Asymptotic properties of radial A+B→C reaction fronts

Abstract

If two initially separated solutions of reactants are put in contact and a simple A + B → C reaction takes place, reaction-diffusion profiles develop due to the coupling of reaction and diffusion. The properties of such fronts are well known in the case of an initially planar contact line between the two solutions. In this study one of the reactants is injected at a constant flux from a point source into a miscible solution of the other reactant so that the reaction front expands out radially. Both the leading order large time and small time asymptotic limits of the reactant concentrations and reaction front position are obtained analytically. Just as in the planar reaction front case, the position of the reaction front scales like t 1/2 and the width of the reaction front scales with t 1/6. In the large Péclet number limit the large time asymptotic properties of the radial reaction front are found to be similar to those of the planar front except that the profiles are advected with the fluid flow. The distance between the contact line and the position of the radial reaction front is 1/ √ 2 of the distance that a planar reaction fronts travels. Further, the length scales inside and outside of the reaction zone are reduced by factors of 2 1/6 and √ 2, respectively, compared to the planar reaction front.

Dynamics ofA+B → Creaction fronts under radial advection in three dimensions

“…The maximum production rate is thus constant for t 1 and the larger γ , the larger R max E which is logical as more B is then available for the reaction for a fixed concentration of A. Remarkably, this is the same behavior as for rectilinear [23,24] and 2D radial [28] RD fronts. Actually, this result should be quite general since it does not depend on the particular form of a.…”

“…For a relatively fast reaction, such that τ = 10 −3 s, a dimensionless flow rate Q = 100 corresponds thus to a flow rateQ 10 −6 ml/s for D = 10 −9 m 2 /s. Similarly, a slower reaction, such that τ = 10 3 s, leads to a flow rateQ 10 −3 ml/s for the same values of Q and D. Note also that the PDEs (3) are similar to those of the 2D radial case [27,28] except for a factor 2 in the Laplacian operator and for the fact that v r ∼ r −2 in three dimensions whereas v r ∼ r −1 in two dimensions. The coupled nonlinear PDEs (3) must be solved with the initial condition a(r > 0, 0) = c(r, 0) = 0 and b(r > 0, 0) = γ and the boundary conditions a(r → 0, t ) = 1, a(r → ∞, t ) = b(r → 0, t ) = c(r → 0, t ) = c(r → ∞, t ) = 0 and b(r → ∞, t ) = γ .…”

“…Recently, the dynamics of RDA fronts in two dimensions in the presence of radial injection of A into B has been analyzed [27,28]. Such two-dimensional (2D) RDA models are relevant, for example, in studies on infectious disease spreading [29], precipitation patterns obtained under radial injection conditions [30][31][32][33], and, more generally, material synthesis in nonequilibrium conditions [34][35][36].…”

Chemo-Hydrodynamic Patterns and Instabilities

“…This will, in turn affect the related mobility profiles if the chemical species have an active effect on density or viscosity. Note that, in flow conditions such that the reactants are passively advected, the properties of the RD fronts are recovered in a rectilinear geometry while, in radial geometries, the properties of the front can be tuned by varying the flow rate (Brau et al (2017), Trevelyan & Walker (2018)). Let us now review what happens if the chemical species are actively changing the flow, starting with the effect on viscosity before addressing the changes in density.…”

“…Note that control of the viscosity profile via extrema to obtain stabilization of fingering or on the contrary destabilization of an otherwise stable displacement can be further tuned via the addition of nanocatalysts acting on the reaction rate (Ghesmat et al (2013), Sabet et al (2017Sabet et al ( , 2018, Dastvareh & Azaiez (2019)), or the local production of foams (Kahrobaei et al (2017)) which enlarges the range of action. Interestingly, the geometry also matters as it has been shown that, in a radial injection, the fact that the local speed decreases with the radius from injection influences the production of C (Brau et al (2017), Trevelyan & Walker (2018)) and thus changes the fingering of finite size samples (Sharma et al (2019)). In the case where the reactants and the chemical product all have different viscosities, the wealth of possible different dynamics of course increases (Hejazi & Azaiez (2010a)).…”