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Abstract

The study of boundary layer flows over an irregularly shaped needle with small horizontal and vertical dimensions is popular among academics because it seems to have a lot of uses in fields as different as bioinformatics, medicine, engineering, and aerodynamics. With nanoparticle aggregation, magnetohydrodynamics, and viscous dissipation all playing a role in the flow and heat transmission of an axisymmetric $T i O_{2} - C_{2} H_{6} O_{2}$ nanofluid via a moving thin needle, this article provides guidance on how to employ a boundary layer for this purpose. In this case, we utilized the similarity transformation to change the dimensional partial differential equation into the dimensionless ordinary differential equation. We utilize MATHEMATICA to include shooting using RK-IV methods after identifying the numerical issue. Several characteristics were measured, leading to the discovery of a broad variety of values for things like skin friction coefficients, Nusselt numbers, velocity profiles, and temperature distributions. Velocity profile decreases with increasing values of $ϕ, M, e$ and increases against $ε .$ Temperature profiles enhances with increasing values of $ϕ, M, e, ε$ , and $E c .$ The reduction in skin friction between a needle and a fluid can be observed when the values of M and $ϕ$ are boosted. Furthermore, it was also noticed an increase in heat transfer on needle surface dramatically when $ϕ, e$ , and M were raised, whereas $E c$ displayed the opposite effect. The findings of the current study are compared with prior findings for a particular instance in order to confirm the findings. Excellent agreement between the two sets of results is found.

Keywords

Steady flow MHD viscous dissipation slender needle aggregation effect

Introduction

Nanoliquids are used to manage the process of heat transfer in a variety of practical applications, including heat pumps, cooling systems, biochemical mechanisms, and digital equipment. This is possible owing to the higher thermal efficiency of nanoliquids. Nanofluids are defined as liquids created by the uniform dispersion of metal or metal-based microscopic particles at the nanoscale.¹ According to the findings of both experimental and numerical investigations,^2,3 the dynamics of the host fluid are significantly impacted by the nanomaterials that are included inside the host fluid. Researchers in the contemporary day are very interesting since nanofluids may be used in such a broad variety of scientific and technological applications. The temperature of the host fluid increased as a result of the presence of a mixture of nano-sized particles within the fluid. This happened because the small particles were good at moving heat around, which can speed up the rate of heat transfer.⁴ The small solid particles’ varying forms have an effect on the dynamics of the host fluid, as well as a significant impact on the temperature of the host fluid. The many particles that may be found inside of nanofluid flows have been the subject of investigation by the scientific community on all three fronts (numerically, analytically, and empirically).⁵ In recent years, a number of scientists have studied the different properties of nanoparticles in order to improve the thermal properties of the host fluid.^6–10 They have used a wide range of approaches, geometries, and methods to do this. Mixing minute solid particles with the base fluids can improve the thermal properties of a number of fluids.^11,12

Aggregates are a term used to describe the ways in which particles or molecules come together to create extended structures. Colloidal science and engineering place a large emphasis on the aggregation mechanism, which is a process that cannot be reversed. The concepts of aggregation and agglomeration, on the other hand, are not quite the same. The process of assembling molecules in a specific sequence with strong bonding is referred to as aggregation, while the process of building molecules in a pattern that is only weakly attached and may be disrupted by mechanical force is referred to as agglomeration. Using fractal geometry, one may determine the nature of this aggregation structure. There is a lot of controversy surrounding the idea of increasing nanoliquids’ thermal conductivity. Recent research indicates that the aggregation of nanocomposite particles plays a significant role in the thermal performance of nanofluids. According to Keblinski et al.,¹³ as related to Brownian movement of the particles, major upgrade factors include the size of the particles, nanoparticle aggregation, and the liquid–molecule interface. Wang et al.¹⁴ showed evidence that the aggregation kinematics was a good way to create thermal conductivity. The research carried out by Cai and colleagues¹⁵ shows that fractal theory is capable of providing an accurate description of nanoparticles and their aggregation. See the work of Mackolil and Mahanthesh, and others^16–20 for recent publications of many different works that make an effort in this area.

Because magnetohydrodynamic (MHD) flow phenomena are so important, they have gotten a lot of attention because they are used in many engineering and industrial fields, such as fusion reactors, optical fiber, crystal growth, and stretching plastic sheets. Lorentz forces result from the interaction between magnetic fields and electrical currents. Consequently, MHD describes the behavior of a conductor's magnetic field and fluid. Research should be conducted to determine the impacts of MHD flow on industrial and technological sectors. As a result of the fields of magnetism and electrically conducting fluids, the cooling rate has a significant impact on the final product. In Carreau nanoliquid, Sandeep and Ashwinkumar²¹ investigated both the nanoparticle shape and the MHD stagnation point flow. A quantitative study of Sutter by fluid flow confined at a stagnation point with an inclined magnetic field and thermal radiation impacts has been published by Sabir et al.²² In Sarada et al.,²³ nonlinearities and temperature spikes influence nanofluid flow and heat transfer in non-Newtonian MHD through sheets that are stretched thin.

