Page 1 of 17
European Journal of Applied Sciences – Vol. 12, No. 3
Publication Date: June 25, 2024
DOI:10.14738/aivp.123.17017.
Mba, N. U., & Abbey, T. M. (2024). The Biomechanics of Green Plants in The Presence of Environmental Thermal Perturbations.
European Journal of Applied Sciences, Vol - 12(3). 424-440.
Services for Science and Education – United Kingdom
The Biomechanics of Green Plants in The Presence of
Environmental Thermal Perturbations
Nnenna Ude Mba
Department of Mathematics, Abia State University, Uturu
Tamunoimi Michael Abbey
Department of Physics, Applied Mathematics and
Theoretical Physics Group, University of Port-Harcourt, Nigeria
ABSTRACT
This study mimics the flow model of mineral salt solution in green plants when the
environmental temperature suddenly increased. The usual Navier stokes equation
in cylindrical coordinate is used to formulate the governing system of equations
for the flow model. The analysis is then carried out by means of the homotopy
perturbation method (HPM). The result obtained shows that the increase in the
thermal parameters caused a corresponding increase in the temperature, velocity
and the quantity of fluids being transported to the leaves of the plant. It further
reveals that when the aspect ratio of the plant is very large, that is for plant whose
length is far greater than its diameter, the flow profiles tend to be streamline,
laminar and Poiseuille. This understanding shows why some green plants of this
nature tend to strive during the period of heat waves and drought. It was further
seen that when the thermal parameter γ = 0, the model returns to that of Rand [1]
and Bestman [2].
Keywords: Biomechanics, Green plant, xylem flow, homotopy perturbation method
(HPM)
INTRODUCTION
In recent times there has been a growing concern among governmental organizations,
agencies, scientist, environmentalist and climatologist as well as medical experts on the issue
of global warming as a result of thermal perturbations; which has brought about
environmental temperature rise. The global change in environmental temperature has
resulted in climate changes as well as the ocean rise and water flooding. (Abbas et al. [3];
Shahzad [4]). In fact, the impact of global thermal perturbation on biotic life within the
biomass cannot be over emphasized. Its effect can be seen in areas where heat waves and
drought have caused a lot of impact on the livelihood of man, animals and plants.
The usefulness of plants, particularly green plants are numerous. In addition to serving as
source of food and energy for man and animals, they are source of air quality balance in the
atmosphere. Their economic value as a source of timber for building and infrastructure
development has enhanced the GDP of most countries.
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Mba, N. U., & Abbey, T. M. (2024). The Biomechanics of Green Plants in The Presence of Environmental Thermal Perturbations. European Journal
of Applied Sciences, Vol - 12(3). 424-440.
URL: http://dx.doi.org/10.14738/aivp.123.17017
Due to the above usefulness, several researchers in the literature have turned attention to the
study and investigation of plants and their applications. For example, Bestman [2,5]
investigated the effect of aspect ratio (valve) (which is the ratio of the length of the plant to its
diameter) and porosity on the concentration field of a fully developed xylem and phloem flow
in green plants using the Laplace transform method. For the case of non-fully (ie partially)
developed flow, the method of perturbation and finite Fourier sine and cosine technique were
adopted. It was discovered that the velocity near the walls of the plant were nearly zero for
the fully developed case. It was also observed that for very large values of the porosity
parameter, the concentration near the wall of the valve reduced and then increased when it
was far away from the wall. More so, for smaller values of the porosity parameter, the
concentration increased near the wall of the valve. More so, it was noticed that for the non- fully developed case and at a larger aspect ratio, the concentration was slightly less than that
of the fully developed flow when their aspect ratios were equal. They further observed that as
the aspect ratio increases, the concentration increases. The velocity of the non-fully developed
flow on the other hand was observed to be higher than that of the fully developed velocity
when the aspect ratio equals one. ( ie R = 1).
