EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6505
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Thermocapillary and Photocapillary Effects on a
Spherical Droplet Embedded in a Brinkman Medium

A. S. Aamer1, Baraa A. Ahmed2, Samar A. Mahrous2,∗

1 Department of Mathematics and Computer Science, Faculty of Science,
Alexandria University, Alexandria, Egypt
2 Institute of Basic and Applied Science, College of Engineering, Arab Academy for Science,
Technology and Maritime Transport, Alexandria, Egypt

Abstract. This study develops an analytical formulation to investigate the thermocapillary and
photocapillary migration dynamics of a spherical fluid droplet suspended within a saturated, ho-
mogeneous, and isotropic porous matrix, assuming local thermal equilibrium. A uniform external
thermal gradient, parallel to an adiabatic boundary, drives the migration. The analysis oper-
ates under low Péclet (Pe ≪ 1) and Reynolds (Re ≪ 1) numbers, implying negligible advective
heat transport and inertial effects, respectively. Consequently, the thermal field is governed by
Laplace’s equation, internal droplet fluid motion by Stokes equations, and interstitial flow by the
Brinkman equation. Analytical expressions for the droplet’s translational velocities and associated
hydrodynamic forces are derived, examining the influence of the Brinkman parameter (α), thermal
conductivity ratio (k), and viscosity ratio (σ) on droplet kinematics. Furthermore, the limiting
Stokes (high permeability) and Darcy (low permeability) flow regimes are analyzed.

2020 Mathematics Subject Classifications: 76S05, 76D07, 76T30

Key Words and Phrases: Brinkman flow, Thermocapillary, Photocapillary, Spherical droplet

1. Introduction

The mobility of droplets and bubbles driven by gradients in interfacial tension—commonly
referred to as thermocapillary or Marangoni migration—is a phenomenon of fundamen-
tal importance in several scientific and practical applications. Such interfacial tension
gradients often occur from spatial changes in temperature or concentration, creating a
tangential stress at the droplet surface that leads in directed motion [1]. Thermocapil-
lary migration plays a crucial role in materials processing, microfluidics, environmental
remediation, and increased oil recovery [2–5].

The standard theoretical foundation for thermocapillary migration was provided by
Young et al. [6], who produced an analytical expression for the velocity of a spherical

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6505

Email address: samar.mahrous@aast.edu (S. A. Mahrous)

https://www.ejpam.com Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

bubble or drop migrating in an unbounded fluid subject to a uniform temperature gradient.
Assuming a creeping flow and negligible convective heat transfer, their model revealed that
the migration velocity is proportional to the thermal gradient and the surface tension’s
temperature coefficient. Although immensely important, this concept applies exclusively
to isolated drops in idealized situations.

In practical applications, migrating droplets often engage with adjacent interfaces, such
as confining walls or other droplets, which considerably modify their movement. Over the
past few decades, a vast variety of analytical and numerical investigations has addressed
such confinement effects [7–11]. Meyyappan et al. [7] and Sadhal [12] examined the
migration of gaseous bubbles near planar surfaces, utilizing spherical bipolar coordinates
to elucidate quasi-steady dynamics. Meyyappan and Subramanian [13] expanded this
concept to examine bubble migration parallel to a barrier, demonstrating a monotonic
reduction in velocity relative to the unbounded scenario as the bubble neared the interface
[14, 15].

Subsequent inquiries examined the thermocapillary movement of fluid droplets adja-
cent to planar surfaces. Barton and Subramanian [16] and Chen and Keh [17] expanded
the analysis to liquid droplets, whereas Barton and Subramanian [18] offered experimental
validation for droplets traveling perpendicular to a horizontal surface. Comparable theo-
retical findings were achieved through boundary collocation by Chang and Keh [19], and
for spherical cavities by Lee and Keh [20, 21]. Asymptotic methods, including lubrication
theory [22] and the method of reflections [23, 24], elucidated the impact of proximity to
interfaces on droplet mobility.

Recently, sophisticated numerical methods have facilitated more profound inquiries.
Choudhuri and Raja Sekhar [25] examined thermocapillary drift through decoupled solu-
tions for thermal and momentum transfer. Chiu and Keh [26] examined droplet motion in
cylindrical tubes, demonstrating that the droplet’s velocity is influenced by both viscous
drag and wall-induced alterations to thermal gradients. Chennuri and Prakash [27] inves-
tigated a concentric compound drop in arbitrary flow, developing closed-form formulas for
migration velocity and hydrodynamic forces, while emphasizing the effects of Marangoni
number and inner drop radius.

Although extensive research has been conducted on thermocapillary migration in trans-
parent fluids, comparable phenomena in porous media have received comparatively scant
attention. The interaction between interfacial thermal gradients and hydrodynamic re-
sistance caused by the porous structure poses a distinct challenge in these environments.
Although the Darcy and Darcy–Brinkman equations have been employed to predict flow in
porous media [28], previous research has largely overlooked the influence of temperature-
dependent surface tension in these systems.

This study examines the thermocapillary and photocapillary movement of a spherical
fluid droplet within a Brinkman medium. Our objective is to comprehend how porous
permeability influences both flow and heat fields, therefore affecting migration velocity.
The Darcy–Brinkman model is adopted to bridge the classical Darcy flow regime (low per-
meability) with Stokes flow (high permeability), enabling direct comparison with existing
clear-fluid results. To the best of our knowledge, this is one of the first analytical studies



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

to consider thermocapillary (and photocapillary) effects in a Brinkman medium.

2. Modeling of Thermal Energy Transport Through a Porous Medium

In this section, we present the energy conservation equation to describe heat transfer
within a porous medium based on the first law of thermodynamics. The porous medium
is assumed to be homogeneous and isotropic. Let Ts and Tf denote the temperatures
of the solid and fluid phases, respectively, with the assumption that there is thermal
equilibrium between these two phases such that Ts = Tf = T . This assumption implies
that at every point in the medium, the solid matrix and the interstitial fluid are at the
same temperature. Furthermore, we neglect several effects to simplify the analysis of the
current problem, including; heat generation due to radioactive decay, viscous dissipation,
and work done by pressure changes. Additionally, the Péclet number of the fluid phase is
assumed to be small, which implies that conductive heat transfer dominates over convective
heat transfer. Under these assumptions, and considering a steady-state regime with no
internal heat sources, the governing energy equation for the porous medium reduces to:

∇ · kp∇T = 0, (2.1)

where kp represents the effective thermal conductivity of the porous medium, defined as:

kp = (1− ϕ)ks + ϕkf , (2.2)

here, ks and kf are the thermal conductivities of the solid and fluid phases, respectively,
and ϕ is the porosity of the medium.

3. Thermocapillary Driven Migration of a Spherical Droplet Embedded
in a Porous Medium

We examine a spherical droplet, denoted as Sd, with radius a and thermal conductivity
kd, situated within an infinite porous medium having effective thermal conductivity kp.
Let (r, θ, ϕ) represent a system of spherical coordinates with origin O located at the center
of the droplet Sd. The corresponding unit vectors in this coordinate system are e⃗r, e⃗θ,
and e⃗ϕ. At large distances from the droplet, the porous medium is subjected to a uniform
and constant temperature gradient, denoted by E∞, directed parallel to the z-axis, as
illustrated in Fig. 1. In the current formulation, we assume that all thermal properties
of both the droplet and the porous medium are constant and do not vary with position
or temperature. The problem is analyzed in two stages: first, we drive the temperature
distribution fields within the droplet and the surrounding porous medium; and second, we
develop the corresponding flow field induced by thermocapillary effects on the droplet and
the Brinkman porous medium.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Figure 1: Geometrical sketch for a spherical droplet Sd embedded in a porous medium.