Due to its significant practical value, researchers have paid close attention to the investigation of the behavior of the thermal energy transfer through boundary layers toward a thin needle. The first numerical results for a thin needle were subjected to boundary layer flow computation and explanation by Lee.²⁴ When the needle disappeared, he observed that displacement thickness and drag per unit length gradually decreased until they reached zero. A flow and a heat source for transmission over a thin needle in a variety of liquids were then investigated. As part of one of these studies, Grosan and Pop²⁵ examined the flow of fluid properties and the transfer of heat properties when a thin needle was immersed in nanofluids. In the study, it was found that nanoparticle volume or size significantly influenced heat transmission and the characteristics of flow. In addition, Tiwari, Das, and Buongiorno models were used to investigate the effect of MHD nanofluid flow by means of a thin needle that moves rapidly with frictional heating (see Sulochana et al.²⁶). Krishna et al.²⁷ has developed this idea further by investigating the boundary layer and heat transfer properties of $C u$ nanoparticles embedded in methanol, as well as Ag-water nanofluids and Ag-Kerosene nanofluids. The stability valuation was also performed to determine the stability of the acquired solutions. Salleh et al.^28,29 studied the movement of thin needles in nanofluids under various circumstances, including heating and the presence of a slippery wall. They observed that raising the convective and slip parameters speeds up the heat transfer along the local wall. Recent flow along boundary layers investigations around hybrid nanofluid needles have extended to investigating the flow of heat and its behavior when needles are immersed in nanofluids (see Aladdin et al.³⁰ and Mousavi et al.³¹).

Since viscous dissipation has so many applications in industry, a growing number of academics are beginning to focus their attention on studying its influence. Some examples of these applications include the development of electric ovens, the manufacture of paper, and the growth of crystals. When Pal et al.³² studied the MHD flow of nanoliquid in a heated sheet with viscous dissipation, they found that the heat transfer gradient decreased with the existence of the Eckert number. Hsiao³³ used numerical analysis to study the MHD and Ohmic dissipative flow of Maxwell liquid on a stretched sheet when thermal radiation was present. The results of their research showed that when the Eckert number goes up, the mass transfer gradient gets stronger. Butt and colleagues³⁴ investigated the effect that viscous dissipation and MHD flow of viscous liquids with entropy production had on a viscous liquid as it flowed through a cylinder implanted in a porous medium. They found that increasing the Eckert number resulted in an increase in the thickness of the thermal boundary layer. Alshehri and Shah³⁵ performed a computational study to investigate the issue of the viscous dissipative flow of a hybrid nanoliquid accompanied by radiation. As the Eckert number was increased, they found that there was an improvement in the nanoliquid thermal field. In their research on Sisko nanoliquids suspended with gold nanoparticles, Tang et al.³⁶ showed that increasing the Eckert number causes a rise in the temperature of the liquid being studied. The effects of viscous dissipation and joule heating, along with other physical properties, across a permeable stretched sheet were investigated by Rafique et al.³⁷

According to a study of the relevant published research, which shows that nanoparticle aggregation is a factor, the flow and heat transmission around tiny needles moving in parallel streams of nanofluid have not received any attention. Because of this, the focus of the current research is on the transport phenomena of titanium dioxide nanoparticle aggregates in ethylene glycol-based fluids when an externally applied magnetic field and viscous dissipation are present. The process of aggregation can make it possible for nanoparticles to have much better thermal conductivity. In addition, nanoparticles provide a variety of challenges that must be overcome. One of the hardest things for engineers to figure out is how the interactions between molecules in a system, like the aggregation kinetics, affect the thermophysical properties of the fluid. Aggregation kinetics, which is something that is spoken about in the research literature, has an effect on heat transfer, the temperature that is created, and thermal conductivity. Based on the results of previous studies, we decided to look into how the kinetics of aggregation affect thin needles moving in parallel streams of nanofluid while Lorentz force and viscous dissipation are acting on them. The main goal of this paper is to find out what role the grouping together of nanoparticles might play in making the thermal conductivity of the host fluid better. The shooting with RK-IV, a well-known computational strategy, is used in order to solve the environment of MATHEMATICA. When the effect of nanoparticle aggregation is considered, it becomes clear that the temperature function improves, even though aggregation might sometimes result in a decrease in velocity. This is the case even if the velocity can sometimes increase. So, this shows how important and influential aggregates are, as well as how useful they are as a theoretical tool that could be used in the future in the industrial and technical fields. Through the use of RK-IV, solutions have been found for the many distinct forms of boundary layer flow that are responsible for a variety of fluid-related issues. This method provided a quick, accurate, and quick solution to the fluid concerns.