Bestman [6] also considered the transient case of the problem due to daily changes in the
stem diameters resulting from dehydration, neglecting environmental thermal differences
using the Laplace transform and Fourier sine and cosine transform method. Results showed
that concentration decreased exponentially with time. Thus, at lesser time, concentration
increased from zero and then decreased as time increases. The velocity on the other hand
became fully developed at small value of the aspect ratio. Similarly, Rand and Cooke [7]
studied the flow of viscous fluid through sieve tubes with sieve plates in the phloem of green
plants using an idealized single pore axisymmetrical model. The pressure drops encountered
by the viscous fluid moving through the series of sieve tube and plate were expressed using
the resistance formula. Result showed that the resistance formula gave a very small value for
pressure drop and was about two times smaller than the value obtained using the more
realistic idealized axisymmetric case. In the same vain, Rand et al. [8] used an approximate
formula to study the same flow problem. Their result gave a more exact solution for the
pressure drop.
Similarly, Hoad [9] studied the translocation of plant hormones in the phloem of higher
plants. Problems associated with collection of sieve tube exudates and the analysis of samples
were discussed. More so, possible functions of the hormones were investigated. From the
study, it was established that mobile hormones played a part in controlling the structure of
the plant as their concentration in sieve tube have been shown to be influenced by the
environment at developmental stage. Peuke et al. [10] is reported to have used the nuclear
magnetic resonance (NMR) spectrometry to study the measurement of rates of flow in xylem
and phloem. The effects of light regime on water flows on xylem and phloem were monitored
using this same approach. It was observed that the presence of light did not change the flow of
water in the phloem very much whereas, it increased the velocity of flow in the xylem.
Transport of solutes were mainly affected by the solute concentration during loading and thus
resulting to changes in concentration. Rengel [11] studied the transport of micronutrients
(manganese and zinc) from leaves to roots, leaves and stems to developing grains and then
from one root to another in the xylem and phloem of a developing plant species. It was
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discovered that zinc solute was more mobile in phloem while manganese had poor mobility in
the phloem and therefore occurred mainly on the xylem vessel. Also, Jensen et al. [12]
presented an experimental and theoretical study of transient osmotically driven flows
through pipes with semi-permeable walls. The study considered the transport and decay of a
sudden loading of sugar in a closed water filled pipe. The equations of motion for the sugar
concentration and water velocity were solved using the method of characteristics. Result
showed that the concentration and velocity front decayed exponentially with time and the
dimensionless Munch number M. The applicability of the results to plants were discussed. It
was observed that the Munch mechanism could account for only short distance transport of
sugar in plants. Furthermore, Pitterman [13] studied the evolution of plant vascular system,
highlighted the recent developments that contributed to a better understanding of the xylem
evolution, discussed the functions of vascular structure in terms of support, drought and
freeze-thaw stress resistance and also discussed in details the impacts of plant transport on
hydrology and climate. Cabrita [14] investigated the magnitude of radial fluxes in the stem
(that is, water and solute exchanges along the long pathway) and what controlled them using
experiment and theory. A steady state model of phloem transport was constructed using the
Navier-Stokes and convection-diffusion equations. It was observed from the model that, radial
water exchange that affects the pressure gradient and solute exchange which depends on the
permeability of the phloem also affects the pressure gradient. Woodruff [15] examined how
the height of a tree together with water stress characteristics, that is, the sieve cell structural
characteristics and phloem sap composition influenced phloem transport capacity. It also
provided information on the relative contributions of changes in each of these characteristics
towards the impact of water stress on conductivity. De schepper et al. [16] carried out a
review on mechanisms and controls in phloem transport. Important differences on phloem
related data measured between trees and herbaceous species were revealed. It was observed
that not only do their loading strategies differ, but also the pressure was much higher in
herbaceous plant than in trees. The study gave a vital understanding of the phloem system on
how photosynthates/assimilates content is exchanged between sources and sinks and also,
how growth is regulated in plants. Payvandi et al. [17] studied the transport of water and
nutrient in xylem vessels of a wheat plant. Solutions to the transport of the nutrients were
obtained considering convection and diffusion. The diffusive transport showed a significant
effect. Jovanic et al. [18] investigated the effects of high pressure on the leaves of two green
plant species. It was observed that as pressure increased, the effectiveness of the
photosynthetic device decreased slowly and nonlinearly for both plant species, thus damaging
the leaf tissue. Knoblauch et al. [19] tested the Munch hypothesis of long-distance phloem
transport in plants. The aim of the study was to investigate whether the conductance of the
sieve tubes and sieve plate pore were sufficient to allow pressure flow. Observation showed
that as the distance between the source and the sink increased, the sieve tube conductance
and turgor increased dramatically thus providing a strong support for the Munch hypothesis.
