Flow Over a Thin Needle Moving in a Casson Fluid
D. SRINIVASACHARYA*, G. SARITHA
Department of Mathematics, National Institute of Technology Warangal,
Hanumakonda 506 004, Telangana,
INDIA
*Corresponding Author
Abstract: - This study examines the boundary layer flow across a thin, horizontal needle moving in a Casson
fluid. The underlying equations are initially converted into a set of ordinary differential equations using
similarity transformations, and thereafter successive linearization is applied. The Chebyshev collocation
technique is applied to find the solution of the linearized equations. The temperature and velocity profiles,
together with the skin friction coefficient and the Nusselt number, are illustrated graphically for different values
of the needle size and Casson fluid parameter.
Key-Words: - Boundary Layer flow, Casson fluid, Thin needle, Similarity transformations, Heat transfer rate,
Skin friction coefficient, Successive linearization, Chebyshev collocation method.
Received: April 25, 2023. Revised: November 15, 2023. Accepted: January 14, 2024. Published: April 1, 2024.
1 Introduction
Several investigators have been examining the flow
and heat transfer in different geometries, such as a flat
plate, stretching sheet, horizontal cylinder, stretching
cylinder, disc, stretching disc, sphere, elastic sheet, etc.
Applications for the flow and heat transmission
corresponding to a thin needle can be found in a wide
range of fields, including biomedicine, microstructure
electronic tools, hot wire anemometers, lubrication and
power generation, aerodynamics, blood flow,
microscale cooling devices, cancer therapy, wire
coating, and many more. Thin needle geometry refers
to the smearing surface produced by rotating a
parabola around its axis. [1], was the first to introduce
a boundary layer flow across a tiny moving needle in a
parallel free stream. Thereafter, the similarity
solutions for convective flow over a needle were
presented in [2], [3], [4]. The forced laminar
convective nanofluid stream past a horizontal needle
was analysed in [5]. The flow around a small needle
with changeable viscosity and thermal conductivity
was examined in [6]. The consequences of hybrid
nanoparticles on the flow across a hot needle has been
reported in [7]. In a fluid containing hybrid
nanoparticles, [8] studied the influence of viscous
dissipation, magnetic fields, and radiation on fluid
flow across a moving needle.
The study of non-Newtonian fluid flow attracted
the interest of several researchers due to its relevance
to a wide range of engineering problems. These
include tarry fuel abstraction from petroleum-based
goods, the manufacture of plastic materials, crystal
growth, the freezing of nuclear reactors, etc. Further,
the boundary layer flow of non-Newtonian fluid
around the thin needle has important physical practical
applications such as in marine structures and marine
vehicles, torpedoes, water towers, bridges, and many
more. There is no single fluid model that can
accurately describe the properties of non-Newtonian
fluids. As a result, various fluid models were described
in the literature over the previous century to explain
real fluid dynamics. [9], presented the Casson fluid
model, which is a shear-thinning fluid. The viscosity
of these fluids is infinite at no shear rate and non-
existent at infinite shear rates. It is worth noting that if
the yield stress is less than the shear stress, this model
reduces to a Newtonian liquid. It offers an easy way to
calculate the two parameters, Casson viscosity, and
apparent yield stress, for use in real-world
applications. In recent years, the Casson fluid model
has been employed in a variety of theoretical and
computational studies due to its extensive applicability
in drilling operations, metallurgy, food processing,
polymer processing industries, synthetic lubricants,
biomedical fields, the preparation of printing ink, etc.
Although Casson fluid flow across a thin needle is
significant, very little research has been published. The
influences of radiation and cross diffusion on the
magnetic Casson nanofluid along a needle were
scrutinized by [10]. [11], analyzed the MHD effects
on the Casson fluid with nanoparticles over a thin
needle. The thermally radiative magnetic Casson fluid
with nanoparticles on the needle under the Navier-Slip
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D. Srinivasacharya, G. Saritha
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effect has been explored in [12]. In the presence of
internal heat and nonlinear thermal radiation, [13],
examined the Casson fluid flow with nanoparticles
around a thin needle. The influence of injection,
temperature-dependent viscosity, and thermal
conductivity on the natural convection Casson fluid
flow from a spinning cone in a porous medium were
considered in [14]. The consequence of Soret effect
and viscous heating with thermal and solutal
dispersion on the double diffusion convection Casson
fluid flow along a vertical plate was analysed by [15].