3.1. Thermal Distributions of The Spherical Droplet and The Porous
Medium

According to the assumptions stated earlier, the energy equation governing the temper-
ature distribution Td(r, θ) for the viscous fluid inside the spherical droplet, with thermal
conductivity kd, is given by the steady-state Laplace equation:

∇2Td = 0, r < a. (3.1)

Similarly, for the porous medium surrounding the droplet with effective thermal conduc-
tivity kp, the temperature distribution satisfies:

∇2Tp = 0, r > a. (3.2)

It should be noted that equations (3.1) and (3.2) omit the convective heat transfer term.
This is justified under the assumption of low Reynolds number and small droplet size,
where the fluid motion is slow and the corresponding Péclet number is small. In such
regimes, heat conduction dominates over convection [29], allowing the convective effects
to be neglected. However, if the Péclet number is of order 0.1 or higher, convective effects
may become significant and should not be disregarded [30]. The boundary conditions at
the droplet surface (r = a) require that both the temperature and the normal heat flux be
continuous across the interface. Additionally, far from the droplet, the fluid temperature
should approach the linear profile imposed in the ambient medium, while inside the droplet
it remains finite. Now the boundary conditions are stated as:

Tp =Td, r = a,

kp
∂Tp
∂r

=kd
∂Td
∂r

, r = a,

 (3.3)

Td is finite, r < a. (3.4)

We place the origin of the coordinate system at the droplet center. A linear temperature
field T∞(z) is imposed in the far field of the surrounding fluid, described by:

T∞ = T0 + E∞z = T0 + E∞r cos θ, (3.5)



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

where E∞ is unvarying and T0 is also a constant that represent the reference temperature
at the droplet center. The capillary number Ca is assumed to be small (that is, the surface
tension dominates viscous forces), ensuring that the droplet maintains a spherical shape
under thermocapillary migration. Solving the Laplace equations with the above boundary
conditions yields the following expressions for the temperature distributions in the porous
medium and inside the droplet:

Tp = T0 + E∞

[
1− k − 1

2 + k

(a
r

)3]
r cos θ, (3.6)

Td = T0 +
3E∞
2 + k

r cos θ, (3.7)

where k = kd/kp is the ratio of thermal conductivities of the droplet to the porous medium.

3.2. Fluid Flow Inside Both The Droplet and The Porous Medium in The
Case of Thermocapillary Migration

The governing equations for the slow, steady, and incompressible flow of a viscous
fluid through a porous medium are derived from the Brinkman model. This model is a
modification of the classical Darcy’s law and incorporates viscous shear effects by including
a Laplacian term. Assuming the absence of body forces and negligible inertial effects (i.e.,
low Reynolds number flow), the equations simplify and can be written as:

∇ · q⃗ = 0, (3.8)

µ̃∇2q⃗ − µ

K
q⃗ = ∇P, (3.9)

where q⃗ is the volume-averaged fluid velocity vector, P denotes the pore-averaged pressure,
µ is the dynamic viscosity of the fluid, and K is the permeability of the porous medium.
The term µ̃ represents the effective viscosity, which accounts for both the fluid and the
porous matrix interactions. These equations model the fluid dynamics inside the porous
domain, where the Laplacian term

(
∇2q⃗

)
captures the internal viscous stresses, and the

term
( µ
K q⃗
)
models the Darcy drag due to the porous structure. The associated stress

tensor Π within the Brinkman framework is formulated as:

Π = −PI + 2µ̃∆, (3.10)

where I is the identity tensor and ∆ is the rate-of-strain tensor given by:

∆ = 1
2

(
∇q⃗ + (∇q⃗)T

)
. (3.11)

This formulation is particularly suitable for modeling flow at micro-scales, such as in
thermophoretic or thermocapillary scenarios where the transport is driven by temperature
gradients and the effects of convective inertia are neglected as in the current study. Let U



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

be the thermocapillary velocity of the droplet in the z− direction to be determined. The
motion is axisymmetric in which

q⃗(r, θ) = qre⃗r + qθe⃗θ. (3.12)

From the incompressibility equation (3.8), accordingly we obtain the velocity components
qr and qθ in terms of the Stokes’ stream function ψ, as

qr = − 1

r2 sin θ

∂ψ

∂θ
, qθ =

1

r sin θ

∂ψ

∂r
. (3.13)

Substituting in the field equation (3.9), we obtain an equation satisfied by the stream
function in a dimensionless form as:

E2
(
E2 − α2

)
ψ(p) = 0, r > 1, (3.14)

where,

E2 ≡ ∂2

∂r2
+

1− ζ2

r2
∂2

∂ζ2
, ζ = cos θ, α =

√
µa2

µ̃K
.

The pressure within the porous medium, is given by integrating the following relations

∂P (p)

∂r
= − µ̃

r2 sin θ

∂

∂θ

(
E2 − α2

)
ψ(p),

∂P (p)

∂θ
=

µ̃

sin θ

∂

∂r

(
E2 − α2

)
ψ(p).

 (3.15)

Whereas, the vorticity vector is given by:

ω⃗p =
e⃗ϕ

2r sin θ
E2ψ(p). (3.16)

Now, inside the droplet, given the same conditions applied as the flow inside the porous
medium, the governing equation would be the simplified Stokes equation, that is, by
removing the Darcy’s term from equation (3.9). Similarly, by incorporating the stream
function ψ(d), the Stokes equation becomes:

E4ψ(d) = 0, r < 1. (3.17)

The porous stream function ψ(p) can be defined as ψp = ψ
(p)
1 + ψ

(p)
2 , where E2ψ

(p)
1 = 0

and
(
E2 − α2

)
ψ
(p)
2 = 0, on that, equations (3.15) and (3.16) become

∂P (p)

∂r
=

µ̃α2

r2 sin θ

∂ψ
(p)
1

∂θ
,
∂P (p)

∂θ
= − µ̃α2

sin θ

∂ψ
(p)
2

∂r
, (3.18)

ω(p) =
α2ψ

(p)
2

2r sin θ
. (3.19)



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

To complete the formulation of the present problem, the boundary conditions for the
fluid velocity components along the droplet/porous interface must be specified. Here, we
consider thermal capillarity along the surface of the interface at r = 1:

q(p)r = q(d)r , (3.20)

q
(p)
θ = q

(d)
θ , (3.21)

q(p)r = U cos θ, (3.22)

τ
(p)
rθ − τ

(d)
rθ = −

(
∂γ

∂T

)
1

r

∂Tp
∂θ

, (3.23)

as r → ∞,
ψ(p)

r2
→ 0, (3.24)

where γ is the surface tension. The general solution of equation (3.17), that is bounded
for r → 0, is given by:

ψ(d) =
(
A(d)r4 + C(d)r2

)
sin2 θ. (3.25)

Also, the general solution of equation (3.14), that satisfy condition (3.24), is:

ψ(p) =

(
A(p)

r
+B(p)

(
1

r
+ α

)
e−α(r−1)

)
sin2 θ. (3.26)

Using (3.13), the components of the velocities of the flow through the porous medium and
inside the droplet are stated below;
equation (3.13) and equation (3.25) give,

q(d)r = −2(A(d)r2 + C(d)) cos θ, (3.27)

q
(d)
θ = (4A(d)r2 + 2C(d)) sin θ. (3.28)

equation (3.13) with (3.26) give,

q(p)r = −2

(
A(p)

r
+B(p)

(
1

r3
+
α

r2

)
e−α(r−1)

)
cos θ, (3.29)

q
(p)
θ = −

(
A(p)

r3
+B(p)

(
1

r3
+
α

r2
+
α2

r

)
e−α(r−1)

)
sin θ. (3.30)

The shear stress expression in terms of the velocity components, is given by:

τrθ = µ

[
1

r

∂qr
∂θ

− qθ
r

+
∂qθ
∂r

]
,

for the fluid droplet:

τ
(d)
rθ = µ[6A(d)r sin θ], (3.31)



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

and for the porous medium:

τ
(p)
rθ = µ̃

[(
6A(p)

r4
+B(p)

(
6

r4
+

6α

r3
+

3α2

r2
+
α3

r

)
e−α(r−1)

)
sin θ

]
. (3.32)

We have also,
1

r

∂Tp
∂θ

= − 3E∞
2 + k

. (3.33)

Applying the boundary conditions (3.20)-(3.23), we obtain the following set of simultane-
ous equations for determining the unknown constants:

A(d) −A(p) + C(d) − (1 + α)B(d) = 0, (3.34)

4A(d) +A(p) + 2C(d) + (1 + α+ α2)B(p) = 0, (3.35)

A(p) +B(p)(1 + α) = −U
2
, (3.36)

and

µ̃

[(
6A(p)

r4
+B(p)

(
6

r4
+

6α

r3
+

3α2

r2
+
α3

r

)
e−α(r−1)

)
sin θ

]
− µ[6A(d)r sin θ]

= −
(
∂γ

∂Td

)(
−3E∞
2 + k

sin θ

)
,

by simplifying the above equation, it becomes:

6A(p) + (6 + 6α+ 3α2 + α3)B(p) − 6σA(d) =
3U (0)

2 + k
, (3.37)

where, σ = µ/µ̃, and characteristic velocity U (0) = βE∞/µ̃, β is the constant (−∂γ/∂Td).
Solving the above system we obtain,

A(p) = −
(
U
3(2 + 3σ)(1 + α) + 3(1 + σ)α2 + α3

2
(
3(1 + σ)α2 + α3

) + U (0) 6(1 + α)

(2 + k)
(
3(1 + σ)α2 + α3

)),
(3.38)

B(p) = 3
2U

(3k + 6)σ + 2k + 4

γ1
+ U (0) 3

γ1
, (3.39)

A(d) = 3
4U

((1 + α)k + 2α+ 2)

γ2
− U (0) 3/2

γ2
, (3.40)

C(d) = −1
4U

(6k + 12)σ + (5α+ 9)k + 10α+ 18

γ2
+ U (0) 3/2

γ2
, (3.41)

where,
γ1 = α2

(
(3k + 6)σ + (α+ 3)k + 2α+ 6

)
,

and
γ2 = (3k + 6)σ + (α+ 3)k + 2α+ 6.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

3.3. Thermocapillary Velocity of The Droplet

The formula for the hydrodynamic drag force exerted by the porous medium on an
axially symmetric droplet has the form

F = µ̃α2

(
UVp + 4π lim

r→∞

rψ(p)

sin2 θ

)
, (3.42)

where Vd is the volume of the droplet. The expression (3.42) can be derived in the same
manner as that obtained by Lawrence and Weinbaum [31] for the axisymmetric motion
of an oscillating body in a viscous fluid. Inserting the expression (3.26) for the stream
function into (3.42) we obtain

F = 4
3πµ̃α

2(U + 3A(p)). (3.43)

Since the droplet is freely suspended in the porous medium, this force must vanish and
equation (3.43) leads to U + 3A(p) = 0, that gives the thermocapillary velocity of the
spherical droplet in the porous medium as

UT =
18(α+ 1)

(2 + k)
(
9(2 + 3σ)(1 + α) + 3(1 + σ)α2 + α3

)U (0). (3.44)

The normalized thermocapillary velocity is therefore

U∗
T =

UT

U (0)
=

18(α+ 1)

(2 + k)
(
9(2 + 3σ)(1 + α) + 3(1 + σ)α2 + α3

) . (3.45)

For a clear fluid, i.e. (α = 0), we have:

U0 =
2U (0)

(2 + k)(2 + 3σ)
. (3.46)

This recovers the result of Luzhang et al. [1]. Knowing the thermocapillary velocity UT

from equation (3.44) into equation (3.26), the porous stream function ψ(p) would be:

ψ(p) =−
[(
12(1 + α)− 6(1 + αr)eα(1−r)

)
U (0) +

(
(2 + k)(6 + 9σ + α(6

+ 9σ + α(3 + 3α+ 3σ)))− 3(2 + k)(1 + αr)(2 + 3σ)eα(1−r)
)
U

]
sin2 θ

δ1
,

(3.47)

δ1 = 2(2 + k)rα2(3 + α+ 3σ).

Also, the vorticity in equation (3.19) becomes:

ω(p) =−
[(
12(1 + α)− 6(1 + αr)eα(1−r)

)
U (0) +

(
(2 + k)(6 + 9σ + α(6

+ 9σ + α(3 + 3α+ 3σ)))− 3(2 + k)(1 + αr)(2 + 3σ)eα(1−r)
)
U

]
sin θ

δ2
,

(3.48)

δ2 = 2(2 + k)r2(3 + α+ 3σ).



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

3.4. Thermocapillary Force Acting on The Droplet

To find the thermocapillary force exerted on the droplet due to the prescribed tem-
perature gradient ∇T∞, we have to consider boundary condition (3.22) with vanishing
particle velocity (i.e. U = 0) into the stream functions given by (3.25) and (3.26). In-
serting (3.27)-(3.33) into the boundary conditions (3.20)-(3.23) (with U = 0), we get four
simultaneous equations in the unknown constants A(p), B(p), A(d), and,C(d), the solution
of which give A(p) in the following form,

A(p) =
3(α+ 1)U (0)

γ1
. (3.49)

Again, by using formula (3.43) (with U = 0), the thermocapillary force acting on the
droplet would be:

FT =
12πµ̃(α+ 1)

(3k + 6)σ + (α+ 3)k + 2α+ 6
U (0). (3.50)

Now, the normalized thermocapillary force acting on the droplet is,

F ∗
T =

FT

12πµ̃U (0)
=

FT

12πE∞β
=

α+ 1

(3k + 6)σ + (α+ 3)k + 2α+ 6
. (3.51)

For very small α, the asymptotic expansion of F ∗
T is given by:

F ∗
T =

1

3(k + 2)(σ + 1)

[
1 + α (1− δ) + α2

(
−δ + δ2

)]
+O(α3) (3.52)

where, δ = 1/(3(σ + 1)). At (α = 0) the force would correspond to a clear fluid and is
given by:

F0 =
1

3(k + 2)(σ + 1)
, (3.53)

which recovers the result of Luzhang et al. [1]. It is obvious that in the presence of a
porous structure, the normalized force is greater than the viscous counterpart.

4. Photocapillary Effects on the Spherical Droplet

In the context of photocapillary phenomena, the temperature distribution within the
porous medium remains governed by the same formulation as in equation (3.2). How-
ever, the temperature inside the droplet—now absorbing electromagnetic radiation—is
described by a new energy balance equation [32, 33]:

∇2T̃d = − 1

kd
Q(r, ζ). (4.1)

Here, Q(r, ζ) represents the volumetric energy generation [34] due to the absorption of
incident radiation. For monochromatic radiation, the heat generation term is given by:

Q(r, θ) =
4πνk̃I

λ
B(ξ, ζ), (4.2)

in this equation:



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

• ν and k̃ are the real and imaginary parts of the droplet’s complex refractive index.

• λ is the incident radiation wavelength.

• B(ξ, ζ) = |E(ξ,ζ)|2
|E0|2 is a dimensionless electric field distribution where E(ξ, ζ) is the

electric field of the particle and E0 is the incident electric field strength.

• ξ = r
a , with a being the droplet radius.