Julius et al. [20] highlighted latest discoveries on the path of phloem loading of sugar
(sucrose) in rice and maize leaves and also discussed phloem loading pathways in stems and
roots and the sugars putatively involved. The study described how heat and drought stress
impact carbohydrate portioning and phloem transport; Illuminated how plant pathogens
hijack sugar transporters to obtain carbohydrate for pathogen survival and how plant
employs sugar transporters to defend against the pathogens and finally discussed the
different roles for sugar transporters in plant biology. Their discoveries provided valuable
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Mba, N. U., & Abbey, T. M. (2024). The Biomechanics of Green Plants in The Presence of Environmental Thermal Perturbations. European Journal
of Applied Sciences, Vol - 12(3). 424-440.
URL: http://dx.doi.org/10.14738/aivp.123.17017
knowledge that eventually helped mitigate the impending societal challenges due to global
climatic changes and a growing population by improving crop yield and enhancing renewable
energy production.
Recently, Uka and Olisa, [21], Uka et al. [22,23] examined the effects of free convective forces,
porosity, aspect ratio, magnetic field, Schmidt and Sherwood number on the upward flow of
soil mineral salt water on velocity, concentration and temperature fields of green plants using
the homotopy perturbation method (HPM). It was discovered that increase in the porosity,
aspect ratio, Schmidt and Sherwood numbers resulted to a decrease in the concentration of
the flow, whereas the flow velocity and temperature increases as porosity and aspect ratio
increases. Increase in the magnetic field strength decelerated the flow velocity, temperature
and concentration. The free convective forces had a positive effect on the flow by increasing
its velocity and concentration thus enhancing the growth and productivity of the plant.
This study therefore is set out to provide a mathematical model on the flow structure of green
plants in the presence of thermal perturbations or temperature rise. The following format will
be adopted. The physics of the problem and the mathematical equations (the governing
equations) is presented in section two. Section three has the method of solutions while
section four presents the analysis and discussion of the results obtained from the model for
various physical parameters.
THE PHYSICS AND MATHEMATICAL FORMULATION OF THE PROBLEM
FIG 1: The Physical Representation of a Green Plant
Various studies in the literature have shown that plant cells are distributed in such a way so
that pores are formed in between them. The number of such pores per unit volume of the
plant body is known as the porosity. Therefore, plants are generally porous in nature. Also,
plants strive in an environment. The condition of any environment will subsequently affect
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plants and animals in such environment. A sudden change in temperature has been reported
to cause a change in the amount of heat in the environment.
Experts in Agronomy have shown that heat and temperature perturbation in most cases
result in drought which may subsequently affect the internal and physical wellbeing of the
plant. It is also observed that fluid transport from the roots to the branches and leaves of a
plant is basically done by means of osmotic pressure differentials, suction pressure,
convective forces, chiefly as a result of temperature differentials. Studies have shown that
when there is excessive environmental temperature rise, more fluid are transported from the
root via the stem and then evaporated to the atmosphere via the leaves. Okuyade and Abbey
[24].
