Influence of Nanoparticles and Magnetic Field on the Laminar Forced

Convection in a Duct Containing an Elastic Fin

ABDERRAHIM MOKHEFI1, EUGENIA ROSSI DI SCHIO2, PAOLO VALDISERRI2,*,

CESARE BISERNI2

Faculty of Technology,

Bechar University,

B.P.417, 08000, Bechar,

Abstract: - In the present paper, an investigation of the effect of a magnetic field and nanoparticles suspended

in pure water on the forced flow in a duct containing an elastic rectangular fin is performed. The nanofluid, i.e.,

CuO nanoparticles suspended in water, flow in the duct with an inlet fully developed velocity profile and a cold

temperature. The lower boundary of the duct is kept at a hot temperature, while the upper boundary is adiabatic.

According to the ALE formulation, numerical simulations of the laminar flow are carried out, by employing the

software package Comsol Multiphysics, to solve the governing equation system: mass, momentum, energy, and

deformation. The behavior of the Nusselt number, of the temperature and velocity fields as well as of the stress

Received: June 12, 2022. Revised: May 12, 2023. Accepted: June 21, 2023. Published: July 28, 2023.

Elastic wall effects make the fluid dynamic problem

very hard to treat theoretically; indeed,

computational simulations for such configurations

fins or elastomeric structures and their sensibility

especially in the industrial field, several previous

works have been carried out in the literature. In, [5],

the authors studied numerically the problem of

unsteady natural convection inside an inclined

square cavity equipped with a flexible impermeable

membrane. They showed that in the case of a low

Rayleigh number, the membrane shape is a function

of the imposed body force. In, [6], the authors

presented a theatrical study based on the numerical

heat transfer of a power-law non-Newtonian fluid in

a cavity. The Arbitrary Lagrangian-Eulerian (ALE)

along the moving mesh method is employed to

model the deflection of the structure inside the fluid

fluid flow and a deformable (elastic) wall in the

presence of a discrete heat source. They found that

the presence of elastic walls enhances heat transfer

efficiency compared to standard walls. In, [9], the

authors reported techniques destined for the

modelization of fluid-structure interactions using

Comsol Multiphysics software. They illustrated the

step followed to study the flow in a continuously

deforming geometry based on the Arbitrary

Lagrangian-Eulerian (ALE) technique and

different types of fluids including nanofluid. Indeed,

the geometry of channels with an enclosed cavity

has been investigated in, [10], [11], with reference

to Couette flow, underlining the effects of

thermophoresis and Brownian diffusion. In

particular, in, [11], MHD effects are considered as

well. In both papers, Buongiorno’s model is

employed, [12], [13].

The forced convection of nanofluids in the

presence of MHD effects has been investigated in

with elastic walls/fins, attention has been paid

especially to mixed convection, [16]; concerning

forced convection, recently the arbitrary

Lagrangian–Eulerian (ALE) technique has been

used to numerically determine the interactions and

movements of the nanofluid and fins, [17]. The

investigated geometry was a rectangular enclosure,

with two fins. As a result, the authors highlighted an

increase in the heat transfer rate due to the

oscillation of the fins. However, as far as the authors

water nanofluid is presented. In the literature, in,

model the presence of an elastic fin. The geometry

under investigation is a channel that displays an

enclosure; the nanofluid enters the channel with a

fully developed velocity profile and after a short

entrance region a variation of the cross-section

occurs, due to the presence of a cavity, and the fluid

encounters an elastic fin. The effect of thermal

boundary conditions is investigated as well.

2.1 Geometry Description

Let us investigate the effect of a vertical elastic fin

inserted in a channel. The geometry under

investigation is sketched in Fig. 1. We assume that

after a short inlet region, with height H, where the

nanofluid (water-CuO) has a fully developed

parabolic profile velocity, the channel displays an

increased height 2H. Namely, a cavity or forward

The thermophysical properties of the CuO

nanoparticles and the base fluid are shown in Table

1.

where w and σ denote the local velocity and solid

stress tensor, respectively. A detailed description of

the following set of dimensionless variables has

been introduced:

2.3 Initial and Boundary Conditions

The governing differential equations system is

solved together with the following initial-boundary

conditions.