The hydromagnetic flow of Casson nanofluid in
Armory production was considered by [16].
This work considers the steady flow of a Casson
fluid along a horizontal, thin needle. The flow
equations are initially converted into a set of ordinary
differential equations, then linearized by applying
successive linearization and solved via the Chebyshev
collocation technique.
2 Problem Formulation
Consider the flow of Casson fluid with uniform
velocity U over a thin needle moving horizontally
with a velocity Uw. Assume that flow is steady,
laminar, and incompressible. The coordinate system
and schematic of the problem is shown in Figure 1.
The equation for the radius of the needle is r = R(x).
It is supposed that the needle is thin while the
needle's thickness is less than that of the boundary
layer surrounding it. The temperature of the needle
is Tw, whereas the temperature of the surrounding
fluid is T, where Tw > T and Cw > C .
Fig. 1: “Coordinate system and physical flow
model”
With the above assumptions and invoking
boundary layer postulations, the flow equations are:
(1)
(2)
(3)
where u and v denote the axial and redial velocity
components, represents the fluid temperature,
represents the viscosity, represents the Casson
fluid parameter, represents the fluid density and α
represents the thermal conductivity.
The conditions on the surface of the needle are:
v = 0, T = , u = C = at r=R(x)
T , u , C at r (4)
The similarity transforms are defined as:
= ), () =
, =
(5)
where is the composite velocity and
is stream function.
If the equation , where ‘a’ is
dimensionless constant, represent the needle wall,
the surface of the needle, using Eq. (5), can be
written as R=
which characterizes the
shape and size of the needle.
Making use of similarity transformations given
in (5) in equation (1) to (3), we get:
(6)
(7)
The modified conditions on boundary are:
(a)=
,
, ,
, (8)
where is the velocity ratio parameter,
denotes the Prandtl number.
The non-dimensional form of skin friction
coefficient (Cf) and the heat transfer rate (Nusselt
number (Nu)) are:
and
(9)
3 Problem Solution
The set of differential Eqns. (6) and (7) are linearized
by means of the successive linearization method
(SLM), [17], [18]. The solution of these linearized
equations is obtained by utilizing the Chebyshev
collocation method. In SLM, it is supposed that the
unidentified functions can be
taken as:
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+
(10)
where is undetermined function
and is an estimate. This estimate can be
calculated by solving the linearized set of equations
generated by applying equation (10) in the equations
(6) and (7). The basic idea is that, even if becomes
large, become very small and hence non-linear
terms in and their differential can be neglected.
Substituting (10) in the equations (6) to (7) and
neglecting nonlinear terms containing, , and ,
we get the following equations
++= (11)
++= (12)
where
= 2[1 +
],
=2[1 +
] +
,
=
= - 2 [1 +
]
-2[1 +
]
,
= , +
,
= -
-
-
The set of linearized equations (11) and (12) is
solved by means of the Chebyshev collocation
method, [19]. For this problem, the range of the
solution, is adjusted to [0, B], here B is a
selected to obtain the conditions far away from the
body. To implement this approach [0, B] is again
changed to [-1, 1] by using the transformation:
(13)
The Gauss-Lobatto collocation points, [19], on [-
1 1] are given by:
cos
(14)
and , and are estimated at the above points
as:
,
(15)
where is the order Chebyshev polynomial.
Similarly, the differentials of , and are
guesstimated as:
(16)
where
with D is the Chebyshev derivatives
matrix.
Equations (15) - (16) are substituted into
equations (11), and (12) to get the equation in matrix
form as:
(17)
where represents order square
matrix and and represents order
column matrices given by:
(18)
Here
,
,
A11 = ++, A12 = O,
A21 =, A22 = [+
=,
=,
where the [ ]T stands for transpose, denotes the
zero, denotes the identity matrix.