Description of the problem

When an asymmetric MHD flow of ethylene glycol-based fluids is studied, the effects of nanoparticle aggregation and viscous dissipation on the boundary layers are taken into account. An axisymmetric flow is a type of flow in two dimensions that can be recognized by symmetrical streamlines around an axis or along a line of symmetry along a needle surface. An illustration of the current study is shown in Figure 1. The needle wall is thought to be set at a constant temperature $T_{n}$ where $T_{n}$ is higher than the temperature of the unrestricted stream $T_{\infty}$ . $x$ and r are the coordinates of radial and axial axes, respectively, a scenario in which needle radius indicated through $r = (ν_{f} e x / U)^{1 / 2} = R (x) .$ Assume that surface of needle moves uniformly $U_{n}$ in the same and opposite directions as constant velocity fluid motion $U_{\infty}$ . The transverse magnetic field $B (x) = B_{0} x$ , which is replicated normal to the stretching/shrinking needle, is calculated using the applied magnetic field intensity $B_{0}$ . Based on the assumptions, the governing equation for system can be written as: (see^12,38)

\frac{\partial (r u)}{\partial x} + \frac{\partial (r v)}{\partial r} = 0

(1)

u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial r} = \frac{μ_{n f}}{ρ_{n f}} r^{- 1} (r u_{r})_{r} - \frac{σ_{n f}}{ρ_{n f}} B_{0}^{2} (u)

(2)

u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial r} = \frac{k_{n f}}{{(ρ C_{p})}_{n f}} r^{- 1} (r T_{r})_{r} + \frac{μ_{n f}}{{(ρ C_{p})}_{n f}} {(\frac{\partial u}{\partial r})}^{2} .

(3)

The following are the boundary conditions:³⁸

u = U_{n}, v = 0, T = T_{n} a t r = R (x),

(4)

u \to U_{\infty}, T \to T_{\infty}, a s r \to \infty

Figure 1.

Coordinate system and physical configuration.

In this case, u and v components of velocity along $x, r$ is likewise, $μ$ denotes dynamic viscosity, $ρ$ indicates density, $α$ represents thermal diffusivity, and T temperature of fluid. Subscript term “ $n f$ ” is used for “nanofluid.” The thermophysical characteristics of $T i O_{2} - C_{2} H_{6} O_{2}$ and their correlation coefficient may be found in Tables 1 and 2, respectively (see^16–18,20).

Table 1.

Base fluid $C_{2} H_{6} O_{2}$ with nanoparticles $Ti O_{2}$ thermophysical characteristics (see¹⁶).

Properties	$Ti O_{2}$	$C_{2} H_{6} O_{2}$
$ρ (kg / m^{3})$	$4250$	$1114$
$C p (J / kgK)$	$686.2$	$2415$
$σ (S / m)$	$2.38 \times 10^{6}$	$1.07 \times 10^{- 6}$
$k (W / mK)$	$8.9538$	$0.252$

Table 2.

Efficacious models of nanoliquids with regard to their thermophysical qualities (see^16–18,20).

Effective property	Without aggregation	With aggregation
Density	$\frac{ρ_{n f}}{ρ_{f}} = (1 - ϕ) + ϕ \frac{ρ_{S}}{ρ_{f}}$	$\frac{ρ_{n f}}{ρ_{f}} = (1 - ϕ_{a}) + ϕ_{a} \frac{ρ_{S}}{ρ_{f}}$
Dynamic viscosity	$\frac{μ_{n f}}{μ_{f}} = \frac{1}{{(1 - ϕ)}^{2.5}}$	$\frac{μ_{n f}}{μ_{f}} = {(1 - \frac{ϕ_{a}}{ϕ_{m}})}^{[η] ϕ_{m}}$
Specific heat capacity	$\frac{{(ρ C_{p})}_{n f}}{{(ρ C_{p})}_{f}} = (1 - ϕ) + ϕ \frac{{(ρ C_{p})}_{S}}{{(ρ C_{p})}_{f}}$	$\frac{{(ρ C_{p})}_{n f}}{{(ρ C_{p})}_{f}} = (1 - ϕ_{a}) + ϕ_{a} \frac{{(ρ C_{p})}_{S}}{{(ρ C_{p})}_{f}}$
Thermal conductivity	$\frac{k_{n f}}{k_{f}} = \frac{(k_{S} + 2 k_{f}) - 2 ϕ (k_{f} - k_{S})}{(k_{S} + 2 k_{f}) + ϕ (k_{f} - k_{S})}$	$\frac{k_{n f}}{k_{f}} = \frac{(k_{a} + 2 k_{f}) - 2 ϕ_{a} (k_{f} - k_{a})}{(k_{a} + 2 k_{f}) + ϕ_{a} (k_{f} - k_{a})}$
Electrical conductivity	$\frac{σ_{n f}}{σ_{f}} = 1 + \frac{3 (\frac{σ_{n f}}{σ_{f}} - 1) ϕ}{(\frac{σ_{n f}}{σ_{f}} + 2) - (\frac{σ_{n f}}{σ_{f}} - 1) ϕ}$	$\frac{σ_{n f}}{σ_{f}} = 1 + \frac{3 (\frac{σ_{n f}}{σ_{f}} - 1) ϕ_{a}}{(\frac{σ_{n f}}{σ_{f}} + 2) - (\frac{σ_{n f}}{σ_{f}} - 1) ϕ_{a}}$

