In the long time limit, the cavity flow becomes stationary and all variables β
) averaged over a period of villus motion become independent of time:
The time-averaged variable
averaged again over horizontal planes depends only on the vertical coordinate, y
Molecules are transferred from the source in the lid (representing bulk nutrient molecule concentration) to the lower villi-covered surface where they are absorbed at the rate
(per unit depth). The vertical flux of molecules is a summation of molecular diffusion and advection by local vertical velocity υ
. In the stationary limit, the total flux of molecules towards the lower surface, averaged in time and across horizontal planes, is independent of y
and is proportional to
In what follows, we examine the relationships between vertical flux of molecules from molecular diffusion and advection arising from interactions between the outer macro-scale flow generated by the moving lid and the inner micro-scale flow generated by the moving villi.
(a) Overview of macro–micro-scale interactions and the MML
gives an overview of the macro–micro-scale interactions that will be examined in detail. In this simulation the upper lid moves as described in §2a
, the villus length is ν
=200 μm and the villus-to-outer scale frequency ratio is fν
=40. We analyse the flow in the stationary state.
Figure 2. Fields averaged over one period of villi motion with moving lid. (a) Streamlines. (b) Isocontours of the molecular concentration field. (c) Vertical concentration flux by advection. (d) Vertical concentration flux by diffusion. (e) and (f) show time-averaged (more ...)
The time-averaged streamline patterns of a show both a macro-scale outer eddy formed by the moving lid, and micro-scale eddies generated by the oscillatory motions of the two-dimensional villi. Comparing with the concentration field in b, it is apparent that the outer eddy enhances the diffusive flux of molecules by advecting high concentration fluid at the lid to the lower surface where the molecules come into contact with the villi-induced micro-scale eddies. We also observe the presence of low concentration fluid between and on the right side of villi groups. Relatively higher concentrations exist on the left side of the villi groups with low concentration adjacent to the villi surfaces from immediate absorption. Compare the concentration and streamline fields, shown together in the inset to a. We observe that, on average, an upward flow is produced between the villi groups that carries low molecular concentrations vertically. This villi-driven upward flow interacts with the outer eddy creating an average streamline pattern that drives higher concentration fluid originating in the outer flow towards the villi on the downstream sides of the villi groups. Tentatively, it appears that lower concentration fluid is forced upward by the flow between villi groups while higher concentration fluid is forced downward over the villi by the interaction between micro- and macro-scale flows. The layer adjacent to the villi surface over which this macro–micro-scale interaction takes place is what we refer to as the MML.
Consistent with the conclusion above, c shows that in the MML the time-averaged advective flux in the micro-flow is from the outer flow towards the villi in the regions over the villi groups and from the inner flow towards the bulk flow in the gaps between the villi groups. This pattern is modulated by a macro-scale flux from the bulk flow towards the villi on the right (red isocontours) and from the villi to the bulk flow on the left (blue isocontours), enhancing transport to/from the villi in the downward/upward flow regions of the macro-scale eddy, respectively. What is particularly interesting, however, is the comparison between diffusive flux in d and advective flux in c. Whereas advection-dominated micro-scale eddies are generated just above the villi, diffusion-dominated eddies are formed in the layer just above that. Whereas the advective flux is of both signs, the diffusive flux is fully downward, and is concentrated above and between the upward/downward advective eddies that are induced by the oscillatory motion of the villi.
When averaged over horizontal planes, the summation of diffusive and advective fluxes is independent of y
)). However, as shown in f
, there is a tradeoff between advective and diffusive flux that is associated with macro–micro-scale interactions. In this flow, advective flux is overall larger than diffusive flux everywhere except in a diffusion-dominated ‘unstirred water layer’ (UWL) adjacent to the villi surfaces (red concentration in d
), and adjacent to the upper lid source. Advective flux peaks in a layer just above the UWL, while a peak in diffusive flux occurs in a layer just above that.
We define the y locations of the peaks in advective and diffusive fluxes, relative to the top of the villi surface, as δA and δD, respectively. e shows how the vertical mean concentration gradient is larger in a transitional layer between a micro-scale mixed layer below and a mixed layer above associated with the macro-scale outer flow from the outer-scale eddying motion. The thickness of the region of influence of the MML appears to scale as δD.