The boundary conditions remain similar to those discussed in the thermocapillary effects
section, with a new outer condition as follows; matching the temperature and heat flux at
the droplet interface (ξ = 1):

T̃p = T̃d, kp
∂T̃p
∂ξ

= kd
∂T̃d
∂ξ

, (4.3)

with
T̃p → T∞ as r → ∞. (4.4)

We expand the internal temperature field as:

T̃d =
aI

kd

∞∑
n=0

Fn(ξ)Pn(ζ). (4.5)

Substituting into the heat equation from (4.1) yields:

d

dξ

(
ξ2
dFn

dξ

)
− n(n+ 1)Fn = −4πνk̃a

λ
ξ2Ωn(ξ), (4.6)

where,

Ωn(ξ) =
1

2
(2n+ 1)

∫ 1

−1
B(ξ, ζ)Pn(ζ) dζ. (4.7)

Solving via variation of parameters, the complete solution is given as:

Fn(ξ) = A(ph)
n ξ−(n+1) +B(ph)

n ξn + Sn(ξ), (4.8)

where, A
(ph)
n and B

(ph)
n are unknown constants to be determined, Sn(ξ) is the particular

solution of equation (4.6), that is given by:

Sn(ξ) =
4πνk̃a3

λ(2n+ 1)

[
ξn
∫ 1

ξ
t−(n−1)Ωn(t)dt+ ξ−(n+1)

∫ ξ

0
tn+2Ωn(t)dt

]
. (4.9)

Now the temperature profiles (as solutions of equations (3.2) and (4.1)) are as follows:

T̃p = T∞ +
aI

k

∞∑
n=0

B(ph)
n ξnPn(ζ), (4.10)



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

T̃d = T∞ +
aI

k

∞∑
n=0

(A(ph)
n ξ−(n+1) + Sn(ξ))Pn(ζ). (4.11)

Applying the boundary conditions at the droplet interface (4.3), the unknowns A
(ph)
n and

B
(ph)
n are found to be:

A(ph)
n =

nSn(1)− S′
n(1)

nk + n+ 1
, B(ph)

n =
−(n+ 1)Sn(1) + kS′

n(1)

nk + n+ 1
, (4.12)

where, S′
n = dSn/dξ and k = kd/kp. The first term (at n = 1) in the series of the

temperature profile at equation (4.11) would give A
(ph)
1 as:

A
(ph)
1 =

J1
k + 2

, (4.13)

where,

J1 = S1(1)− S′
1(1) =

3πνk̃a

λ

∫ 1

0

∫ π

0
ξ3B(ξ, θ)dθdξ. (4.14)

4.1. Fluid Flow Equations in Photocapillary Regime

We now repeat the same procedure as outlined in section (3.2), which encompasses
applying the same boundary conditions. However, we replace every occurrence of ψ(d)

with ψ̃(d), and every ψ(p) with ψ̃(p), this yields

ψ̃(d) =
(
Ã(d)r4 + C̃(d)r2

)
sin2 θ, (4.15)

ψ̃(p) =

(
Ã(p)

r
+ B̃(p)

(
1

r
+ α

)
e−α(r−1)

)
sin2 θ. (4.16)

To determine the unknowns Ã(d), C̃(d), Ã(p), and B̃(p), we apply the boundary conditions
(3.20)-(3.23) using the updated thermal distribution results for the photocapillary case.
This leads to the following system of four simultaneous equations:

Ã(d) − Ã(p) + C̃(d) − B̃(p)(1 + α) = 0, (4.17)

4Ã(d) + Ã(p) + 2C̃(d) + B̃(p)(1 + α+ α2) = 0, (4.18)

Ã(p) + B̃(p)(1 + α) = −U
2
, (4.19)

and,

µ̃

[(
6Ã(p)

r4
+ B̃(p)

(
6

r4
+

6α

r3
+

3α2

r2
+
α3

r

)
e−α(r−1)

)
sin θ

]
−µ
[
6Ã(d)r sin θ

]
= −

(
∂γ

∂T

)
1

r

∂T̃p
∂θ

,



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

implying the following:
1

r

∂T̃p
∂θ

=
IJ1 sin θ

k(k + 2)ξ3
,

the fourth equation becomes:

6Ã(p) + B̃(p)(6 + 6α+ 3α2 + α3)− 6σÃ(d) =
U

(0)
PhIJ1

k(k + 2)
, (4.20)

where σ = µ/µ̃, and characteristic velocity U
(0)
Ph = 1

µ̃ (∂γ/∂T ). Now, we proceed to solv-
ing this linear system of equations to determine the unknowns, resulting the following
expressions:

Ã(p) =− 1

2
U
(
k2 + 2k

)
α3 (4.21)

+
((3k2 + 6k)(σ + 1))α2 + ((9σ + 6)k2 + (18σ + 12)k)(α+ 1)

γ3
− 1

2
U

(0)
Ph

2IJ1(α+ 1)

γ3
,

B̃(p) =
3

2
U
k2(3σ + 2) + 2k(3σ + 2)

γ3
+ U

(0)
Ph

IJ1
γ3

, (4.22)

Ã(d) =
3

4
U
(α+ 1)k2 + 3(α+ 1)k

γ4
− 1

2
U

(0)
Ph

IJ1
γ4

, (4.23)

C̃(d) = −1

4
U
(5k2 + 10k)α+ 6σk2 + 9k2 + 12σk + 18k

γ4
+

1

2
U

(0)
Ph

IJ1
γ4

, (4.24)

where,
γ3 = α2k(α(k + 2) + 3k(σ + 1) + 6(σ + 1)),

γ4 = k(α(k + 2) + 3k(σ + 1) + 6(σ + 1)).

4.2. Photocapillary Velocity of The Droplet

Again, we repeat the same procedure described in Section (3.3) to derive the following
relation: U+3Ã(p) = 0. From this, we obtain the expression for the photocapillary velocity
as:

UPh =
6IJ1(1 + α)

(α3 + 3α2(σ + 1) + 9(α+ 1)(3σ + 2)) k(k + 2)
U

(0)
Ph , (4.25)

The normalized photocapillary velocity is then given by:

U∗
Ph =

UPh

IJ1U
(0)
Ph

=
6(1 + α)

(α3 + 3α2(σ + 1) + 9(α+ 1)(3σ + 2)) k(k + 2)
. (4.26)

For a clear fluid (α = 0), this simplifies to:

U
(Ph)
0 =

6IJ1U
(0)
Ph

(27σ + 18)k(k + 2)
. (4.27)



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

4.3. Photocapillary Force Acting on The Droplet

To determine the photocapillary force, we follow the same steps as in Section (3.4). We
assume that the particle velocity vanishes (i.e., U = 0) in the boundary condition (3.22),
hence we solve equations (4.17)–(4.20) under this new condition. Consequently, we obtain
Ã(p) in the form:

Ã(p) =
IJ1(1 + α)

γ3
U

(0)
Ph . (4.28)

Inserting (U = 0) into equation (3.43), the resulting photocapillary force is given by:

FPh =
4IJ1(1 + α)

γ4
U

(0)
Ph . (4.29)

And the normalized photocapillary force is simply:

F ∗
Ph =

1 + α

γ4
. (4.30)

5. Results and discussion

5.1. Thermocapillary

In this section, we present the results of the normalized thermocapillary velocity U∗
T

given by (3.45), and the normalized thermocapillary force F ∗
T given by (3.51) of a spherical

droplet embedded in a porous medium. The parameters of the problem are as follows:

1. The permeability of the porous medium that characterize the nature of the porous
medium, α =

√
µa2/µ̃K. Values of α << 1 means high permeability medium, that

is K >> 1, with α = 0 representing a viscous clear fluid, and α >> 1 reduces the
medium to a Darcian medium of low permeability K << 1.