When the fractal structure of aggregates is taken into consideration, the updated nanoparticle volume fraction $ϕ_{a}$ is defined as (see^16–18,20): $(ϕ_{a} = ϕ {(\frac{r_{a}}{r_{p}})}^{3 - D})$ . Where $r_{p}$ signifies the radius of primary NPs, $r_{a}$ symbolizes the radius of aggregates, and D represents the fractal index. For $T i O_{2} - C_{2} H_{6} O_{2}$ nanoliquid, the radius ratio value that was calculated via experimentation is 3.34, and D equals 1.8; this value corresponds to the spherical aggregates (see^16–18,20).

For the purpose of simplifying equations (1) to (4), dimensionless quantities are introduced (see³⁸):

ψ (x, r) = ν_{f} x f (η), η = \frac{U r^{2}}{ν_{f} x^{\cdot}}, θ (η) = \frac{T - T_{\infty}}{T_{n} - T_{\infty}}

(5)

In this case,

ψ

is a representation of an axisymmetric stream function, and u and v are denoted as:³⁸

u = r^{- 1} \frac{\partial ψ}{\partial r} a n d v = - r^{- 1} \frac{\partial ψ}{\partial x}

(6)

And

v_{f}

is the kinematic viscosity of the fluid. Further,

U = U_{n} + U_{\infty}

combined velocity of free stream and the needle,

η

represents variable that measures similarity, and

θ (η)

represents a temperature that is dimensionless. The non-dimensional momentum and energy equations shown below can be obtained by substituting Equation (5) and applying Equation (6) to Equations (1) to (4):

2 \frac{μ_{n f} / μ_{f}}{ρ_{n f} / ρ_{f}} (η f^{″'} + f^{″}) + f f^{''} - \frac{σ_{n f} / σ_{f}}{ρ_{n f} / ρ_{f}} M f^{^{'} 2} = 0,

(7)

\frac{2}{P r} \frac{k_{n f} / k_{f}}{{(ρ C_{p})}_{n f} / {(ρ C_{p})}_{f}} (η θ^{''} + θ^{'}) + f θ^{'} + \frac{μ_{n f} / μ_{f}}{{(ρ C_{p})}_{n f} / {(ρ C_{p})}_{f}} E c (f^{″})^{2} = 0.

(8)

And restrictions of boundary are:

f (e) = \frac{ε}{2} e, f^{'} (e) = \frac{ε}{2}, θ (e) = 1,

f^{'} (η) \to \frac{1 - ε}{2}, θ (η) \to 0. a s η \to \infty

(9)

About the prime, the derivatives are said to be

η

Other important factors to consider include Prandtl number $P r$ , needle thickness e, as well as the flow-to-needle velocity ratio parameter $ε$ , $E c$ for Eckert number parameters and M is magnetic parameter which may be written as follows:

P r = \frac{v_{f}}{α_{f}}, η = e, ε = \frac{U_{n}}{U}, E c = \frac{b^{2}}{ν_{f}}, M = \frac{σ_{f} B_{0}^{2} a^{2}}{2 ρ_{f}} .

