^{1}

^{*}

^{2}

^{3}

The study of the parameters influencing the absorption of atmospheric humidity not condensed by plants with aerial roots is a challenge in the current context of climatic disturbance where methods of adapting plants to water stress have become the concern of many scientific researchers. Based on Poiseuille and Fick laws, leaf water potential depending on solar radiation and atmospheric water potential around aerial roots; the influence of temperature, relative humidity, inclination angle of roots and solar radiation wavelength on the radial and axial water flow through a root segment is simulated. The various simulations show that a low temperature of the air surrounding the root favors a significant entry of water into the root as well as a relative humidity of the high air approaching 100%. The angle of inclination has an influence on the quantity of water entering the root and that rising in the xylem. These different streams of water are optimized for root tilting of 60 <sup>° </sup>. The wavelength of the red radiation received by the leaves favors more entry and rise of water in the root. Future studies for transgenes should be taken into account the influence of all these parameters.

Plants need water to grow. The water mobilization organs vary from one plant to another. Although many plants uptake water by underground roots, some of them absorb water by aerial roots. This is the case of Orchidaceae, Morgeniusae, Araceae, Liliaceae, Amaryllidaceae) [

Velamen absorbs atmospheric humidity; it is supposed that a balance is established between the velamen and the atmospheric air in which the aerial root lives once the velamen is full of water. The water potential of the velamen can therefore be approached by the atmospheric water potential in this work.

The force responsible for the entry of water into the root is the water potential gradient. According to Fick’s law [

J r = D r ∂ Ψ ∂ r (1)

Thus,

J r = k r ( Ψ S − Ψ x y l ) (2)

with: k r = D r e : Radial conductibility; Ψ S : Atmospheric water potential ; Ψ x y l : Xylem water potential.

The atmospheric water potential is given by the relation:

Ψ s = R T V ln H r (3)

The movement of water in the xylem is determined from the Navier Stokes equation for a circular tube.

ρ [ ∂ v ∂ t + ( v ∇ ) ⋅ v ] = − ∇ P + μ ∇ 2 v + ρ g cos θ k (4)

In steady state, without convection:

v ( r ) = r 2 4 μ [ ∂ P ∂ z − ρ g cos θ ] (5)

Axial water flow in the xylem:

J Z = ∫ R 0 2 π r r 2 4 μ [ ∂ P ∂ z − ρ g ] d r = − π R 4 8 μ [ ∂ P ∂ z − ρ g cos θ ] (6)

Replacing P by Ψ x y l which takes into account water pressure, osmotic potential and gravitational potential; we have:

J Z = − π R 4 8 μ [ ∂ Ψ x y l R ∂ z − ρ g cos θ ] (7)

This axial flow of water in the xylem is Poiseuille’s law [

Conservation of the water flow gives:

d 2 Ψ x y l R d z 2 = k 2 ( Ψ x y l R − Ψ s ) (8)

with: k 2 = 2 π r k r k x .

According to [

d 2 Ψ x y l T d z 2 = 0 (9)

Boundary conditions:

∂ Ψ x y l R ∂ z − ρ g cos θ = 0 ; Z = Z 2

Continuity of flows and water potential:

k x y l ∂ Ψ x y l R ∂ z = k x y l ∂ Ψ x y l T ∂ z et Ψ x y l R = Ψ x y l T ; Z = Z 1

Ψ x y l T = Ψ f ; Z = 0

The leaf water potential in this work is supposed to be related to photosynthesis. The sun, the plant are two thermodynamic systems (

To describe the energetic state of plant water, Slatyer and Taylor [

d U = T d S − P d V + ∑ k = 1 N μ k d n k (10)

d G = − S d T + V d P + ∑ k = 1 N μ k d n k (11)

U the internal energy, G Gibbs free energy, T the temperature, P the pressure, S the entropy, μ the chemical potential and n the number of moles. The sun is considered as thermal reservoir [

d U S = T S d S S (12)

d G S = 0 (13)

The photosynthesis reaction is isothermal, the system (leaf) is assumed at constant pressure and volume [

d U P = T P d S P + ∑ k = 1 N μ k d n k (14)

d G P = ∑ k = 1 N μ k d n k (15)

Plants need 60 photon molecules to synthesize a molecule of glucose [

Δ U S = − 60 N h c λ (16)

And the corresponding entropy is:

Δ S S = − 60 N h c λ T S (17)

The part of energy released by the sun and absorbed by the leaves.