If therefore, p is the suction pressure responsible for the upward motion of the fluid in the
green plant; g is the downward gravitational pull as shown in fig (1) ; ρ is the density of the
fluid being transported; T is the temperature of the environment due to thermal perturbation;
C is the quantity of fluid per volume being transported; (T∞, C∞) are the temperature and
concentration of the fluid in the given plant at equilibrium, while d; (Tw, Cw) are the
temperature and concentration at the wall of the plant. In the absent of thermal perturbation;
(u
′
, v
′
, w
′
) are the orthogonal velocity component vector in the (r
′
, θ
′
, z
′
) coordinate (stem) of
the cylindrical tube of the green plant; Assuming the flow is fully developed and the velocity is
symmetrical about the θ
′
axis such that the variations about θ
′
is zero. The coordinate and
velocity vectors become (r
′
, z
′
) and (u
′
, w
′
) respectively. Then the mathematical model
describing the motion of the fluid in the stem of the plant, considering Boussinesq
approximation and the various conservative laws are as follows:
1
r
∗
∂
∂r
∗
(r
∗u
∗
) +
∂w∗
∂z
∗ = 0, (1)
For the mass balance
0 = −
∂p
∗
∂r
∗ + ʋ (
∂
2u
∗
∂r
∗2 +
1
r
∗
∂u
∗
∂r
∗ −
u
∗
r
∗2 +
∂
2u
∗
∂z
∗2
) −
ʋ
א
u
∗
, (2)
And
0 = −
∂p
∗
∂z
∗ + ʋ (
∂
2w
∗
∂r
∗2 +
1
r
∗
∂w
∗
∂r
∗ +
∂
2w
∗
∂z
∗2
) −
ʋ
א
w
∗
+ρgβt
(T − T∞) + ρgβc
(C
∗ − C∞) −
σB0
2u
∗
ρ
, (3)
For the momentum balance, while
ρCp (u
∗
∂T
∂r
∗ + w
∗
∂T
∂z
∗
) = αt (
∂
2T
∂r
∗2 +
1
r
∗
∂T
∂r
∗ +
∂
2T
∂z
∗2
) +
Q0
ρCp
(T − T∞) + ∇qr
, (4)
Is for the thermal energy balance of the system and
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Mba, N. U., & Abbey, T. M. (2024). The Biomechanics of Green Plants in The Presence of Environmental Thermal Perturbations. European Journal
of Applied Sciences, Vol - 12(3). 424-440.
URL: http://dx.doi.org/10.14738/aivp.123.17017
(u
∗
∂C
∗
∂r
∗ + w
∗
∂C
∗
∂z
∗
) = D (
∂
2C
∗
∂r
∗2 +
1
r
∗
∂C
∗
∂r
∗ +
∂
2C
∗
∂z
∗2
) + Kr
,
(C
∗ − C∞), (5)
For the quantity of fluid transported through the system by means of the xylem vessels.
The problem stated above shall be solved subject to the boundary conditions
(u
∗
, w
∗
)(r
∗
, z) = (U0
); T(r
∗
, z) = T∞ and C
∗
(r
∗
, z) = C∞at r
∗ = 0, (u
∗
, w
∗
)(r
∗
, z) =
(0,0); T(r
∗
, z) = Tw and C
∗
(r
∗
, z) = Cw at r
∗ = r0. (6)
א is the permeability, βt and βc are the coefficient of volume expansion for temperature and
concentration respectively, ʋ is the kinematic viscosity, Cp is the heat capacity, αt
is the
thermal conductivity, k0 is the thermal diffusivity, Q0 is the heat absorption coefficient of the
plant, σ is the Stefan Boltzmann constant, U0 is the constant velocity of the fluid in the z
direction, Kr
,
is the rate of chemical reaction of the fluid transport from the soil via root to the
stem mineral salt solution, r0 is the radius of the cylindrical plant stem, D is the mass diffusion
coefficient and qr
is the environmental heat radiation factor.
The presence of the thermal perturbation term ∇qr
in equation (4) calls for an additional
equation in order for the problem to be well posed. Therefore, following Abbey and Bestman
[25], Abbey and Mbeledogu [26], Okuyade and Abbey [27] and Ogulu and Abbey [28] and the
generalized Roseland radiative heat transfer due to thermal perturbation can be given as:
∇. (∇qr
) − 3αr
2qr − 16σαrT
3∇T = 0 (7)
Where αr
is the skin friction, which defines the optical property of the fluid. According to
Bestman [2] for a transparent atmosphere αr
is far less than one (ie αr ≪ 1). In this case,
equation (7) reduces to
∇. (∇qr
) − 16σαrT
3∇T = 0
Such that on integration
(∇qr
) = 4σαr
(T
4 − T∞
4
).
If therefore the temperature rises or difference between adjacent layers of the environmental
atmosphere is not much different from each other so that
T ≅ T∞ + ε
Where is the small temperature correction factor,
Then
(∇qr
) = 16σαrT∞
3
(T − T∞) (8)
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Mba, N. U., & Abbey, T. M. (2024). The Biomechanics of Green Plants in The Presence of Environmental Thermal Perturbations. European Journal
of Applied Sciences, Vol - 12(3). 424-440.