(10)

where

ε > 0

implies that needle travels parallel to the direction of free flow, and

ε < 0

implies that the needle is moving in the opposite direction of the free flow. The physical amounts at stake, the coefficient of skin friction and transfer of heat rate (local Nusselt number) as follows:³⁸

C_{f} = \frac{τ_{n}}{ρ_{f} u^{2}}, N u_{x} = \frac{x_{q n}}{k_{f} (T_{n} - T_{\infty})}

(11)

where

τ_{n}

is the wall shear stress, and

q_{n}

is the wall heat flux, which may be expressed as below:

τ_{n} = μ_{n f} {(\frac{\partial u}{\partial r})}_{r = e}, q_{n} = - k_{n f} {(\frac{\partial T}{\partial r})}_{r = e}

Equation plugging (5) and (6) into Equation (12) and using Equation (11), the following can be obtained:

R e_{x}^{1 / 2} C_{f} = 4 \frac{μ_{n f}}{μ_{f}} e^{\frac{1}{2}} f^{″} (e), R e_{x}^{- 1 / 2} N u_{x} = 2 \frac{k_{n f}}{k_{f}} e^{\frac{1}{2}} θ^{'} (e),

where

R e_{x} = \frac{u_{x}}{v_{f}}

is the local Reynolds number.

Method of solution with code validation

While trying to find a solution for a set of nonlinear partial differential equations (8–9) with linked boundary conditions (10), one encounters a great deal of difficulty. A numerical technique is recommended above any other method for solving these problems. It is possible to arrive at a numerical solution for the transformed system of equations (8) and (9) by using a shooting technique in conjunction with a Runge-Kutta structure of the 4th order and boundary conditions (10). The shooting method turns boundary value problems into a set of nonlinear first-order ordinary differential equations with known initial conditions that can be solved numerically. In order to find a numerical solution that is accurate to the fifth decimal place and has $η_{m a x} = 15$ , the step size $Δ η = 0.001$ is used as a criterion of convergence. To satisfy convergence and boundary restrictions,¹⁰ it is important to use the following beginning approximations.

m_{1} = f, m_{2} = f^{'}, m_{3} = f^{″}, n_{1} = θ, n_{2} = θ^{'},

(13)

By Equation (13)

m_{1}^{'} = f^{'}, m_{3}^{'} = m_{4} = f^{″'}, n_{1}^{'} = θ^{'}, n_{2}^{'} = θ^{″} = n_{3},

(14)

We obtained the following findings from Equations (13) and (14).

m_{1}^{'} = m_{2}, m_{2}^{'} = m_{3}, m_{4} = f^{'''}, n_{2}^{'} = n_{3} = θ^{″}

(15)

We obtain the values of

f^{″'}

and

θ^{″}

that appear in Equation (15):

m_{4} = - 2 \frac{ρ_{n f} / ρ_{f}}{η μ_{n f} / μ_{f}} [m_{3} + m_{1} m_{3} - \frac{σ_{n f} / σ_{f}}{ρ_{n f} / ρ_{f}} M m_{2}^{2}]

(16)

n_{3} = - P r \frac{{(ρ C_{p})}_{n f} / {(ρ C_{p})}_{f}}{η k_{n f} / k_{f}} [n_{2} + m_{1} n_{2} + \frac{μ_{n f} / μ_{f}}{{(ρ C_{p})}_{n f} / {(ρ C_{p})}_{f}} E c {(m_{3})}^{2}] .

(17)

m_{1} = ε, m_{2} = r, n = 1, m_{2} \to 1, n \to 0

(18)

For all of our numerical simulation and graphical analytic needs, we turn to Mathematica, which is a piece of software developed by MathWorks.

Code validation

A comparison of the results shear stress values $f^{″} (e)$ from bvp4c and RK4 was used to figure out how accurate the methods used to calculate were. Using the RK4 method, the governing set of equations is added to this part of the shooting approach. Also, deficient starting conditions are modified by utilizing the shooting strategy until the system achieves the required accuracy. In Table 3, the results of this study are compared to the results that Soid et al.³⁹ and Salleh and colleagues³⁸ talked about. When we worked with shooting in Mathematica, Salleh³⁸ used the bvp4c function in MATLAB. As can be observed, the values of $f^{″} (e)$ are completely consistent with the investigations that are currently being conducted. As a result, we have every reason to believe that our numerical technique is accurate.

Table 3.

Shear stress comparison values for various values of e when $P r = 1.0, ε = - 1.0, ϕ = 0$ .