2. The viscosity ratio σ = µ/µ̃, which represents the ratio between the viscosity of the
fluid inside the droplet and the effective viscosity of the porous medium, 0 ≤ σ <∞.
The value σ = 0 represents a bubble embedded in a porous medium, while σ → ∞
represents a rigid particle.

3. The conductivity ratio k = kd/kp, which represents the ratio between the thermal
conductivity of the fluid inside the droplet and the overall thermal conductivity of
the porous medium. Note that the value of k = 0, which does not exist in practice, is
covered in our graphs for comparison purposes, despite its non-existence in practice.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Figure 2: Normalized Thermocapillary velocity U∗
T versus permeability parameter α for

different values of thermal conductivity ratio k at specific values of viscosity ratio(σ =
0 , σ = 3).

Figure 3: Normalized Thermocapillary velocity U∗
T versus permeability parameter α for

different values of viscosity ratio σ at specific values thermal conductivity ratio (k =
0 , k = 3).



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Figures 2 and 3 show the plots of U∗
T against the permeability parameter characterizing

the porous medium for various values of viscosity ratio σ, and the thermal conductivity
ratio k. As depicted in Fig. 2 and Fig. 3, the presented results reveal a clear dependence
of the dimensionless thermocapillary velocity U∗

T on the porous medium’s permeability
α, the droplet-to-medium viscosity ratio σ, and the droplet-to-medium thermal conduc-
tivity ratio k. A consistent attenuation of U∗

T is observed with increasing permeability
parameter α, signifying the escalating hydrodynamic drag imposed by the porous matrix;
this effect is most pronounced as α transitions from values near unity to higher values,
culminating in negligible velocity in the Darcy limit (α >> 1) where matrix resistance
dominates, contrasting with maximal velocity in the Stokes limit (α = 0) where such re-
sistance is minimal. The physics underlying this trend is the increased energy dissipation
due to the solid matrix obstructing flow. Furthermore, for fixed α and σ, an increase in
the thermal conductivity ratio k leads to a reduction in U∗

T . This is attributable to the
droplet becoming more thermally conductive relative to the surrounding medium, which
tends to homogenize the temperature across the droplet’s surface, thereby diminishing the
interfacial temperature gradients that are crucial for generating the Marangoni stresses
responsible for migration. Conversely, at constant α and k, U∗

T increases as the viscosity
ratio σ decreases (i.e., the droplet becomes less viscous relative to the external fluid).
Reduced internal droplet viscosity promotes stronger internal circulation and surface mo-
bility due to interfacial thermocapillary stresses, leading to a higher overall translational
velocity. These interconnected influences highlight how the balance between the thermo-
capillary driving force, shaped by thermal properties, and the resistive forces, governed
by both fluid viscosities and the porous matrix structure, dictates the droplet’s migratory
behavior.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Stokes flow Darcian flow
Figure 4: Normalized Thermocapillary Force F ∗

T versus permeability parameter α for
different values of thermal conductivity ratio k at specific values viscosity ratio (σ =
0 , σ = 3).

Figures 4 and 5 show the plots of the non-dimensional thermocapillary force F ∗
T against

the permeability parameter α for various values of the viscosity ratio σ, and the thermal
conductivity ratio k. The observed behavior of the non-dimensional thermocapillary force
F ∗
T highlights the crucial interplay between internal fluid dynamics, interfacial phenomena,

and overall droplet propulsion. F ∗
T increases monotonically with the permeability param-

eter α because higher permeability signifies reduced internal resistance to flow, leading to
pronounced internal circulation and superior surface mobility, thereby enhancing the net
propulsive force. Consequently, F ∗

T is minimized as α approaches zero, where the droplet
behaves akin to a rigid particle with suppressed internal convection, and its asymptotes to
a constant value as α approaches infinity, indicating that internal permeability is no longer
the limiting factor and F ∗

T becomes governed by the magnitude of the interfacial temper-
ature gradient (influenced by thermal conductivity ratio, k) and external fluid properties.
Furthermore, for any given α and k, F ∗

T increases with a decreasing viscosity ratio σ,
as lower internal viscosity reduces internal viscous dissipation, enabling the Marangoni
stresses to more efficiently convert thermal energy into kinetic energy of internal and
surface flows, thus augmenting the overall thermocapillary force.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Stokes flow Darcian flow
Figure 5: Normalized Thermocapillary force F ∗

T versus permeability parameter α for differ-
ent values of viscosity ratio σ at specific values thermal conductivity ratio (k = 0 , k = 3).

-3

Figure 6: Normalized Thermocapillary force F ∗
T versus viscosity ratio σ for different values

of permeability parameter α at specific values of thermal conductivity ratio k at (k =
1 , k = 10) .



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

The inverse relationship between the non-dimensional thermocapillary force F ∗
T and the

viscosity ratio σ, as depicted in Fig. 6, is primarily attributable to the augmented internal
hydrodynamic resistance within the droplet. An elevated σ, corresponding to an increased
internal droplet viscosity, intensifies the viscous dissipation of mechanical energy derived
from the temperature-gradient-induced Marangoni stresses at the interface. This increased
internal dissipation not only impedes the establishment of robust internal convective flows,
but also diminishes the droplet’s surface mobility. Given that the net thermocapillary
force is a manifestation of the efficient conversion of these interfacial stresses into coherent
internal and surface kinematics, an elevated internal viscosity inherently diminishes the
efficiency of converting interfacial energy into directed motion, leading to an attenuated
propulsive force irrespective of the specific values of the thermal conductivity ratio k or
permeability α.

Figure 7: Normalized Thermocapillary force F ∗
T versus thermal conductivity ratio k for

different values of permeability parameter α at specific values of viscosity ratio σ at (σ =
1 , σ = 10) .



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Figure 8: UT /U0 versus permeability parameter α for different values of viscosity ratio σ.

The observed dependencies of the non-dimensional thermocapillary force F ∗
T and the

normalized translational velocity UT /U0 on the thermal conductivity ratio k, permeability
parameter α, and viscosity ratio σ, as illustrated in Figures 7, 8, and 9, provide critical
insights into the physics governing thermocapillary migration within a porous medium,
which align broadly with and extend established principles of thermocapillary migration
and droplet dynamics in complex environments. Specifically, Fig. 7 reveals that F ∗

T de-
creases monotonically with an increasing thermal conductivity ratio k. This phenomenon
arises because a higher k implies that the droplet is more thermally conductive relative to
its surroundings. Consequently, heat diffuses more readily through the droplet bulk, which
can lead to a homogenization of the droplet’s internal temperature and thereby reduce the
magnitude of the temperature gradient sustained along its interface. A diminished inter-
facial temperature gradient directly translates to weaker Marangoni stresses, which are
the driving mechanism for thermocapillary motion, thus resulting in a lower propulsive
force and velocity, which is consistent with foundational theories [4, 6].



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Figure 9: UT /U0 versus viscosity ratio σ for different values of permeability parameter α .

Figure 10: Normalized thermocapillary velocity U∗
T versus thermal conductivity ratio k

for different values of permeability parameter α at specific values of viscosity ratio(σ =
0.5 , σ = 2).



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Figure 11: Normalized thermocapillary velocity U∗
T versus thermal conductivity ratio k

for different values of viscosity ratio σ at specific values of permeability parameter(α =
0 , α = 20).

