Abstract
The present article summarized the effects of double magnetic dipole for chemically reactive viscoelastic fluid in the presence of two different ferromagnetic particles, namely, nickel zinc ferrite and magnetite ferrite (). Due to double magnetic dipole, an external magnetic field is applied normal to the flow. Blood is used as base fluid due to its viscoelastic fluid properties. The Cattaneo-Christov heat flux model is used for heat transport phenomena. The physical model is formulated in the form of partial differential equations which are then converted into ordinary differential equations using the suitable transformations. The system is solved numerically using shooting method along with Runge-Kutta-Fehlberg method. The characteristics of different parameters like the strength of homogeneous-heterogeneous reactions (), ferrohydrodynamic interaction (), Schmidt number (), Deborah number (), and thermal relaxation time () on velocity, temperature, and concentration profiles are analyzed through graphs and in tabular form. It has been observed that as magnetic dipole creates a force which attracts the ferrite particles, hence, it slows down the velocity profile. Concentration field depresses due to the presence of strength of heterogeneous reaction parameter .It is also noted that by expanding values of thermal relaxation time (), the temperature profile shows a reverse behavior.
1. Introduction
Ferrofluids lie in the category of smart materials those consist of micron-sized colloidal magnetic nanoparticles that are saturated in a nonmagnetic base fluid. The most fascinating feature of these fluids is its highly magnetizing ability when an external magnetic field is applied. Pappell [1] initially highlighted the characteristic of ferrofluid in 1963. He utilized ferrofluid in a weightless atmosphere in terms of liquid rocket fuel, which is pinched near a pump inlet by applying an external magnetic field. Ferrofluids have many appealing applications in electrical instruments such as in hard disk, rotating X-ray tubes, shafts, and rods. The role of ferrofluids in biomedical equipment’s is no doubt incredible, which is helpful in the process of wound treatment, asthma treatment, removal of cancer with hyperthermia, and many more. For these kinds of procedures, the homogeneous and heterogeneous have great importance. The homogenous and heterogeneous reactions take places in the bulk and occur on the catalyst surface, respectively. Whether the process is homogeneous or heterogeneous depends upon the situation that they exist in the absolute majority of the fluid or a part of catalytic surfaces. In essence, the homogeneous process is continuously intact within the given phase, whereas heterogeneous reactions have restricted boundaries. There are also a variety of chemically reacting structures that involve both h-h reactions termed as such as catalysis, burning, and biochemical reacting systems. In this flow, due to an external flow applied on the outer region of boundary layer, there exists a reaction in that region as explained by [2, 3]. Initially, Andersson and Valnes [4] studied the viscous ferrofluid with magnetic dipole effects on the stretching surface. They examined the influence of magnetothermomechanical coupling on the fluid. Numerical studies related to ferrofluid inside a channel with cold walls focusing on a line source dipole delineated by Ganguly et al. [5]. In a porous medium, Sharma et al. [6] specified the attributes of convection by focusing dust particles on a ferromagnetic fluid. The results corresponding to characteristics of heat transfer in a Darcy number ferrofluid flow with porous wall is analyzed by Strek [7]. The dipole effects in a ferrofluid flow origin below the channel with isothermal wall were studied by Strek and Jopek [8]. They focused on time dependent heat transfer and analyzed the spatial alteration creating magnetization because of gradient effects, depending temperature magnetic susceptibility for ferrofluids. Sadiq et al. [9] studied the Casson fluid model to study the thermal performance under the Brownian and thermoheretic effects of hybrid nanoparticles. The dipole’s magnetization in the series flow of ferrofluid taking elongated surface highlighting thermal radiation is explored by Makinde and Aziz [10]. Aminfar et al. [11] implement a non-acting magnetic field inside a vertical tube during the numerical studies of mixed convection of ferrofluid. During his studies, he used both + and - types of magnetic gradients; the (-) gradient of magnetic field reacts the same as the buoyancy forces and boosts the Nusselt number, whereas the magnetic field with a (+) gradient decelerates it. A stagnation point flow along elongated sheet in the presence of heterogeneous-homogeneous reactions that is analyzed by Bachok et al. [12] analyzed a chemically reactive stagnation point flow over the elongated sheet. In his paper, he discussed the case of fluid having less kinetic viscosity in which he observed that when the extending velocity is low as compared to free steam velocity, a boundary layer is achieved, whereas the inverted boundary layer attains in the case when stretching the velocity exceeds the free steam velocity. Affixing 2-phase mixture model and effecting control volume method, Aminfar et al. [13] executed the transversal non acting magnetic field accomplishing electric current during the inspection of flowing ferrofluid inside a duct. They showed the flourishing behavior of average heat transfer coefficient. The shrinking surface is taken by Kameswaran et al. [14] during the investigation of homogeneous-heterogeneous reaction in a porous medium. By taking into account the radiation effect in an extended sheet, Titus and Abraham [15] used ferrofluid. It also concludes that due to magnetic field local vortex vary the advection energy transport also boosting heat transfer ability. Sheikholeslami et al. [16] conferred the aspects of non-uniform magneto hydrodynamic flow of ferroliquid utilizing convective heat transport. Hayat et al. [17] also detailed the effects of MHD nanofluid with heterogeneous-homogeneous reaction and velocity slip condition. It is concluded that in the case of concentration, both homogeneous and heterogeneous parameters show a reverse behavior. Imtiaz et al. [18] reflected the impact of homogeneous and heterogeneous reactions in the examination of MHD flow in a curved stretchable surface. He depicted the increasing behavior between curvature parameter and fluid velocity. Due to many applications, the boundary layer flow of non-Newtonian fluids with different effects has added an enormous attraction in the recent years [19–33].