Needle size, $e$	Soid et al.³⁹	Salleh et al.³⁸	Present study
$0.01$	$26.599394$	$26.599394$	$26.59396$
$0.1$	$3.703713$	$3.703713$	$3.70711$
$0.2$	$2.005424$	$2.005424$	$2.005421$

Results and discussion

The intension of this research is to determine the significance of shear stress, heat transfer, nanoparticles aggregation effect, viscous dissipation, and magnetic parameters on slender needle for $T i O_{2} - C_{2} H_{6} O_{2}$ flow of nanofluids. As part of this section, we revealed and statistically examined outcomes of $f^{'} (η), θ (η),$ $R e_{x}^{1 / 2} C_{f}$ and $R e_{x}^{- 1 / 2} N u_{x}$ on $ϕ, M, ε, e$ , and $E c$ , respectively. Both with and without aggregating effects are considered in the analysis of $T i O_{2} - C_{2} H_{6} O_{2}$ nanofluid. As control factors change, they are displayed in tables and figures. The values utilized here are determined by the far-field boundary criteria (9). The values of chooses parameters are chosen based on the key sources of flow and alternative solutions, such as $0.0 \leq ϕ \leq 0.04$ (^16–18,20), $0.0 \leq M \leq 1.0$ , $- 1.0 \leq ε \leq - 3.0,$ $0.0 \leq e \leq 0.3$ , and $0.0 \leq E c \leq 1.0.$

The influence of nanoparticle volume fraction $ϕ$ on velocity and temperature profiles is shown in Figures 2 and 3, and it can be seen for both the aggregation model and the one without it. Figure 2 shows that as the nanoparticle volume percent goes up, the nanoparticles’ speed profile goes down. In contrast, the pattern displayed in Figure 3 for the temperature profile is the reverse of what is shown in Figure 2. The influence that the volume fraction has on the flow of the nanofluid is seen in Figure 2. The increased presence of nanoparticles in the liquid causes the viscosity of the liquid to become more viscous. As can be seen in Figure 2, the presence of viscous forces causes a reduction in the flow of the fluid. In addition, the thickening of the momentum barrier layer causes the velocity distributions seen in Figure 2 to drop as the nanoparticle volume fractions increase. This is the cause of the decline. An increase in the volume fraction of titanium dioxide nanoparticles will result in an increase in the thickness of the fluid's thermal boundary layer. Similarly, as shown in Figure 3, an increase in the volume fractions of the nanofluid results in an enhancement of the nanofluid's thermal properties. Figure 3 clearly demonstrates that there is a correlation between an increase in the thickness of the boundary layer and an increase in the temperature distribution. However, this correlation is reversed when the thickness of the boundary layer is large. The temperature profiles become more noticeable when using a model that aggregates the data.

Figure 2.

Velocity profile $f^{'} (η)$ against $ϕ$ .

Figure 3.

Temperature profile $θ (η)$ against $ϕ$ .

The influence of magnetic parameter M is displayed in Figures 4 and 5 for the velocity and temperature profiles. As the value of the magnetic parameter is increased, there is a corresponding drop in the velocity profile, but there is an increase in the temperature profile. According to Figure 4, the flow of liquid decreases when the magnetic field M is increased. When magnetic fields are introduced into a flow system, a Lorentz force is produced that acts in opposition to the velocity of the flow system; thus, the velocity of the flow system decreases as the magnetic parameters are increased $(M)$ . Magnetic fields are another factor that may affect the flow characteristics of nanofluids. Due to this physical phenomenon, according to Figure 5, increasing M enhances energy conduction in nanofluid nanoparticles. As a direct consequence of this physical phenomenon, improvements to the Lorentz force and ultimately higher layers of the thermal boundary are produced. Because of this, elevating the magnetic parameter results in an enhancement of the nanofluid's thermal characteristics. Figure 5 demonstrates that the aggregation model has a higher influence.

Figure 4.

Velocity profile $f^{'} (η)$ against M.

Figure 5.

Temperature profile $θ (η)$ against M.

Figures 6 and 7 illustrate the effect that the velocity ratio parameter $ε$ has on the velocity and temperature curves, respectively. As the values of $ε$ are reduced, it is possible to see an increase in the velocities and temperatures of the profiles. In this scenario, one would be expected to behave in the complete opposite manner. When the value of $ε$ changed, the magnitude of the temperature and velocity fields both increased as a direct consequence. This is something that should go without saying. It has a detectable influence in the sense that as time passes, changes in the value of $ε$ cause the layer of momentum and thermal barrier to get progressively thinner. This indicates that it does have an effect. This is an actual repercussion of the action. This is one of the ways it presents itself in the physical world. The aggregation model has a positive influence in this scenario.

Figure 6.

Velocity profile $f^{'} (η)$ against $ε$ .

Figure 7.

Temperature profile $θ (η)$ against $ε$ .

A look at impact of needle thickness parameter e on velocity and temperature profile is shown in Figures 8 and 9. As the values of e increases, velocity profile decreases and temperature profile increases. As can be seen in Figure 8, the velocity distributions have a downward trend when thicker needles are considered. This decrease is affected when the thickness of the momentum boundary layer at the needle surface goes up more. As the thickness of the needle increases, so does the thickness of the thermal boundary layer. This makes it possible for heat to move from the needle wall into the fluid around it (see Figure 9). Figure 9 shows that an increase in the thickness of the thermal boundary layer has a big effect on the dispersion of temperatures throughout the flow.