- At the initial time τ = 0:

In the nanofluid domain

- At the time τ ≥ 0 :

2.4 Heat Transfer Rate

Concerning nanofluid studies, the local Nusselt

Moreover, an averaged Nusselt number is

defined as the average of the local Nusselt number

along the hot wall:

A Galerkin finite element method is used to solve

the set of nonlinear governing equations, employing

ALE techniques. For this purpose, Comsol

Multiphysics software has been implemented to

simulate the studied fluid-structure interaction

phenomenon. Non-uniform triangular elements are

used to build the mesh of the nanofluid domain. The

fully coupled approach is used to couple the

thermal, momentum, structure, and displacement of

the mesh. The Computational domain mesh is

Fig. 2: Computational domain mesh.

3.1 Mesh Check and Numerical Validation

In order to obtain precise numerical results from the

simulation, a study of mesh independence has been

performed. The objective is to choose a suitable

number of elements beyond which the results of the

different global quantities remain unchanged

depending on the mesh size. For this goal, we chose

the average Nusselt number on the hot wall and the

The mesh check has been performed in the case

of Re = 100, Pr = 6.8, Ca = 10-4, φ0 = 0.04, and Ha =

0. It has been opted for a mesh corresponding to an

element number equal to 14866 beyond which there

was no change in terms of computational Nusselt

and the new value of the dependent variable

becomes less than 10-6.

Fig. 3: Comparison of isotherms and streamlines (a)

and of the average Nusselt number (b) with [20]

To evaluate the verification of the

computational code, the numerical results of the

present work have been compared with the

numerical results of, [20]. In this reference, laminar

forced convection of a nanofluid in the presence of a

Moreover, to solidify the validation of our work,

we validated it also with previous work concerning

fluid-structure interaction. Thus, we referred to the

work of, [7], in which natural laminar convection

was studied in the presence of an elastic fin. Ra =

105, Pr = 10, Et = 1010 and n = 1. The isotherms and

streamlines are presented according to the chosen

parameters, Figure 4. Moreover, the average Nusselt

number for different Rayleigh numbers is shown in

Table 3.

and mainly the discussion, will be presented by

evaluating separately, in dedicated subsections, the

occurring effects, i.e. inertia, nanoparticle

concentration, magnetic field, and elasticity. The

control parameters are the Reynolds number, the

volume fraction of CuO nanoparticles, the

Hartmann number, and the Cauchy number. The

reference thermo-physical properties of the elastic

fin are fixed at: ρr = Cpr = 1 and kr = 100.

the direction of flow. Indeed, under the applied load,

all the points forming the structure except those of

the embedding area (fin) tend to move in the

direction of the nanofluid flow with an increasing

number, the maximum stress applied to the walls of

under the effect of the inertia.

This is due to the increase of the flow intensity

of the recirculation zone formed downstream of the

stability state (reverse flow direction). Furthermore,

decrease in heat transfer becomes abrupt in the

immediate vicinity of the structure as the Reynolds

number increases toward the value of 200. Overall,

along the line of measurement, the highest heat

transfer rate depends on the large value of the

Reynolds number. Since the Nusselt number is a

this area on the one hand, and the thermal

that of the base fluid on the other.

Fig. 6: Maximal fin displacement for various

Reynolds numbers

Fig. 7: Local and average Nusselt number for different values of the Reynolds number

4.2 Effect of the Volume Fraction

One of the main objectives of the present study is to

highlight the influence of the volume fraction of

CuO nanoparticles suspended in pure water on the

thermal and hydrodynamic behavior of the

nanofluid in the extended pipe and the thermal and

section, we analyze the impact of the addition of

CuO nanoparticles on the mentioned behaviors.

Figure 8 shows the temperature and stream

in the flow velocity. Due to the increase in pressure

of the nanofluid as well as its viscous drag, the force

imposed on the walls of the elastic structure

increases relatively compared to the case of pure

water flow.

Fig 9. Maximal fin displacement for various

nanoparticles volume fraction

This leads to an increase of the stress and

consequently, it leads to a higher bending.

Moreover, the maximum dimensionless

with the increase of the volume fraction of CuO

nanoparticles, especially downstream far from the

structure and in the immediate upstream proximity

of the fin. This situation informs about the

development of the hydrodynamic state as a

function of the volume fraction of CuO

nanoparticles.

Indeed, far from the vortex zones where the

fluid velocity is important, the convection rate

grows and the impact of the nanoparticles on the

a function of the addition of CuO nanoparticles

reaches 21.33% compared to the use of pure water,

as shown in Figure 10 (b).

Fig. 10: Local and average Nusselt number for various nanoparticles volume fraction