(b) The micro-mixing layer
In , we illustrate two related phenomena associated with the MML. Consider the time evolution of instantaneous streamlines in a. Where villi groups are moving together, mass conservation forces fluid out from the gaps between the groups, creating vertical jets that scale on the gap in width. At the same time, where the groups move apart fluid is forced into the gaps. The combination creates a single eddy over each villus group at each instant in time. However, the strengths of the eddies vary with time so that, as shown in a, on average the outward-directed motions are stronger than inward motions and the average flow is outward in each gap with a single eddy that is distorted by interaction with the macro-scale bulk flow.
Figure 3. (a) Instantaneous streamlines in the micro-mixing layer at discrete time instants over one period of villi oscillation. (b) Particle trajectories over about 50 periods of villi oscillation overlayed with particle concentrations averaged over one period. (more ...)
As a consequence of the above, the pathlines of b are consistent with the average streamline pattern of a. The particle trajectories show clearly how the MML interacts with the outer macro-scale flow and concentration fields. Fluid particles originating between villi groups are ultimately forced vertically. However, one observes from the dark blue pathlines that fluid particles are sucked into the gap as well. After several periods, the particles move vertically, and as the particles move away from the gaps and over the villi groups, the vertical oscillations in particle motion transition to horizontal oscillatory motions with the villi tips.
The fluid particles are at low concentration when they leave the UWL where rapid absorption across the villi surfaces has depleted this region of molecules. As they move into the higher concentration fluid driven towards the villi by the macro-scale eddy, their concentrations increase by local diffusion. In this flow, it took roughly 15 periods of villi oscillation for fluid particles to reach their apogee before being driven towards the villi tips by a combination of the micro-scale eddying motions and interaction with the macro-scale outer flow. The fluid particles eventually make their way to the downstream gap between villi groups where the process repeats itself, albeit at lower concentrations due to overall depletion of molecules in the macro-scale eddy by absorption. For the combination of inner and outer flow parameters in this simulation, the fluid particles passed from one gap to the next over approximately 50 periods of villi motion. We conclude that the enhancement of absorption from villi motion requires time scales of the order of the large-eddy time scale, H/Ulid.
The MML is formed, in part, by the interaction of the outer macro-scale eddying motions that, in the gut, are generated by the contractions of the wall muscle layer (muscularis propria) and the inner micro-scale motions of the villi (controlled from within the muscularis mucosae). To develop additional insight into this coupling, compare with an equivalent simulation in with the upper surface fixed, thus removing the influence of the outer macro-scale eddy. Immediately apparent from b,e is the absence of an outer macro-scale mixed layer; the outer region is fully diffusive. More interesting is a which shows that, in the absence of an interaction with an outer advection-dominated mixing flow, the time-averaged MML consists of pairs of micro-scale eddies over each villi group. Comparison with a indicates that, when the outer eddy is sufficiently strong, macro–micro-scale interactions suppress the co-rotating micro-scale eddies while enhancing the counter-rotating eddies. The consequence is that the transport of higher-concentration fluid to the villi surface by the micro-scale motions is enhanced and shifted from the centre of the villi groups to the downstream side by the macro-scale eddies.
Figure 4. Fields averaged over one period of villi motion with fixed lid. (a) Streamlines. (b) Isocontours of the molecular concentration field. (c) Vertical concentration flux by advection. (d) Vertical concentration flux by diffusion. (e) and (f) show time-averaged (more ...)
Perhaps more important from a functional perspective is the comparison of parts (c), (d) and (f) in and . Whereas the advective component of the MML is relatively unaffected by the lack of the outer macro-scale eddy—the location of peak advective flux δA is unchanged—the outer margin of the MML, as defined by peak diffusive flux δD, is nearly twice what it was in the presence of an outer-scale eddy. Furthermore, the strengths of the advective and diffusive eddies are much reduced: the inclusion of an outer macro-scale eddy within the MML increases the absorption rate by a factor of 3.6. Thus, the macro–micro-scale interactions force the MML closer to the villi, increase the concentration gradient at the villi surface, enhance the flux of molecules to the lower surface, and increase the rate of absorption.
(c) The effect of the MML on the unstirred water layer and absorption rate
In the simulation discussed in – we find that, compared with no mixing motions of any kind, macro or micro (i.e. pure diffusion), the existence of the macro-scale eddy in the absence of the MML increases the absorption rate by a factor of 5 while the addition of the MML together with the macro-scale eddy increases the absorption rate by a factor of 8. Thus, whereas both the macro-scale and micro-scale motions enhance absorption rate, the interaction between the two can provide major additional increases in the effectiveness of absorption. With our two-dimensional model and with the parameters explored thus far, advective flux is the dominant contributor to the absorption of molecules at the villi surface. However, the generation of a MML by villi motility increases both advective and diffusive contributions to flux. Here, we explore variations in villi motility parameters in the structure of the MML and enhancement of absorption by the MML.
