3.1 Residence times for equatorial release

We first investigate the residence time $\unicode[STIX]{x1D70F}$ that the substance remains in the flow before it is absorbed by the endothelial wall. In the context of reaction kinetics $\unicode[STIX]{x1D70F}$ would be denominated the first-passage time (see e.g. Hänggi, Talkner & Borkovec Reference Hänggi, Talkner and Borkovec1990). The probability distribution $p(\unicode[STIX]{x1D70F})$ is defined as

where $j_{r}=\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{n}_{cylinder}$ denotes the flux through the cylinder wall. Note that $\int _{0}^{\infty }p(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}=1$ for an initially unit amount of substance.

In figure 2(a) we show the probability distribution $p(\unicode[STIX]{x1D70F})$ of the residence time at $Pe=100$ . In the presence of the sphere the distribution is rather narrow with a sharp peak at $\unicode[STIX]{x1D70F}\approx 0.002$ . Comparing with the case when a substance is released at the same position ( $r=0.9$ ) in pure Poiseuille flow the peak height is approximately twice as large. This can be understood by the reflecting boundary conditions on the sphere surface.

The influence of the flow can be seen by varying the Péclet number as in figure 2(b). When removing the flow ( $Pe=0$ ), the distribution remains almost unaltered. Increasing the flow to $Pe=1000$ slightly smears out the distribution, but in general the effects of $Pe$ on the residence time distribution are not as strong as they are on the reaction position as will be shown below.

This is also seen in figure 2(c) where the mean residence time $\unicode[STIX]{x1D70F}_{1}$ is plotted as function of $Pe$ for the two situations with and without sphere shown in figure 2(a). In the absence of the sphere $\unicode[STIX]{x1D70F}_{1}$ (which is scaled by the radial diffusion time $r_{c}^{2}/D$ ) is constant and independent of $Pe$ since the pure Poiseuille flow has no radial components. In the presence of the sphere, however, this simple scaling breaks down and two plateaus at low and high $Pe$ can be discerned. Interestingly, even for very large $Pe$ the mean residence time in the presence of the sphere does not reach the one without the sphere. This observation may seem surprising at first since the very fast flow should quickly pull the substance away from the sphere leading to unhindered diffusion. Inspection of figure 1(c) however shows that the effect can be explained by two characteristics of the flow profile in the region behind the sphere into which the substance is dragged: (i) the radial flow is directed outwards and will thus push the substance in direction of the cylinder wall and (ii) if the solute molecule slightly diffuses towards the channel centre, the flow pushes it back onto the reflecting surface of the sphere. We will come back to this point in more detail in § 3.3.

Figure 2.

(a) Residence time probability distribution at $Pe=100$ for substance released from the equator of a flowing sphere compared to release at $r=0.9$ in pure Poiseuille flow. (b) While lowering the flow speed to $Pe=0$ has only a slight effect, increasing it to $Pe=1000$ leads to a reduction of the peak height. (c) Without the sphere the mean residence time $\unicode[STIX]{x1D70F}_{1}$ is independent of $Pe$ , while in the presence of the sphere two plateau values at small and high $Pe$ can be observed. Dotted lines represent analytical solutions for diffusion between two infinitely long cylinders at $Pe=0$ .

We finally give, for $Pe=0$ , an estimate of the mean residence time using a theoretical development for diffusion of a substance between two infinitely long concentric cylinders with radii 1 and $r_{i}<1$ . As shown by Szabo, Schulten & Schulten (Reference Szabo, Schulten and Schulten1980) and Deutch (Reference Deutch1980) (see also the review by Hänggi et al. (Reference Hänggi, Talkner and Borkovec1990)), the mean residence time when starting at $r$ can be obtained from the solution to the one-dimensional differential equation

with boundary conditions

Taking the radius of the inner cylinder the same as the sphere radius, $r_{i}=R_{s}=0.9$ , the analytical solution of (3.2) yields $\unicode[STIX]{x1D70F}_{1,cyl}=0.0048$ . If the sphere is removed by setting $r_{i}=0$ but the substance is still released at $r=0.9$ , equation (3.2) can be used to compute the corresponding mean residence time $\unicode[STIX]{x1D70F}_{r=0.9}=0.0475$ . Both values are in very good agreement with the numerical solution of the full Fokker–Planck equation at $Pe=0$ as can be seen by the dotted lines in figure 2(c) in the presence/absence of the sphere. Even up to fairly large $Pe\approx 100$ the solute dispersion in the gap between the sphere and the wall is fairly uninfluenced by the flow.