URL: http://dx.doi.org/10.14738/aivp.123.17017
Equations (11) – (13) now become
RK = w
′′ +
1
r
w
′ − α
2w + Grθ
(0) + Gc∅
(0)
, (16)
−PrRγw = θ
(0)
′′
+
1
r
θ
(0)′ + γ
2θ
(0)
, (17)
−ScRγw = ∅
(0)
′′
+
1
r
∅
(0)
′
+ δ
2∅
(0)
, (18)
where K is the constant pressure gradient in z direction, η is a constant, R is the aspect ratio, γ
is the environmental heat exchange parameter due to thermal perturbation, δ is the chemical
reaction parameter, α is sum of the magnetic field parameter and the porosity parameter
(M + x ), Gr is the thermal Grashof number, Gc is the concentration Grashof number, Sc is the
Schmidt number, Pr is the Prandtl number.
The above non-dimensional equations shall be solved subject to the underlisted non- dimensional boundary conditions
∅
(0)
(0) = 0; ∅
(0)
(1) = 1, w(0) = 1; w(1) = 0, θ
(0)
(0) = 0; θ
(0)
(1) = 1 . (19)
Method of Solution
Using HPM as prescribed by He [29,30,31] the homotopy form of (16)- (18) can be
constructed as follows;
(1 − p){w
′′
} + p [w
′′ +
1
r
w
′ − α
2w + Grθ
(0) + Gc∅
(0) − RK] = 0, (20)
(1 − p) [θ
(0)
′′
] + p [θ
(0)
′′
+
1
r
θ
(0) + γ
2θ
(0)
′
+ PrRηw] = 0, (21)
(1 − p) [∅
(0)
′′
] + p [∅
(0)
′′
+
1
r
∅
(0)
′
+ ScRηw + δ
2∅
(0)
] = 0. (22)
Such that w, θ
(0)
and ∅
(0)
can be taken as
w = w0 + pw1 + p
2w2 + ⋯, (23)
θ
(0) = θ
(0)
0 + pθ
(0)
1 + p
2θ
(0)
2 + ⋯, 7 (24)
∅
(0) = ∅
(0)
0 + p∅
(0)
1 + p
2∅
(0)
2 + ⋯, (25)
Substituting equations (23) – (25) into equations (20) - (22), Simplifying and rearranging
based on the powers of p-terms together with its boundary conditions, we have;
p
0
; w0
′′ = 0 with w0
(0) = 1; w0
(1) = 0,
θ
(0)
0
′′
= 0 with θ
(0)
0
(0) = 1; θ
(0)
0
(1) = 1,
∅
(0)
0
′′
= 0 with ∅
(0)
0
(0) = 1; ∅
(0)
0
(1) = 1, (26)
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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 3, June-2024
and
P
1
; w1
′′ = −
1
r
w0
′ + α
2w0 − Grθ
(0)
0 − Gc∅
(0)
0 + KR with w1
(0) = 0; w1
(0) = 0,
θ
(0)
1
′′
= −
1
r
θ
(0)
0
′
− PrRηw0 − γ
2θ
(0)
0 with θ
(0)
1
(0) = 0; θ
(0)
1
(1) = 0,
∅
(0)
1
′′
= −
1
r
∅
(0)
0
′
− ScRηw0 − δ
2∅
(0)
0 with ∅
(0)
0
(0) = 0; ∅
(0)
1
(1) = 0, (27)
while
P
2
; w2
′′′ = −
1
r
w1
′ + α
2f1 − Grθ
(0)
1 − Gc∅
(0)
1 with w2
(0) = 0; w0
(1) = 0,
θ
(0)
2
′′
= −
1
r
θ
(0)
1
′
− PrRηw1 − γ
2θ
(0)
1 with θ
(0)
2
(0) = 0; θ
(0)
2
(1) = 0,
∅
(0)
2
′′
= −
1
r
∅
(0)
1
′
− ScRηw1 − δ
2∅
(0)
1 with ∅
(0)
2
(0) = 0; ∅
(0)
2
(1) = 0, (28)
Equations (26) – (28) bring the mathematical statement of the problems to completion. Next
to be considered are their solutions
The Solution
Equations (26) – (28) presented above are the well-known second order ordinary differential
equations whose solutions following any standard text can be given as;
w0 = H1r + H2, (29)
w1 = −H1
(r In r − r) +
1
6
(α
2H1 − GrH3 − GcH5
)r
3
+
1
2
(α
2H2 − GrH4 − GcH6 + KR)r
2 + H7r + H8, (30)
w2 =
H1
2
(r(In r)
2 − 2r) − H7
(r In r − r) −
α
2H1
2
(
r
3
In r
3
−
r
3
9
)
+
α
2H1
12 r
3 +
Gr H3
2
(
r
3
In r
3
−
r
3
9
) −
Gr H3
12 r
3 +
Gc H5
2
(
r
3
In r
3
−
r
3
9
)
−
Gc H5
12
r
3 + L1r
5 + L2r