Figure 8.

Velocity profile $f^{'} (η)$ against e.

Figure 9.

Temperature profile $θ (η)$ against e.

The significance of Eckert's number $E c,$ as seen in Figure 10, is readily apparent. The temperature profile rises as the Eckert number is increased to higher and higher values. According to Figure 10, this is because when the temperature rises, there is also an increase in the amount of thermal energy that is transported, which causes the nanofluid to become increasingly hot. The temperature profile has been shown to go up, which means that the temperature gradient around the stretching needle is higher than the temperature of the free stream. It was seen that the temperature profile was getting better, which led to this fact being found out. The situation in which $E c$ comes within the range of $0.0 \leq η \leq 1.0$ , which can be seen in the picture, illustrates that the model that does aggregate data gives better results than the model that does not. This can be observed since $E c$ falls within this range. The large amount of heat that is released is the main reason why the temperature inside the nanofluid stretching needle rises right away. Since the effect of the dissipation factor has reached its peak, the nanofluid particles have started to bump into each other. This is because the value of the parameter has exceeded its potential.

Figure 10.

Temperature profile $θ (η)$ against $E c$ .

Figures 11 and 12 show how the volume percentage of nanoparticles and their velocity ratio influence the amount of friction experienced by the skin and the Nusselt number. The addition of nanoparticles into the base fluid would result in an increase in the force of drag. This would be caused by interactions between suspended particles in the flow. As a result, there will be a decrease in the coefficient of friction on the wall. When the model is evaluated without considering aggregation, the skin friction coefficient is much higher. Because of the narrow surface of the needle and the thin wall of the titania-aggregated nanofluid, there is a tiny area for suspended particles to collide, which results in a reduced skin coefficient of friction. The rate of heat transmission quickens with the incorporation of nanoparticles into the system. The higher the temperature, the more intense the intermolecular forces become, which may help explain this phenomenon. In addition to this, the temperatures must be very high in order to disrupt the bonds that hold the nanoparticle molecules together. Figure 12 illustrates that an increase in the rate of heat transfer takes place when more nanoparticles are placed in the flow. In addition, the rate of heat transmission is greater for needles with thinner walls as well as titania nanoparticles with thinner walls. Heat may be physically transported more readily through a thin wall than through a thick wall. Because the thickness of the thermal boundary layer is reduced, the model without aggregation has a greater heat transfer rate than a model with a lower density. This is because the face of the wall that is exposed to the fluid is reduced, which in turn reduces the face of the wall that is exposed to the fluid.

Figure 11.

Skin friction of $ϕ$ towards $ε$ .

Figure 12.

Nusselt number of $ϕ$ towards $ε$ .

Figures 13 and 14 show how the needle thickness e and the velocity ratio parameter $ε$ affect the skin friction and the local rate of heat transfer for models with and without aggregation. When the needle thickness parameter is changed, the skin friction and Nusselt number profiles both go up. Figure 13 demonstrates that the surface shear stress rises with a raised value of e, as can be seen in the figure. When the value of e goes up, the thickness of the momentum boundary layer on the wall goes down. This is what causes the behavior we see. When e is given a larger value, there is a corresponding rise in the local heat flow. In fact, the decrease in the thickness of the thermal boundary layer that occurs alongside a rise in the needle thickness prevents heat from escaping through the needle wall and into the fluid that is located around it. In addition to this, an increase in the thickness of the thermal boundary layer also improves the temperature distribution throughout the flow. Because of this circumstance, there is an increase in the amount of local heat flux at the needle wall.

Figure 13.

Skin friction of e against $ε$ .

Figure 14.

Nusselt number of e against $ε$ .

Figures 15 and 16 show the variations in the skin friction coefficient and the heat transfer coefficient that occur for various values of the parameter $ε$ in conjunction with four distinct values of the parameter $M = 0.25, 0.5, 0.75, 1.0$ . According to the illustration, a rise in the value of M results in a drop in the skin friction coefficients while simultaneously leading to an increase in the Nusselt number. As a consequence of this, the existence of M narrows the range of solutions that are conceivable and creates a considerable barrier for fluid particles. This circumstance results in the production of heat, and an increase in Lorentz force occurs during the electrodynamic movement of the boundary layer. Nanofluid is drawn to a thin needle by means of this force, which, in turn, results in a reduction in friction.

Figure 15.