We finally consider the amount of substance remaining in the cylinder at a given time $t$ after release. This quantity can be calculated directly from the residence time distribution as

The result in figure 3 shows that (i) the presence of the sphere strongly accelerates the clearance of substance from the channel and (ii) stronger convection slows down, albeit only slightly, the clearance process.

Figure 3.

The amount of substance remaining in the channel after a time $t$ .

3.2 Reaction position for equatorial release

We now turn to investigate the probability distribution of the reaction position $\unicode[STIX]{x1D701}$ on the cylinder wall which is defined as

which is again a normalized quantity with $\int _{-\infty }^{\infty }p(\unicode[STIX]{x1D701})\,\text{d}\unicode[STIX]{x1D701}=1$ . While the mean reaction position is rather irrelevant due to the translational invariance along the $z$ -direction, the second central moment (standard deviation) $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ represents an important quantity as it determines how much the biochemically active substance is diluted before it reaches the endothelium. A small $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ implies the existence of a hot spot with a high concentration.

Figure 4(a) shows the distribution $p(\unicode[STIX]{x1D701})$ for substance released on the equator of a flowing sphere compared to the case without sphere. The effect of the sphere is to narrow the distribution due to the reflecting boundary condition on the sphere surface which closely mirrors the reduced residence time shown in figure 2(a). As can be seen in figure 4(b) the influence of the flow speed on the reaction position $\unicode[STIX]{x1D701}$ is much more pronounced than on the residence time $\unicode[STIX]{x1D70F}$ . At $Pe=0$ the system possesses a symmetry about the $z=0$ plane and accordingly the distribution turns out to be symmetric and rather narrow. At $Pe=100$ , the distribution is somewhat smeared out with a strong tendency to higher $\unicode[STIX]{x1D701}$ due to advection by the flow. At $Pe=1000$ , the peak height lowers quite dramatically and the distribution is much more smeared out.

Indeed, as can be seen by a systematic investigation of the standard deviation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ as function of $Pe$ shown in figure 4(c), $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ is almost uninfluenced by the flow until $Pe\approx 100$ , but then rises sharply and finally reaches a regime where it scales linearly with $Pe$ . Without the sphere the qualitative picture is the same, yet the transition to the linear regime occurs much earlier and more smoothly.

Figure 4.

(a) Probability distribution of the reaction position $\unicode[STIX]{x1D701}$ for substance released at $Pe=100$ on the equator of a flowing sphere compared to release at $r=0.9$ in pure Poiseuille flow. (b) Lowering the flow speed to $Pe=0$ sharpens the distribution, while increasing it to $Pe=1000$ leads to a significant reduction of the peak height. (c) Width of the reaction position distribution as function of $Pe$ . For large $Pe$ the width $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ grows linearly with $Pe$ in the presence or absence of the sphere. This behaviour is similar to the Taylor–Aris solution. In absolute numbers, the presence of the sphere reduces the width by a factor of approximately 7 compared to the Taylor–Aris solution in the limit of small $Pe$ and by a factor of 2.5 in the limit of large $Pe$ .