Furthermore, the corresponding trends observed for UT /U0 with respect to the perme-
ability parameter α, Fig. 8 and the viscosity ratio σ, Fig. 9, mirroring the behavior of F ∗

T ,
underscore the direct translation of induced force into droplet kinematics. An increase
in α, signifying reduced internal resistance to flow or enhanced ease of navigating the
porous matrix, facilitates more powerful internal circulation and droplet deformation in
response to Marangoni stresses, thereby augmenting UT /U0. The intriguing limit where
limα→0 UT /U0 = 1 indicates that as the droplet’s internal resistance to deformation be-
comes exceedingly high (or its ability to effectively utilize internal flows to navigate the
pore space diminishes), its normalized velocity in the porous medium approaches that ob-
served in an unconfined, clear viscous fluid U0; this suggests that U0 acts as a baseline for a
droplet whose motion is primarily limited by its intrinsic properties rather than the specific
additional impedance of the porous structure under such limiting conditions. Finally, the
inverse relationship between F ∗

T and UT /U0 and the viscosity ratio σ is well-documented
[6, 35]; higher internal droplet viscosity leads to increased internal viscous dissipation,
damping both internal convective flows and surface mobility, thereby inherently limiting



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the efficiency with which interfacial thermal energy is converted into kinetic energy for
propulsion. These collective findings underscore the intricate coupling between interfacial
thermophysics, internal droplet fluid mechanics, and the hydrodynamics imposed by the
porous medium in dictating thermocapillary-driven transport.

Fig. 10 illustrates the dependence of the normalized thermocapillary velocity U∗
T on

the thermal conductivity ratio k for varying permeability parameter α at specific values of
viscosity ratio (σ = 0.5, σ = 2). The parameter U∗

T consistently exhibits a characteristic
non-linear decrease as k increases, featuring an initial phase of relatively stable values
at low thermal conductivity ratio (k < 10−1) where the droplet effectively sustains an
external temperature gradient. This is followed by a significant reduction in U∗

T as k
increases (10−1 < k < 102) due to enhanced internal thermal homogenization, with U∗

T

ultimately approaching zero at high thermal conductivity ratio (k > 102). The viscosity
ratio markedly influences velocity magnitude: (σ = 0.5 - less viscous droplet) results
in substantially higher U∗

T values (reaching initial stable values near 0.28 for α = 0)
compared to (σ = 2 more viscous droplet, with initial stable values near 0.125 for α =
0), a difference attributed to reduced viscous resistance at lower σ. Interestingly, the
permeability parameter α demonstrates a varied impact depending on σ. For (σ = 0.5),
U∗
T is largely independent of α, while for (σ = 2), increasing α from 0 to 20 systematically

lowers U∗
T . This suggests that for more viscous droplets, higher permeability may induce

internal flows that either disrupt the externally imposed interfacial temperature gradient
or enhance internal thermal equilibration, thereby diminishing thermocapillary propulsion.

Fig. 11 shows the normalized thermocapillary velocity, U∗
T , as a function of the ther-

mal conductivity ratio, k, for varying viscosity ratios (σ = 0, 0.1 , 1, and σ → ∞) at two
distinct levels of the permeability parameter: (α = 0 impermeable droplet, upper set of
curves) and (α = 20 highly permeable droplet, lower set of curves). A consistent trend
observed across all conditions is the decrease of U∗

T with increasing k, transitioning from
an initial region where U∗

T exhibits near-constant values at low thermal conductivity ratio
(k < 10−1) to near-zero values at high thermal conductivity ratio (k > 102). This general
behavior reflects the shift from a scenario where the droplet effectively sustains an interfa-
cial temperature gradient (low k) to one where enhanced internal thermal homogenization
diminishes this gradient (high k). For an impermeable droplet (α = 0, upper curves), the
viscosity ratio σ exerts a pronounced and systematic influence. U∗

T is maximized for an
inviscid droplet (σ = 0), reaching a near-constant value of approximately 0.5 in the low
k region. As σ increases to 0.1 and then to 1, this near-constant velocity progressively
decreases to around 0.43 and 0.2, respectively. In the limit of an infinitely viscous droplet
(σ → ∞, effectively a rigid particle), U∗

T reaches its minimum for the (α = 0) case. This
monotonic decrease in U∗

T with increasing σ is due to greater internal viscous dissipation,
which increasingly impedes the internal fluid circulation driven by Marangoni stresses,
thereby reducing the overall translational velocity. A striking observation is the dominant
effect of high permeability when (α = 20 - lower curves). In this regime, U∗

T is extremely
attenuated across all investigated viscosity ratios, with values consistently remaining very
low. Furthermore, the sensitivity of U∗

T to the viscosity ratio σ is substantially dimin-
ished when permeability is high; the curves for different σ values at (α = 20) are closely



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

grouped. This indicates that high permeability facilitates such efficient internal thermal
equilibration likely through enhanced internal advective heat transfer pathways enabled
by the permeable structure that it effectively short-circuits the externally imposed ther-
mal gradient. Consequently, the thermocapillary driving force is significantly reduced,
irrespective of the droplet’s internal viscosity.

Fig. 12 shows the streamlines of the two flows under study, the flow through the porous
medium outside form the droplet (r > a) and the flow within the droplet itself (r < a).
The (a), (b) and (c) figures show the variations in the streamlines for both regions with
different values of the porosity parameter α, the viscosity ratio σ and the conductivity
ratio k, while the last figure (d) show the case of a solid particle by employing the limiting
case of σ reaching infinity.

(a) (b)

(c) (d)

Figure 12: The streamline function is scaled with respect to Ua2 for translational motion:
ψ(i)/Ua2 = C1, i = d, p, where C1 is a constant. Each contour is found by assigning a
specific value to the constant C1, with (a) α → 0, σ = 0.5, k = 1, (b) α = 5, σ = 1, k =
0.155, (c) α→ ∞, σ = 3, k = 2, (d) α→ 0, σ → ∞, k = 2 (Solid Particle).



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5.2. Photocapillary

The investigation focuses on the non-dimensional photocapillary velocity U∗
Ph and force

F ∗
Ph for a spherical droplet within a porous medium, as defined by expressions (4.26) and

(4.30) respectively. The dynamics are primarily governed by the permeability parameter
(α =

√
µa2/µ̃K), which quantifies the porous medium’s resistance to flow, the thermal

conductivity ratio (k = kd/kp) between the droplet and the surrounding medium, and the
viscosity ratio (σ = µ/µ̃) between the droplet and the surrounding medium.

Stokes flow Darcain flow
Figure 13: Normalized photocapillary velocity U∗

Ph versus permeability parameter α for
different values of thermal conductivity ratio k at specific values of viscosity ratio(σ =
0 , σ = 0.5).



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Stokes flow Darcain flow
Figure 14: Normalized photocapillary velocity U∗

Ph versus permeability parameter α for
different values of viscosity ratio σ at specific values thermal conductivity ratio (k =
1 , k = 0.3).

Figures 13 and 14 graphically depict how the thermocapillary velocity changes in re-
sponse to variations in the permeability parameter. These illustrations further explore
this relationship under several different conditions of thermal conductivity ratio and vis-
cosity ratio. These figures show that U∗

Ph generally decreases as α increases, across the
entire range of thermal conductivity and viscosity ratios studied. In the Stokes limit
(α = 0), where the porous medium’s influence is negligible, U∗

Ph reaches its maximum
value. This observation is consistent with classical analyses of thermocapillary migration
in unbounded fluids, such as the seminal work by Young [6], where motion is driven by
Marangoni stresses unimpeded by a porous matrix. For small permeability parameters
(α < 1), U∗

Ph demonstrates a pronounced insensitivity, maintaining a relatively constant
value. This regime signifies that the porous structure offers minimal resistance to the
droplet’s motion, mirroring the fluid dynamics within a highly permeable or sparsely con-
stituted porous system. A sharp decline in U∗

Ph is observed in the transitional regime
(1 < α < 100), indicating a significant increase in Darcy drag exerted by the porous
medium. This behavior is analogous to the transition from Stokes flow to Darcy flow for
rigid particles, though modified by the internal circulation and deformable interface of the
droplet [36]. Finally, in the Darcian limit (α > 100), where the porous medium’s resistance
dominates, the photocapillary velocity effectively vanishes. This is expected, as the pres-
sure drop required to drive flow through a dense porous medium at a given velocity scales
with decreasing permeability (increasing α). Figures 10 and 11 also demonstrate that for



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a fixed α, U∗
Ph increases with a decrease in both the thermal conductivity ratio k and the

viscosity ratio σ. A lower k (e.g., droplet less conductive than the medium, or medium
more conductive than the droplet) can lead to steeper interfacial temperature gradients,
enhancing the Marangoni effect. A lower σ (droplet less viscous than the surrounding
fluid) allows for more strong internal circulation within the droplet in response to surface
tension gradients, translating to a higher translational velocity [6, 37].

