Comparison with other buoyancy-driven convections

The chemotaxis–diffusion–convection system has many features similar to other well known buoyancy-driven flows, such as the double diffusive and Rayleigh Bénard convec-tion.

Double diffusive convection occurs in a fluid containing at least two components with different diffusivities. A destabilizing component diffuses faster than the stabilizing one [146]. The distinct diffusivities yield a density difference capable of driving the motion of fluid [147].

Comparison with a classical example of double diffusive phenomenon in oceanogra-phy can be carried out by considering two superposed fluid layers with a specific combi-nation of the temperature (T ) and the solute concentration (s), namely the salinity. The system of dimensionless equations of the double diffuse problem is the following:

∂u

5.4. NUMERICAL RESULTS AND DISCUSSION

(a) (b)

A_t t G A_t t G

0.18 0.16 0.18 0.16

0.19 0.18 0.46 0.20 0.18 1.07 0.20 0.19 0.76 0.21 0.19 1.07 0.21 0.20 0.92 0.23 0.20 1.99 0.23 0.21 1.53 0.28 0.21 3.53 0.26 0.22 2.91 0.37 0.22 8.13 0.34 0.24 6.44 0.57 0.24 16.34 0.51 0.25 13.81 0.78 0.25 17.12 0.72 0.26 17.81 0.87 0.26 8.13 0.83 0.27 9.67 0.93 0.27 4.91

Table 5.8: The predicted growth rate G of the plume amplitudes A_tfor two simulations subject to the random initial condition (5.22).

where Ra_T and Ra_m are the thermal and mass Rayleigh number, respectively, Pr_T the Prandtl number, and Le_T the Lewis number (table 5.2).

The stability of the system that exhibits a diffusive and a finger regime depends on both Rayleigh number types. In the finger regime, a small perturbation at the interface between the layers develops into a pattern of descending fingers, for instance salt fingers.

Another buoyancy-driven convection is the Rayleigh-Bénard convection that arises by fluid in a reservoir heated from below. Convection results from thermal gradient. The set of equations is as follows:

∂u

∂t+(u · ∇)u−Pr_T ∇²u+Pr_T ∇p = −Pr_T Ra_T T j,

∇ · u = 0,

∂T

∂t + u · ∇T − ∇²T = 0, T = 1 at bottom, T = 0 at top.

(5.24)

Ra_T and Pr_T are the Rayleigh and Prandtl number, respectively. The balance between the gravitational and viscous forces is expressed by the Rayleigh number Ra_T. When Ra_T is larger than a critical value that can be obtained analytically, convective patterns appear [148].

The double diffusive convection equations in (5.23) and the Rayleigh-Bénard con-vection equations in (5.24) are solved (figures 5.17 and 5.18). Plumes formed in the Rayleigh-Bénard convection process are similar in shape to plumes of the chemotaxis–

diffusion–convection. However, double diffusive and Rayleigh-Bénard convections ex-hibit both ascending and descending plumes, whereas chemotaxis–diffusion–convection is only characterized by descending plumes.

5.4. NUMERICAL RESULTS AND DISCUSSION

(a)

(b)

Figure 5.17: Numerical solution s of the double diffusive system (5.23). Red area des-ignates the region rich in s, while yellow area is poor in s. Descending plumes rich in s and ascending plumes poor in s form at the interface of two layers of fluid. The green color designates a region where the concentration of s is slightly higher than that in the yellow area due to diffusion. (a) Initial condition: s(x, y, t = 0) = 2/7, if y < 0.5, and s(x, y) = 1.0, if y ≥ 0.5. (b) s at time t=0.1 for Ra_T = 8000, Ra_m = Ra_T/6.2, Pr_T = 7 and LeT = 0.01.

Figure 5.18: Numerical solution T of the Rayleigh-Bénard system (5.24). Descending plumes of lower temperature and ascending plumes of higher temperature form in the fluid. Ra_T = 6000, Pr_T = 50.

5.4. NUMERICAL RESULTS AND DISCUSSION

I.C. for fingers Layers Any Layers

B.C. for mass Neumann Neumann Dirichlet

Physical regime Diffusive & Diffusive, Diffusive &

convective convective & convective

chemotactic

Table 5.9: Recapitulative of the physical mechanisms involved in the double diffusive convection (DDC), chemotaxis–diffusion–convection (CDC), and the Rayleigh-Bénard convection (RBC)

In each problem, the effective Rayleigh numbers depend on the temperature difference between the opposite fluid domain surfaces, the gradients of temperature and salinity be-tween the fluid layers, and the difference of density bebe-tween the bacteria and solvent, for the Rayleigh-Bénard, double diffusive, and chemotaxis–diffusion–convection systems, respectively. All of the three diffusion–convection processes have similar and distinct features (table 5.9). The involved dimensionless parameters are listed in table 5.2.

Figure 5.3 shows that the arrangement of the convection cell structure can be the same for the three systems mentioned above. A particular position of the domain will be in a clockwise cell in some simulations and counter-clockwise cell in others. This behavior is also observed in Rayleigh-Bénard convection simulations.

In the chemotaxis–diffusion–convection system, chemotaxis plays an essential role in the early stage, as it organizes the fluid domain. In a rectangular domain, starting from a given initial condition, aerotaxis of bacteria builds quasi-homogeneous layers in the

hori-5.5. CONCLUDING REMARKS

zontal direction, similarly to the double diffusive [147]. The multi-layered fluid creates a density gradient between the top of the stack layer and the bottom of the depletion layer that is similar to the temperature gradient set by the Dirichlet boundary condition imposed in the Rayleigh-Bénard case. The chemotaxis–diffusion–convection system evolves itself to a proper condition that leads to instabilities. Chemotaxis, that is not present in the other systems, brings flexibility in the choice of initial as well as boundary conditions.

5.5 Concluding remarks

In this chapter, the chemotaxis–diffusion–convection system was investigated with the focus on the differential system rather than on the experimental settings. Several simulations of the the coupled convective chemotaxis-fluid equations exhibit physically different spatially organized convection patterns.

The model for the chemotaxis–diffusion–convection system interstitial flow is able to describe three physical regimes. The convective regime exhibits the formation of plume patterns. The diffusive and chemotactic regimes are characterized by the stabilizing effect of the chemotaxis system on the fluid. The chemotaxis/fluid coupling gives a method to prevent blow-up of the solution of the chemotaxis. Incorporation of the mastocytes – interstitial fluid interaction in the model proposed in chapter 4 could give more realistic results.

The proposed comparison of buoyancy-driven flows (chemotaxis–diffusion, double diffusive, and Rayleigh-Bénard convection) shows that the dimensionless differential sys-tems and the convection patterns are similar. This analogies between these types of con-vection should launch a further analytical study of the chemotaxis–diffusion–concon-vection problem to gain a better understanding of the role of the chemotaxis in the convection system.