We now draw a connection to the classical Taylor–Aris problem of dispersion in a cylinder with reflecting boundary conditions. The Taylor–Aris solution considers times which are large compared to the diffusive time scale $r_{c}^{2}/(4D)$ which in our dimensionless units becomes $\unicode[STIX]{x1D70F}_{diff}=0.25$ , and at which therefore all radial concentration differences are negligible. The concentration then becomes a function of $z$ only which spreads with a (dimensionless) effective dispersion coefficient

Although not being applicable to perfectly absorbing boundaries, the Taylor–Aris solution nevertheless reproduces one of the characteristic features observed in the present study, namely the linear scaling of $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ with $Pe$ in the limit of large $Pe$ . To show this, we use $D_{TA}$ in a regular diffusion equation and compute the distribution width inside the flow $\unicode[STIX]{x1D70E}_{z}$ at $\unicode[STIX]{x1D70F}_{diff}$ . The result is shown as the dashed line in figure 4(c) and agrees qualitatively with our numerical data at large $Pe$ . Nevertheless, in the large $Pe$ regime, the Taylor–Aris solution overestimates the width by a factor of approximately 2.5. At low $Pe$ , the deviation is even larger. This demonstrates that the presence of the drug releasing sphere is important over the entire range of Péclet numbers.

Figure 5.

(a) Mean residence time as a function of the release position for $Pe=0$ , $Pe=100$ and $Pe=1000$ . While at $Pe=0$ the situation is perfectly symmetric, at $Pe=1000$ an upstream ( $\unicode[STIX]{x1D703}_{0}<\unicode[STIX]{x1D703}_{c}\approx 1.1$ ) and a downstream region ( $\unicode[STIX]{x1D703}_{0}>\unicode[STIX]{x1D703}_{c}$ ) can be distinguished. Analytical solutions for free diffusion and diffusion between two cylinders are shown as dotted lines and approximate surprisingly well the behaviour in the two regions. (b) The width of the reaction position shows similar behaviour. (c) Mean distance from the moving sphere centre for release on the equator, the front and the tail.

3.3 Influence of release position

We now turn to investigate changes to the above observations when the substance is released at other positions than the equator. For this, we denote the positions by their polar angle starting with $\unicode[STIX]{x1D703}_{0}=0$ at the front of the sphere as illustrated in figure 1(b).

In figure 5(a) the mean residence time is shown as function of the release position. At $Pe=0$ the system is symmetric around $\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x03C0}/2$ with the smallest $\unicode[STIX]{x1D70F}_{1}$ at $\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x03C0}/2$ , i.e., for equatorial release. At higher $Pe$ the distribution becomes strongly asymmetric and at $Pe=1000$ separates clearly into two regions with almost constant values. In the upstream region for $\unicode[STIX]{x1D703}_{0}<\unicode[STIX]{x1D703}_{c}\approx 1.1$ , the mean residence time almost equals that of free diffusion in the absence of the sphere while for $\unicode[STIX]{x1D703}_{0}>\unicode[STIX]{x1D703}_{c}$ the mean residence time decreases abruptly to around $\unicode[STIX]{x1D70F}_{1}\approx 0.009$ and is thus only slightly larger than the theoretical value for diffusion between two cylinders $\unicode[STIX]{x1D70F}_{cyl}=0.0048$ computed from (3.2). A similar behaviour is seen for $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ in figure 5(b). This separation into two regions can be understood from the appearance of the separating streamline in front of the sphere and the flow properties in figure 1(c) as follows. (i) For large $z$ the Poiseuille profile needs to be recovered which implies that the velocity on the central axis increases from $U_{s}$ to 1 when going away from the front of the sphere. Consequently, a solute molecule close to the front of the sphere will be dragged away from the sphere. (ii) Due to the symmetry of the flow profile, the same increase from $U_{s}$ to 1 occurs when going to $z=-\infty$ . Thus, a solute molecule released around the tail will, instead of being dragged away, be pushed towards the sphere. This asymmetry in solute behaviour is the cause of the different behaviour in the two regions. (iii) Since the net flow through the gap between sphere and cylinder wall is negative relative to the sphere, a particle released at a position $\unicode[STIX]{x1D703}_{0}$ even slightly smaller than $\unicode[STIX]{x03C0}/2$ , but still above the separating streamline, will be dragged into the region behind the sphere. This latter property explains why the transition between the two regions is not located strictly at $\unicode[STIX]{x03C0}/2$ , but at the smaller value of $\unicode[STIX]{x1D703}_{c}\approx 1.1$ .

Figure 6.