Stokes flow Darcain flow
Figure 15: Normalized photocapillary force F ∗

Ph versus thermal conductivity ratio k for
different values of permeability parameter α at specific values of viscosity ratio σ at (σ =
0 , σ = 1).

The non-dimensional photocapillary force, F ∗
Ph, is examined in Figures 15 and 16

as a function of α for different σ and k values, respectively. For (α < 1), F ∗
Ph shows

no significant variation. As α increases into the range (1 < α < 100), F ∗
Ph increases.

For (α > 100) (Darcian regime), F ∗
Ph tends towards a constant value. This saturation

suggests that the drag exerted by the dense porous medium reaches a limit determined by
the photocapillary driving mechanism. Figures 15 and 16 also indicate that for a given α,
F ∗
Ph increases as σ or k decreases.



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Stokes flow Darcain flow

Figure 16: Normalized photocapillary force F ∗
Ph versus permeability parameter α for differ-

ent values of viscosity ratio σ at specific values thermal conductivity ratio (k = 0.01 , k =
0.03).

Figure 17: Normalized photocapillary force F ∗
Ph versus thermal conductivity ratio k for

different values of permeability parameter α at specific values of viscosity ratio σ at (σ =
1 , σ = 10).



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Figure 18: Normalized photocapillary force F ∗
Ph versus viscosity ratio σ for different values

of permeability parameter α at specific values of thermal conductivity ratio k at (k =
0.01 , k = 0.05).

Fig. 17 illustrates the dependence of the normalized photocapillary force F ∗
Ph on the

thermal conductivity ratio k, for various combinations of viscosity ratio σ and permeability
parameter α. A particularly striking finding highlighted by this figure is the rapid decrease
of F ∗

Ph towards zero within a very narrow range where (k << 1). This suggests that with
the droplet being a considerably less efficient thermal conductor relative to the surrounding
medium (k << 1), the development and sustenance of strong interfacial temperature
gradients, essential for the Marangoni effect, are substantially diminished.

Fig. 18 provides a focused examination of the normalized photocapillary force F ∗
Ph

as it varies with the viscosity ratio σ, specifically under conditions where the thermal
conductivity ratio (k << 1). This condition (k << 1) signifies that the droplet is sub-
stantially less thermally conductive than the surrounding medium. The figure reveals a
critical trend: F ∗

Ph tends towards zero as the viscosity ratio σ becomes large (i.e., for
highly viscous droplets), and this behavior holds true across the entire investigated range
of the permeability parameter α. The physical basis for this is quite direct: a droplet with
high internal viscosity offers significant resistance to the internal fluid circulation that
is essential for the Marangoni effect. These surface-tension-gradient-driven internal flows
are strongly damped by high viscosity, consequently suppressing the propulsive Marangoni
stress and, therefore, diminishing any resultant photocapillary force.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Stokes flow Darcian flow
Figure 19: U∗

Ph/U
(0)
Ph versus permeability parameter α for different values of viscosity ratio

σ.

Figure 20: U∗
Ph/U

(0)
Ph versus viscosity ratio σ for different values of permeability parameter

α.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

The normalized photocapillary velocity, UPh/U
(Ph)
0 , where U

(Ph)
0 is the thermocapil-

lary velocity in a clear viscous fluid (i.e., α = 0), provides further insight. Fig. 19 plots

UPh/U
(Ph)
0 against the permeability parameter α for various viscosity ratios σ. As ex-

pected, in the limit α → 0, the ratio UPh/U
(Ph)
0 approaches 1, recovering the clear fluid

behavior. For (α < 1), UPh/U
(Ph)
0 remains approximately 1 across the entire range of

σ, confirming the negligible impact of the porous medium in this regime. This finding
aligns with studies showing that for very high permeabilities, the Brinkman correction to
Darcy’s law, which accounts for viscous shear, becomes dominant and approaches Stokes
flow behavior [28]. Consistent with the trend ofU∗

Ph, Fig. 19 shows a sharp decrease in

UPh/U
(Ph)
0 for (1 < α < 100), eventually tending to zero for (α > 100), highlighting

the progressive attenuation of motion by the porous matrix. For a fixed α within this

transitional range (1 < α < 100), UPh/U
(Ph)
0 increases as σ decreases. This implies that

droplets with lower internal viscosity (relative to the external fluid) are more efficient at
converting Marangoni stresses into motion, even when normalized against their clear fluid
velocity.

Fig. 20 plots UPh/U
(Ph)
0 against the viscosity ratio σ for various α, corroborates these

findings, illustrating a similar behavioral pattern to Fig. 19: increasing σ (more viscous
droplet) generally increases the normalized velocity, and for a fixed σ, the normalized
velocity reduces as this reduction is more pronounced at higher α values where the porous
medium’s influence is stronger.



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

(a) (b)

(c) (d)
Figure 21: The streamline function is scaled with respect to Ua2 for translational motion:
ψ̃(i)/Ua2 = C2, i = d, p, where C2 is a constant. Each contour is found by assigning a
specific value to the constant C2, with (a) α → 0, σ = 0.5, k = 1, (b) α = 5, σ = 1, k =
0.155, (c) α→ ∞, σ = 0.5, k = 0.01, (d) α→ 0, σ → ∞, k = 2 (Solid Particle).

Fig. 21 similar to Fig. 12 shows the streamlines within the two regions (r > a) and
(r < a). Again, the (a), (b) and (c) figures show the effects of the porosity parameter α,
the viscosity ratio σ and the conductivity ratio k, and the solid particle case is shown in
figure (d).



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Table 1: Comparison between normalized thermocapillary velocity U∗
T and normalized

photocapillary velocity U∗
Ph versus thermal conductivity ratio k, for different values of

permeability parameter α, at σ = 1.

α k U∗
T U∗

Ph

α = 2
0.5 0.129341317 0.086227545
1 0.107784431 0.035928144
1.5 0.092386655 0.020530368
2 0.080838323 0.013473054
2.5 0.071856287 0.009580838
3 0.064670659 0.007185629
3.5 0.058791508 0.005599191
4 0.053892216 0.004491018
4.5 0.049746661 0.003684938
5 0.046193328 0.003079555

α = 10
0.5 0.037804296 0.025202864
1 0.03150358 0.010501193
1.5 0.027003069 0.006000682
2 0.023627685 0.003937947
2.5 0.021002387 0.002800318
3 0.018902148 0.002100239
3.5 0.017183771 0.00163655
4 0.01575179 0.001312649
4.5 0.014540114 0.001077045
5 0.013501534 0.000900102

α→ ∞
0.5 0.000683109 0.000455406
1 0.000569257 0.000189752
1.5 0.000487935 0.00010843
2 0.000426943 7.11572E-05
2.5 0.000379505 5.06006E-05
3 0.000341554 3.79505E-05
3.5 0.000310504 2.95718E-05
4 0.000284629 2.37191E-05
4.5 0.000262734 1.94618E-05
5 0.000243967 1.62645E-05



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Table 2: Comparison between normalized thermocapillary force F ∗
T and normalized photo-

capillary force F ∗
Ph versus thermal conductivity ratio k, for different values of permeability

parameter α, at σ = 1.