In , we plot the time–space averages of concentration together with the relative contributions to molecule concentration flux towards the villi from diffusion and advection over a range of villi-to-outer flow frequency ratios and villi lengths. Note that the plots are a function of distance from the villi surface, so that as villus length is increased in d–f, the distance to the lid decreases. We can immediately draw several general conclusions from . Interestingly, increasing either villus frequency or villus length, while maintaining the outer macro-scale flow the same, changes the MML and macro–micro-scale interactions in similar ways. As both increase, the following changes take place: (i) what, at low frequencies and villus lengths, was a diffusion-dominated layer with monotonic decrease in mean concentration adjacent to the villus surface transitions to a mixed layer with progressively more uniform concentration, higher levels of advective flux and lower levels of diffusive flux, (ii) diffusive flux in the macro-scale region does not change, while advective flux progressively increases, driving molecules from the bulk flow to the villi at a more rapid rate and increasing absorption, and (iii) whereas the strength of the advection-dominated sublayer within the MML increases, its depth does not change significantly while the total MML thickness, defined by the peak in diffusive flux between the macro- and micro-scale flow regions, increases. Thus, as the frequency of oscillation of the villi or as the length of the villi increase, not only does the eddy strength and advective flux increase, driving molecule concentration towards the villi surface at a more rapid rate, the region of influence of the MML also grows, extending deeper into the higher concentration fluid within the macro-scale flow. suggests that significant influence of the MML on concentration flux may not be apparent until the frequency ratio fν/fL exceeds 10–20 and the villus length exceeds 100–200 μm.
Figure 5. Fields averaged over one period of villi oscillation and over horizontal planes with moving lid. (a,d) Concentration. (b,e) Concentration flux by diffusion. (c,f) Concentration flux by advection. (a–c) Varying frequency ratio fν/fL for (more ...)
The variations above are shown explicitly in where variables relevant to absorption are plotted as a function of villus length and frequency. In a
we plot the enhancement in absorption rate as a result of villi-induced micro-scale mixing, where we non-dimensionalize with
, the absorption rate at zero villus length with surface motion defined as per the RHS of equation (2.1
). The result for frequency ratio 0 in a
(solid line) is particularly important. We learn that, contrary to the commonly stated explanation for the existence of villi in the gastrointestinal tract as an enhancer of absorption simply by increasing the exposed surface area of the gut, the existence of villi in the absence of motion does very little to increase the rate of nutrient absorption at the villi surface. At a villus length of 400 μm, the increase in absorption rate is only 7 per cent. However, when the villi are in motion the enhancement can be substantial. a
suggests that when villus length exceeds 100–200 μm, the absorption rate is sensitive to villus length.
Figure 6. Enhancement of absorption rate and alteration of the UWL (diffusion layer) with increasing villus length for various oscillation frequencies. (a) Increase in scalar absorption rate relative to absorption with zero villus length. (b) Reduction in UWL thickness. (more ...)
The enhanced absorption rate with villi motion suggests a reduction in the UWL thickness, as hypothesized by Levitt et al. (1992)
. This is shown explicitly in b
, where the UWL is defined as an effective diffusion layer thickness:
is the average concentration in the cavity. δUWL
is an effective measure of the thickness of the red isocontours and high diffusive flux layer adjacent to the villi surface in d
, respectively. b
shows that the UWL thickness decreases substantially as villus length and villus frequency increase. c
shows quantitatively that, whereas the thickness of the advection-dominated portion of the MML (δA
) is relatively insensitive to villus length and frequency, the overall region of influence of the MML (δD
) grows with increasing length and frequency of villus oscillation.
In addition, in d
we plot the ratio of advective to total flux,
in the advective (filled circles) and diffusive (open circles) regions of the MML separately. d
shows that Rϕ
increases in both regions as villus length and frequency increase. We conclude that the enhancement of absorption rate from macro–micro-scale interactions results from a major enhancement of both advective and diffusive contributions to molecular concentration flux, as well as through the extension of the MML into the region of higher molecular concentration associated with macro-scale transport from the bulk flow to near the villi surface.