(a) Mean residence time as function of the Péclet number for release on the front and tail of a flowing sphere compared to release on the axis in pure Poiseuille flow and for release on the equator. While substance released at the front behaves very much like substance released in pure Poiseuille flow, a significant influence of the sphere is found for substance released around the tail and the equator. (b) The width of the reaction position shows similar behaviour.

We further illustrate this behaviour by picking the front ( $\unicode[STIX]{x1D703}_{0}=0$ ) and the tail ( $\unicode[STIX]{x1D703}_{0}=\unicode[STIX]{x03C0}$ ) of the sphere as the most obvious prototypes for the upstream and downstream regions. Figure 5(c) shows the mean distance in flow direction from the (moving) sphere centre

as a function of time. For front release the substance is advected away from the sphere in positive direction, while for tail release it moves much slower and away from the sphere in the negative direction. Apart from a short initial transient substance released at the equator follows the same behaviour as tail release.

We now turn to investigate the influence of the flow strength on $\unicode[STIX]{x1D70F}_{1}$ and $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ for front and tail release. Since both positions are located on the central cylinder axis, we discern the influence of the sphere by comparing to the case where the substance is released on the axis in pure Poiseuille flow. Figure 6(a) shows the mean residence time $\unicode[STIX]{x1D70F}_{1}$ for these three situations. In pure Poiseuille flow, the value is simply constant and equal to $\unicode[STIX]{x1D70F}_{diff}=0.25$ . As reasoned above, a substance released at the front will quickly be dragged away from the sphere and then experience free diffusion such that $\unicode[STIX]{x1D70F}_{1}$ quickly reaches a plateau identical to the one for free diffusion. For tail release, the situation is very different. A rather strong influence of the Péclet number is seen reducing the mean residence time from ${\approx}0.25$ at $Pe=0$ by almost an order of magnitude at $Pe=10^{5}$ . Near the sphere tail the substance is strongly pressed against the sphere by the incoming flow. Furthermore, in the downstream region a small, but non-negligible outward radial flow is present which further accelerates the substance’s motion towards the cylinder wall. These two effects explain the $Pe$ -dependence of $\unicode[STIX]{x1D70F}_{1}$ in the downstream region.

In an experimental situation it will be difficult to control the precise point of substance release or, equivalently, substance molecules may be released homogeneously over the entire sphere surface. We therefore calculate the mean residence time and the width of the reaction position for a weighted average over release angles $0<\unicode[STIX]{x1D703}_{0}<\unicode[STIX]{x03C0}$ covering the entire sphere surface. This is shown by the dashed black lines in figure 6 and demonstrates the non-negligible influence of the sphere in the present problem.

Figure 7.

Width of the cross-sectionally averaged distribution inside the flow for a reflecting cylinder wall. The green line in (a) shows axial release in an empty channel and is in perfect agreement with three regimes predicted by Latini & Bernoff (Reference Latini and Bernoff2001) (red lines, see main text). The blue line is for a substance released from the equator of a flowing sphere. (b) Substance released from the front and tail of a flowing sphere.

3.4 Reflecting boundaries

We now investigate the influence of the sphere if the channel walls are reflecting, i.e. we replace the boundary condition (2.3) by

In this situation, the mean residence time $\unicode[STIX]{x1D70F}_{1}$ becomes infinite and the reaction position is undefined. We therefore choose the time-dependent width (second central moment) $\unicode[STIX]{x1D70E}_{z}(t)$ of the cross-sectionally averaged concentration inside the flow as our quantity of interest. This quantity is similar, but not identical, to $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D701}}$ for the absorbing channel and allows us to make comparison to a number of earlier works, namely those by Taylor (Reference Taylor1953) and Aris (Reference Aris1956) for long times and by Latini & Bernoff (Reference Latini and Bernoff2001) for short and intermediate times. Figure 7(a) shows $\unicode[STIX]{x1D70E}_{z}(t)$ as a function of time for release on the axis in an empty channel. The dotted, dashed and solid red lines denote purely diffusive scaling $\unicode[STIX]{x1D70E}_{z}=\sqrt{2t}$ , the anomalous prediction by Latini & Bernoff (Reference Latini and Bernoff2001) $\unicode[STIX]{x1D70E}_{z}=\sqrt{8/3}\,Pe\,t^{2}$ and finally the classical Taylor–Aris scaling $\unicode[STIX]{x1D70E}_{z}=\sqrt{2D_{eff}t}$ , respectively. We find perfect agreement with those three predictions. Considering next the situation where the substance is released on the equator of the sphere, we find that for large times the Taylor–Aris regime is recovered. However, the dimensionless times required are of the order of 10 which is much higher than the mean residence times $\unicode[STIX]{x1D70F}_{1}$ found in the case of absorbing boundaries. This underlines again that the Taylor–Aris scenario is not applicable for strongly absorbing boundaries. At short times, the presence of the sphere is seen to accelerate the deviation from the purely molecular diffusive regime which can be attributed to the rather large shear rates in the vicinity of the equator (cf. figure 1 c).