Δ U P = 60 N h c λ (18)

Δ S P = 60 N h c λ T P (19)

The solar energy absorbed by the leaf is the trigger fuel for the process of photosynthesis. This energy is supposed to be totally transformed into chemical energy in the synthesis of glucose; so:

∑ k = 1 N μ k d n k = − 60 N h c λ (20)

The water potential is the variation of water potential per unit of molar volume of the water [

Ψ f = − 60 N h c λ V e a u (21)

Analytical solution:

Ψ x y l R = Ψ a t m + { ρ g cos θ k exp ( − k Z 2 ) + ρ g ( 1 − cos θ ) − 1 Z 1 ( Ψ a t m − Ψ f ) − ρ g cos θ 1 k ( 1 Z 1 − k ) exp k ( Z 1 − Z 2 ) ( 1 Z 1 − k ) exp k ( Z 1 − 2 Z 2 ) − ( k + 1 Z 1 ) exp k ( − k Z 1 ) exp ( − 2 k Z 2 ) } exp ( k z ) + ρ g ( 1 − cos θ ) − 1 Z 1 ( Ψ a t m − Ψ f ) − ρ g cos θ 1 k ( 1 Z 1 − k ) exp k ( Z 1 − Z 2 ) ( 1 Z 1 − k ) exp k ( Z 1 − 2 Z 2 ) − ( k + 1 Z 1 ) exp k ( − k Z 1 ) exp ( − k z ) (22)

Ψ x y l T = 1 Z 1 { ( Ψ a t m − Ψ f ) + ρ g cos θ k exp k ( Z 1 − Z 2 ) + ρ g ( 1 − cos θ ) − 1 Z 1 ( Ψ a t m − Ψ f ) − ρ g cos θ 1 k ( 1 Z 1 − k ) exp k ( Z 1 − Z 2 ) ( 1 Z 1 − k ) exp k ( Z 1 − 2 Z 2 ) − ( k + 1 Z 1 ) exp k ( − k Z 1 ) exp k ( Z 1 − 2 Z 2 ) + ρ g ( 1 − cos θ ) − 1 Z 1 ( Ψ a t m − Ψ f ) − ρ g cos θ 1 k ( 1 Z 1 − k ) exp k ( Z 1 − Z 2 ) ( 1 Z 1 − k ) exp k ( Z 1 − 2 Z 2 ) − ( k + 1 Z 1 ) exp k ( − k Z 1 ) exp ( − k Z 1 ) } Z + Ψ f (23)

The values in

Symboles | Description | Valeurs | Unités | Sources |
---|---|---|---|---|

λ | Wave length of radiation blue and red | 430 - 460 645 - 665 | nm nm | Standard |

h | Planck number | 6.626 × 10^{−34} | m^{2}∙kg∙S^{−1} | |

C | light speed | 3 × 10^{8} | m∙S^{−1} | |

N | Avogadro number | 6.02 × 10^{23} | mol^{−1} | |

R | Constant of the perfect gas | 8.31 | J∙mol^{−1}∙K^{−1} | |

V e a u | Molar volume of water | 18 × 10^{−3} | l∙mol^{−1} | |

ρ | Density of water in the xylem | 10^{3} | kg∙m^{−3} | |

g | Acceleration of gravity | 9.81 | m∙S^{−2} | |

r | Radius of the root | 5 × 10^{−4} | m | [ |

k x | Axial conductivity of the root | 10^{−9} | m^{4}∙Pa^{−1}∙S^{−1} | |

k r | Radial conductivity of the root | 2.5 × 10^{−7} | m∙Pa^{−1}∙S^{−1} |

this is more remarkable for the inclination of 60˚. For the wavelength 665 nm of the band of the red radiation, the various curves keeps the same pace, however, the variation of the flow of water is more remarkable in each for each inclination.

It appears from this analysis that when chlorophyll reacts in the red radiation band, the flow of water entering the root is important for the inclination of 60˚.

^{−6} m∙s^{−1} for 100% humidity for the various values of the simulation.

This analysis shows that the more water vapor in air surrounding aerial root, the more water enters the root.

This analysis shows that the entry of water into the root is favored by the low temperatures of the air surrounding the roots.

It emerges from this analysis that the axial flow does not depend linearly on the angles of inclination of the roots. The roots at an angle of 60˚ or 30˚ favor the rise of water in the xylem of the roots. Taking into account the inclinations angles of roots is important in water mobilization from the root-atmosphere interface to the root xylem.

The aerial roots of plants such as Epiphytic orchids and Ficus microcarpa mobilize atmospheric water i.e. rain, dew, and uncondensed moisture. As non-condensed humidity is an available resource, its direct mobilization by aerial roots can reduce water stress in plants. This mobilization depends on the parameters including the inclination angle of roots, the temperature, the relative humidity of the air surrounding the roots, the wavelength of the light radiation received by the leaves. The results of the various simulations show that these parameters strongly influence the entry and the rise of water in the root. Thus, for studies within the framework of improving the mechanism of plants to directly mobilize uncondensed atmospheric humidity, it is important to take into account the influence of these parameters.

This work was supported by IRITESE/CBRSI.

We authors declare that we have no competing interests.

Alexis, M.E., Basile, K.B. and Clément, K. (2018) Dynamics of Water Flow in the Atmosphere-Aerial Roots Continuum. Open Journal of Fluid Dynamics, 8, 404-415. https://doi.org/10.4236/ojfd.2018.84026