4 − L3r
3 − L4r
2 + H13r + H14, (31)
θ
(0)
0 = H1r + H2, (30)
θ
(0)
1 = −H3
(r In r − r) −
1
6
(γ
2H2 + PrRηH1
)r
3
−
1
2
(γ
2H4 + PrRηH2
)r
2 + H9r + H10, (32)
θ
(0)
2 =
H3
2
(r(In r)
2 − 2r) − H9
(r In r − r) +
γ
2H3
2
(
r
3
In r
3
−
r
3
9
)
+
PrRγH1
2
(
r
3
In r
3
−
r
3
9
) + L5r
5 + L6r
4 − L7r
3 + L8r
2 + H15r + H16, (33)
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437
Mba, N. U., & Abbey, T. M. (2024). The Biomechanics of Green Plants in The Presence of Environmental Thermal Perturbations. European Journal
of Applied Sciences, Vol - 12(3). 424-440.
URL: http://dx.doi.org/10.14738/aivp.123.17017
Fig 10: Effects of R on velocity at Pr = 7. 0, γ = K = δ = Sc = x = M = η = 1. 0,Gc = Gr = 1. 0.
Fig 11: Effects of R on temperature at Pr = 7. 0, γ = K =, δ = Sc = x = M = η = 1. 0,Gc = Gr =
1. 0.
Fig 12: Effects of R on concentration at Pr = 7. 0, γ = K =, δ = Sc = x = M = η = 1. 0,Gc = Gr =
1. 0.
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European Journal of Applied Sciences (EJAS) Vol. 12, Issue 3, June-2024
The analysis further reveals the effect of the aspect ratio on the flow model. The computations
show that the flow fields (velocity, temperature and concentration) increases with increase in
the aspect ratio. These observations are depicted in figures (10) – (12). It is seen that for very
large value of the aspect ratio R (ie R ≫ 1), the temperature and concentration fields become
extremely large compared to the velocity. This suggests that the flow assume a laminar state
and thereby the molecules of the fluid become streamline and Poiseuille in nature (Bestman
[2,5]). It follows therefore that for plants such as iroko, paw-paw, coconut tree, palm tree and
the likes that develop branches at a length which is extremely large compared to the diameter
of the plant. (ie l ≫ D0
)where D_0 is the diameter (Bestman [2]), thermal perturbation will
slightly affect the flow velocity. This may account for why these categories of trees survive
heat waves and extreme drought. The Nusselt number as illustrated in table (1) and (2) below
added credent to this fact.
Table 1: The effect of γ variation on Nusselt number (Nu)
γ R Gr/Gc Pr Nu
0.5 0.5 1.0 7.0 0.569
1.0 0.5 1.0 7.0 0.658
2.0 0.5 1.0 7.0 1.233
3.0 0.5 1.0 7.0 2.961
Table 2: The effect of Pr variation on Nusselt number (Nu)
Pr R Gr/Gc γ Nu
0.1 0.5 1.0 1.0 2.001
5.0 0.5 1.0 1.0 1.945
10.0 0.5 1.0 1.0 1.684
15.0 0.5 1.0 1.0 1.424
CONCLUSION
We have demonstrated in the forgoing a mathematical model for the biomechanics of green
plants with an attendant thermal perturbation when the environmental temperature is high
so as to cause heat waves. The model spelt out the effects of thermal perturbations on the flow
properties of green plants. This research will aid our understanding of the environmental
effects on green plants.
DECLARATIONS
Competing Interest:
The authors have no competing interests to declare that are relevant to the content of this
article.
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