Skin friction of M against $ε$ .

Figure 16.

Nusselt number of M against $ε$ .

The influence of $E c$ with $ε$ on the rate of heat transmission is shown in Figure 17. It should come as no surprise that when Eckert number values climb, the rate of heat transmission slows down. In the meanwhile, a rise in the $E c$ intensity brings about an enhancement in the thermal state of the fluid, which, in response, brings about an increase in the thermal boundary layer, as can be seen in Figure 17. As a result of what has happened so far, the temperature of the fluid as a whole will gradually rise. When there is an increase in the total quantity of dissipation that is occurring, the Nusselt numbers have indeed been observed to be decreasing, but when there is no dissipation going to take place at all, the Nusselt numbers $(R e_{x})^{- 1 / 2} N u_{x}$ have really been determined to be greater. This was demonstrated via a broad range of empirical investigations.

Figure 17.

Nusselt number of $E c$ against $ε$ .

Conclusion

Using nanoparticles that contain $T i O_{2} - C_{2} H_{6} O_{2}$ , this work involves the transfer of heat via a moving horizontal needle that is very thin. The impacts of nanoparticle aggregation, MHD, and viscous dissipation were considered. When a number of certain physical conditions are met, it is feasible to orient the needle so that it is lying horizontally in the nanofluid. RK-IV, in conjunction with shooting, is used so that a solution may be determined for the modeled issue. Graphical models have been used to explain the implications that different emergent factors have on the flow, thermal, and skin friction properties, as well as the local heat transfer rate. Due to the fluid nature of the amounts under investigation, there are only a few important things to report: When the aggregation model is used, the temperature profiles are able to be more clearly distinguished. Velocity profile decreases with increasing values of $ϕ, M, e$ and increases against $ε .$ Temperature profile enhances with increasing values of $ϕ, M, e, ε$ , and $E c .$ The reduction in skin friction between a needle and a fluid can be observed when the values of M and $ϕ$ are boosted. Furthermore, it was also noticed an increase in heat transfer on needle surface dramatically when $ϕ, e$ , and M were raised, whereas $E c$ displayed the opposite effect.

Titania nanoparticles and thin needle walls have been shown to increase the drag forces between the needle surface and fluid particles in a nanofluid made of ethylene glycol. This phenomenon is suitable for certain applications requiring substantial frictional forces because it has the effect of slowing down the moving surface. The current model has a high rate of heat transfer, which results in a surface that cools down very quickly. This is an additional benefit of the model. Possessing this quality is necessary to carry out the necessary cooling activities.

Footnotes

Author contributions

All authors contributed equally to this manuscript.

Data availability

Data will be available on demand from Z.M.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Zafar Mahmood

Author biographies

Bilal Ali a PhD student in School of Mathematics and Statistics, Central South University Changsha, 410083, China. His research focus on applied Mathematics, neural network, computational fluid mechanics.

Sidra Jubair a PhD student in School of Mathematical Sciences, Dalian University of Science & Technology, Dalian, China and working as lecturer mathematics at Department of Mathematics and Statistics, Women University Swabi, KPK, Pakistan. Her area of research is applied mathematics.

Dowlath Fathima is currently working as an assistant professor in College of Science and Theoretical Studies, Department of Basic Sciences at Saudi Electronic University, Saudi Arabia. She received a PhD from B S Abdur Rahman University in 2014. Her research interests span both Computer Science and Mathematics. Much of her work has been on Operations Research and Inventory control, mainly through the application of statistics and functional analysis, fractional calculus. Few of other works involves Differential equation, Mathematical modelling and its application. To her credit, she has many published articles in journals and conferences.

Afroza Akhter is pursuing her PhD in VIT Bhopal-India since 2021 in Mathematics using applications. She has received her Master's (MSc) in Mathematics from University of Kashmir. She is author/coauthor of many papers published both in national and international journals. She has also presented many papers in some international conferences.

Khadija Rafique a PhD student at department of mathematics and Statistics, Hazara University Mansehra, Pakistan. Her area of research is applied mathematics, difference equation, fluid mechanics.

Zafar Mahmood is currently working lecturer mathematics. He received a PhD degree from Hazara University Mansehra, Pakistan. His research works involves Differential equation, Mathematical modelling and its application, computational fluid mechanics. To his credit, he has many published articles in journals.

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MHD flow of nanofluid over moving slender needle with nanoparticles aggregation and viscous dissipation effects

Abstract

Keywords

Introduction

Description of the problem

Method of solution with code validation

Code validation

Results and discussion

Conclusion

Footnotes

Author contributions

Data availability

Declaration of conflicting interests

Funding

ORCID iD

Author biographies

Appendix

References