α k F ∗
T F ∗

Ph

α = 2
0.05 0.182926829 3.658536585
0.25 0.166666667 0.666666667
0.45 0.153061224 0.340136054
0.65 0.141509434 0.217706821
0.85 0.131578947 0.154798762
1.05 0.12295082 0.117096019
1.25 0.115384615 0.092307692
1.45 0.108695652 0.074962519
1.65 0.102739726 0.062266501
1.85 0.097402597 0.052650053
2.05 0.092592593 0.045167118

α = 10
0.05 0.335365854 6.707317073
0.25 0.305555556 1.222222222
0.45 0.280612245 0.623582766
0.65 0.259433962 0.399129173
0.85 0.24122807 0.28379773
1.05 0.225409836 0.214676034
1.25 0.211538462 0.169230769
1.45 0.199275362 0.137431284
1.65 0.188356164 0.114155251
1.85 0.178571429 0.096525097
2.05 0.169753086 0.082806384

α→ ∞
0.05 0.464795214 9.29590428
0.25 0.423480084 1.693920335
0.45 0.388910281 0.864245069
0.65 0.359558562 0.553167018
0.85 0.334326382 0.393325155
1.05 0.312403341 0.297526991
1.25 0.29317852 0.234542816
1.45 0.276182663 0.190470802
1.65 0.261049367 0.158211737
1.85 0.247488361 0.133777492
2.05 0.235266713 0.11476425



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

Figure 22: Comparison between normalized thermocapillary force F ∗
T and normalized

photocapillary force F ∗
Ph versus thermal conductivity ratio k, for different values of per-

meability parameter α, at specific values of viscosity ratio σ = 1 , σ = 10.

Table 1 displays the numeric comparison between the two normalized velocities due
to the effects of thermocapillary U∗

T and the photocapillary U∗
Ph. The two velocities

are found for different values of the thermal conductivity ratio k and the permeability
parameter α, at the viscosity ratio (σ = 1). It is observed that for fixed α, both U∗

T and
U∗
Ph decrease as k increases, that is, a more thermally conductive droplet leads to lower

interfacial temperature gradients, which in turn reduces Marangoni-driven motion, also,
the photocapillary velocity drops more sharply than the thermocapillary velocity with
increasing k, especially at low α. For increasing α, both U∗

T and U∗
Ph decline, approaching

zero in the Darcian limit (α → ∞), and for any fixed k, it is obvious that the values
of both velocities drop by more than two orders of magnitude as α increases from 2 to
∞, confirming the dominant drag of the porous structure. Table 2 shows the comparison
between the normalized thermocapillary force F ∗

T and the normalized photocapillary force
F ∗
Ph, the same parametric attributes as Table 1 are present. The observable is that both

forces decrease significantly with increasing k. For instance, at (α = 2), F ∗
T drops by

half, on the other hand, F ∗
Ph drops by a much larger magnitude. This again confirms



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

that the low thermal conductivity of the droplet enhances temperature gradients, that is,
stronger Marangoni stresses, that is, larger forces. At a fixed k, both forces increase with
α (contrary to the velocities U∗

T , U
∗
Ph). Larger α reflects to more hydrodynamic drag, so a

greater driving force is required to maintain motion. Notably, despite the lower velocity,
the force needed to move the droplet under photocapillary effect can be larger due to
concentrated internal heating leading to stronger local stresses, but not necessarily higher
net motion due to dissipation. A comparative analysis of the normalized thermocapillary
force F ∗

T and photocapillary force F ∗
Ph versus the thermal conductivity ratio k, as depicted

in Fig. 22, reveals distinct behaviors rooted in their mechanisms of interfacial temperature
gradient generation. F ∗

T , arising from an externally imposed thermal field, exhibits a
monotonic decrease with an increasing k; a higher k signifies a more thermally conductive
droplet relative to its surroundings, leading to enhanced heat diffusion through the droplet
bulk, which homogenizes the droplet’s internal temperature and thereby diminishes the
magnitude of the temperature gradient sustained along its interface by the external field,
consequently weakening Marangoni stresses and reducing the propulsive force. Conversely,
F ∗
Ph, generated by internal heat absorption from light, displays a particularly striking

rapid decrease towards zero within a very narrow range where (k << 1). This occurs
because when the droplet is a considerably less efficient thermal conductor (both internally
and relative to the surrounding medium), the heat generated within it cannot effectively
establish and sustain strong interfacial temperature gradients necessary for a significant
Marangoni effect; the poor internal thermal transport delays the development of a non-
uniform surface temperature profile. Thus, at (k << 1), F ∗

T tends to be relatively high
as the droplet’s poor conductivity helps maintain the externally imposed gradient across
its interface, while F ∗

Ph drops because the same poor conductivity prevents internally
generated heat from creating an effective interfacial gradient. The permeability parameter
α primarily impacts F ∗

Ph, where increasing α (from 0 to 10) significantly enhances F ∗
Ph,

likely due to improved internal convective heat transfer which aids in establishing stronger
interfacial temperature gradients from the absorbed light; while its impact on F ∗

T is less
direct unless it significantly alters the bulk effective thermal conductivity. Notably, F ∗

Ph

generally achieves considerably larger magnitudes than F ∗
T , particularly at lower σ and

higher α, highlighting the potential for internal light absorption, aided by permeability,
to generate more potent Marangoni propulsion than external thermal fields, though this
advantage diminishes significantly when the droplet’s thermal conductivity k is extremely
low.

6. Conclusion

In conclusion, this investigation delineates the fundamentally divergent behaviors of
thermocapillary and photocapillary droplet migration within a saturated porous matrix,
establishing a predictive analytical framework based on the Brinkman equation. Our
analysis rigorously demonstrates that the externally-driven thermocapillary velocity U∗

T is
invariably attenuated by increasing matrix resistance (permeability parameter α), droplet-
to-medium thermal conductivity ratio k, and viscosity ratio σ attributable respectively to



A. S. Aamer, B. A. Ahmed, S. A. Mahrous / Eur. J. Pure Appl. Math, 18 (3) (2025), 6505 of

intensified Darcy drag, interfacial thermal homogenization that erodes the driving gradi-
ent, and heightened internal viscous dissipation. Significantly, we reveal that high matrix
resistance not only severely suppresses U∗

T but also diminishes its sensitivity to viscosity
by effectively immobilizing the interface and promoting internal thermal equilibration.
In stark contrast, while the internally-driven photocapillary velocity U∗

Ph is similarly re-
tarded by matrix resistance and high internal viscosity, its propulsive mechanism is criti-
cally predicated on the droplet’s ability to establish its own interfacial thermal gradient.
Consequently, the associated photocapillary force F ∗

Ph is extinguished at low thermal con-
ductivity ratios k << 1 where internal heat distribution is ineffectual. This establishes a
key mechanistic dichotomy: whereas the thermocapillary force F ∗

T monotonically weakens
with increasing k, the photocapillary force F ∗

Ph exhibits a greater sensitivity to matrix
permeability, which modulates the internal convection essential for distributing absorbed
energy, and can achieve superior magnitudes. However, this potential advantage is strictly
contingent upon sufficient droplet thermal conductivity for the efficient conversion of ab-
sorbed volumetric energy into a potent surface tension gradient. Ultimately, this research
provides a comprehensive framework detailing the intricate interplay between porous con-
finement, thermophysical properties, and the locus of the thermal stimulus, offering foun-
dational principles for the design and control of Marangoni-actuated transport in complex
multiphase systems.

7. Conflict of interest

We have no conflict of interest to declare.

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