Imposing the boundary conditions in terms of the
collocation points, the solution is provided by:
(19)
4 Results and Discussion
The effects of three dimensionless parameters are
primarily the focus of the current study. They are
size of the needle (a), the Casson fluid parameter
(), and the velocity ratio parameter (λ). The effects
of these parameters on the velocity and temperature
together with a coefficient of skin friction and the
heat transfer rate (Nusselt Number) are presented
graphically (Figure 2, Figure 3, Figure 4, Figure 5
and Figure 6).
Figure 2 represents the impact of needle size on
velocity and temperature profiles. The velocity
increases as the size of the needle decreases, as
depicted in Figure 2(a). Physically, as the thin
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needle's size decreases, the surface area for the fluid
particle decreases, lowering the force and thus
increasing the velocity. Furthermore, as needle size is
reduced, the boundary layer thickness for the velocity
diminishes. The temperature and its boundary layer
decrease with the thin needle size, as seen in Figure
2(b).
(a)
(b)
Fig. 2: “Effects of needle size on the (a) Velocity
profile (b) Temperature Profile”
(a)
(b)
Fig. 3: “Effect of Casson fluid parameter on the (a)
Velocity (b) Temperature”
The effect of the Casson fluid parameter (β) on
the velocity and temperature is depicted in Figure 3.
It is observed from Figure 3(a) that an increase in
the value of β increases the velocity. Further, as η
increases, the velocity also increases. The impact of
β on the temperature is almost negligible as shown
in Figure 3(b). It is clear that for increasing values
of η, the temperature is decreasing.
(a)
(b)
Fig. 4: Effect of velocity ratio parameter on the (a)
Velocity (b) Temperature”
The variations in the velocity and temperature
for different values of the velocity ratio parameter
are presented in Figure 4. Figure 4(a) indicates that
an increase in the value of λ enhances the velocity
near the wall of the needle and subsequently
decreases as it moves away from the needle's wall.
As shown in Figure 4(b), the temperature is not
affected by the velocity ratio parameter.
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(a)
(b)
Fig. 5: “Effect of needle size on the (a) coefficient
of skin friction (b) Heat transfer rate”
Figure 5 describes the influence of the size of
the needle on the coefficient of skin friction and
Nusselt number. According to Figure 5(a), for a
constant value of needle size and an enhancing
value of , the skin friction decreases. In addition,
as needle size increases, the skin friction coefficient
decreases. The rate of heat transmission decreases
as the needle size increases, as seen in Figure 5(b).
The Casson fluid parameter has no substantial effect
on the Nusselt number, as shown in Figure 5(b).
(a)
(b)
Fig. 6: “Effect of velocity ratio parameter on the (a)
Skin friction coefficient (b) Heat transfer rate”
The consequence of the velocity ratio parameter
on the coefficient of skin friction and Nusselt
number is displayed in Figure 6. It is observed from
Figure 6(a) that the skin friction coefficient is
decreasing for enhancing values of the velocity ratio
parameter. Figure 6(b) reveals that the rate of heat
transfer is increasing with an increase in the velocity
ration parameter.
5 Conclusion
The significance of flow parameters on the
temperature, velocity, heat transfer rate, and the skin
friction coefficient in the flow over a thin horizontal
needle moving in a Casson fluid is examined. The
main findings of the current examination are as
follows
The velocity rises with reducing needle size and
drops with rising Casson fluid parameter.
Temperature declines with the diminishing size
of a thin needle. The effect of the Casson fluid
and the velocity ratio parameters on the
temperature are insignificant
The findings presented in this study open up
intriguing possibilities for future research in the
domain of fluid flow in porous media. This research
explores a broader range of phenomena, including
chemical reactions, Magnetic fields, heat
source/sink, radiation, etc, which could provide a
more comprehensive understanding of the complex
systems.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflicts of Interest
The authors declare no conflict of interest.
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(Attribution 4.0 International, CC BY 4.0)
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