In figure 7(b) we show release from the front and tail positions of the sphere. Similarly as before with absorbing walls, we find that substance released from the front is quickly dragged away from the sphere and therefore the situation without sphere is recovered even before the Taylor–Aris regime is reached. For tail release, the sphere influence is much more pronounced. At short times, there exists a regime during which the dispersion is almost zero illustrating the pushing of the substance towards the sphere by the flow. Afterwards, a sharp increase in the width is seen when the substance starts to diffuse radially away from the sphere surface. Finally, the Taylor–Aris regime is recovered, albeit more slowly than for release from the front.

3.5 Continuous release

In this final part, we consider a sphere continuously releasing substance at a rate $w$ . As the substance molecules are non-interacting, the residence time distribution for a single molecule is identical to the instantaneous release studied above and given by $p(\unicode[STIX]{x1D70F})$ . As a novel observable in the continuous release scenario, we investigate the time-dependent absorption rate observed at a fixed position as the sphere passes by. Without restriction of generality we choose the observation position at $z=0$ and consider a sphere having started its trajectory at $t\rightarrow -\infty$ , passing the observation point at $t=0$ and continuing its trajectory until $t\rightarrow \infty$ . Due to the linearity of the underlying (2.1), the continuous release scenario can be studied in a Green’s function like approach by superposing data obtained for the instantaneous release. For this, we introduce the flux $j^{\ast }$ through the entire cylinder circumference (per unit length in $z$ and per unit time) which is observed at $z$ , $t$ if the sphere has released a unit amount of substance at $z_{0}$ , $t_{0}$ . This can be related to our previous notation by integrating the flux over the azimuthal angle which (for a normalized cylinder radius of 1) yields

The total flux per unit length and time at $z=0$ is then given by superposing the contributions from all releases that have taken place since $t\rightarrow -\infty$ :

Substituting $\hat{t}=t-t^{\prime }$ yields finally

which can conveniently be computed using the instantaneous release condition on a ring studied in the previous sections. Due to mass conservation in a very long cylinder at steady state the total amount of substance absorbed by per unit length of channel wall must equal the amount of substance released per unit length by the sphere, i.e.

In figure 8(a) we show $a(t)$ for the same parameters as in the instantaneous release scenario ( $R_{s}=0.9$ ). We first note that there is a strong asymmetry around the passing moment $t=0$ . Just before $t=0$ the signal rises sharply due to substance diffusing up front relative to the approaching sphere. The maximum signal is reached for $t$ slightly larger than 0 and then decays slowly which is due to convection of the substance into the region behind the sphere as described in detail above. Comparing to a hypothetical continuous release from a moving ring source (without sphere), we observe that the signal picked up at $z=0$ becomes narrower due to the presence of the sphere. This is fully in line with the results derived above for the instantaneous injection.

Finally, in figure 8(b) we show a comparison for three different Péclet numbers. For $Pe=0$ the sphere does not move and therefore the signal has no time dependence. Increasing $Pe$ from 100 to 1000 leads to a smaller and sharper peak.

Figure 8.

The absorption rate per unit length and time observed at $z=0$ for a sphere passing at $t=0$ . In this scenario, the sphere continuously releases substance. (a) The presence of the sphere sharpens the signal. (b) Increasing $Pe$ leads to a smaller peak height in line with (3.14).