently distinguished wall-induced modulations in the peak temperature or solute concentration on the surface of the autophoretic particles considered in the respective cases. Figure 16(a,b) suggest that the minimum temperature location is shifted towards the wall and attains almost a $\theta \approx 180^\circ$ position for both the high coverage angles. Moreover, in stark contrast to the study of Mozaffari et al. (Reference Mozaffari, Sharifi-Mood, Koplik and Maldarelli2016), the maximum location shifts towards the north pole with an increase in the coverage angle from $\varphi _{cap}=120^\circ$ to $150^\circ$. Following the directions of arrows and their altered lengths in figure 16(b) to (a), we find that the contribution of counterclockwise surface traction intensifies. This further causes a net negative rotation leading to $\theta _{p,end}=0^\circ$, and the same can be verified from figure 18(b) in the Appendix. Another difference with their work arises from the decreasing values of $V_x$, as the swimmer approaches the wall (see figure 18a). Subsequently, the micromotor traverses little distance before coming to rest.

Figure 16. Swimmer surface temperature variation $(\mathcal {T}_S)$ for different distances from the wall $(\delta )$. Panels (a) and (b) correspond to coverage angles of $\varphi _{cap}=120^\circ$ and $150^\circ$, respectively. (c,d) The solid-black and green-dashed arrows indicate the direction of surface flow. We have chosen $\varphi _{cap}=120^\circ$ and $150^\circ$, respectively, in the two panels. The background colours denote the scaled temperature field. The other parameters are $\mathcal {K}=1$, $\delta =0.1$ and $\theta _p=135^\circ$. The green and yellow bubbles denote the minimum and maximum temperature locations on the plots, respectively.

When the micromotor is launched from a sufficient height above the planar wall $(h_0 \sim 10)$, the wall-induced temperature distortions can hardly influence the particle velocity. Accordingly, the micromotor feels a weak attractive force from the wall. Along similar lines, a comparison of the phase maps in figure 14(a–c) reveals that the zone of stopping trajectories in the common region of $\theta _p<90^\circ$ and high $\varphi _{cap}$ shrinks as the initial height gets increased.