The ultimate goal of this research is to scrutinize the effects of magnetic double dipole in the examination of Maxwell ferrofluid which is composed of base fluid blood, whereas ferrite particles were used as under the high lightening impact of homogenous and heterogeneous reaction. Also, the concentration levels of these particles are about 20% into the base fluid. The basic theme of including these ferrite particles is to enhance the strength of heat transferring phenomena which is the inspiration of this research. Brief literature survey is summarized in Section 1. Section 2 is focused on the mathematical formulation of problem. In Section 3, the computational procedure is discussed in details. The possible outcomes of the current study are deliberated in Section 4. Section 5 presents the key features of this article.
2. Mathematical Formulation
Consider a steady 2D flow of an incompressible non-Newtonian Maxwell electrically conducting a ferromagnetic fluid running on a flat surface in direction as shown in Figure 1. The location of dipole is set in thedirection as the displacement between the dipole and surface is taken as, while the magnetic fieldgenerated with the presence of magnetic dipole directions as-axis. The scalar potential of permanent magnetic dipole which influences ferrofluid defined in [2] is where is the representation of dipole moment per unit length. The relation between gradient of magnetic scalar potential and applied magnetic field is held as . The components of are
The force field strength has a resultant magnitude , with their element forms that are termed as
Experiencing Equation (1) in Equations (4)-(5), we get the following form by expanding up to order :
The assuming effects of surface’s wall is , with including assumption that We get the transformed form of equation (4)
The change in magnetization can be expressed as
2.1. Analysis of Flow
We have considered a viscoelastic fluid flow over a flat surface with double magnetic dipoles, which are placing at a space () from the wall and perpendicular to the surface. Consider as the length of plate with the wall. The temperature changes linearly with the plate length that is defined as . In the boundary layer flow, the connection between homogeneous and heterogeneous reactions adding 2-species chemically, named, , are taken following Chaudhary and Merkin [2] as
Concentrations are represented by , and are termed as rate constants. The process is considered to be isothermal for both reactions.
The corresponding flow equations in the presence of magnetic dipole are
The admissible flow conditions are where (,) is the velocity components alonganddirections, () is the magnetic permeability, () is the dynamic viscosity, () is the kinematic viscosity of nanofluid, () is the nanofluid density,is the specific heat, () represented as thermal conductivity of the nanofluid, () is the relaxation time of fluid, and () identifies as the relaxation time of heat flux.
2.2. Thermophysical Properties
Using similarity transform as defined by [23],
By implementing similarity variables express in Eq. (12), our modified equations take the subsequent form: with nondimensional boundary conditions:
In Eqs. (13)-(16), is the ferrohydrodynamic interaction, defines the Prandtl number, mean the Deborah number, is represented here as a nondimensional thermal relaxation time, the shows Schmidt number, and describe the strength of homogeneous and heterogeneous reaction parameters, which are defined here as
are equivalent, i.e., ; then, we can write as
Utilizing Eqs. (15)-(16), we get the equation as
The corresponding boundary equation of the concentration field yields the following form:
The conversions of the wall drag and convective heat transfer into a nondimensional form are scaled as
The wall shear stress is termed as
The dimensionless expressions of Eq. (22) are attained: where,
3. Numerical Methodology
The saturation of two ferrite particles into Maxwell fluid under the action of a double magnetic dipole is examined with homogeneous and heterogeneous reactions. The transformed nonlinear systems of ODEs are solved numerically using the shooting method along with the RK-45 algorithm. For the implementation of the shooting method, one should convert the boundary value problem into an initial value problem. The reduce initial value problem is further converted into a system of first order differential equations and then solved by choosing the missing conditions as an initial guess. Therefore, the suitable transformations are used to obtain initial value problem. The new set of variables for first order system of equations is defined as
This substitution yields which are subject to the following conditions:
To solve the above system of Eqs. (27)-(33), the values of , , and are unknown, so by taking a suitable initial guess, the convergent numerical solution is obtained. It is important to note that if the boundary residuals are fewer than the tolerance error 10-6, the calculated solution converges. If the computed results do not satisfy this requirement, the initial estimates are changed by using Newton’s technique, and the procedure is repeated until the solution fulfills the specified convergence threshold. The thermophysical properties of the base fluid and ferrite particles for numerical procedure are defined in Table 1.
4. Results and Discussion
The mathematical model of chemically reactive viscoelastics fluid over the stretching surface is solved numerically. The numerical solutions are checked by applying the grid independence test. The analysis is performed at different tolerance levels for Nusselt number and skin friction coefficient. The numerical values of Nusselt number and skin friction are presented in Table 2.
Table 3 represents the performance of quantity of engineering interest (Nusselt number) against ferrohydrodynamic interaction () and Prandtl number () in both cases nickel zinc ferrite and magnetite ferrite . It is depicted that the increment occurs in the Nusselt number due to the increasing values of, with better results of magnetite ferriteas compared to nickel zinc ferrite. Table 4 displays the response of mass flux coefficient against the strength of homogeneous and heterogeneous parameters and Schmidt number. It is perceived that the mass flux coefficient shows a positive trend towards the strength of homogeneous () and heterogeneous parameters () while an opposite behavior seen in the case of the Schmidt number (). The higher magnitude was observed in the case of nickel zinc ferrite.
The analyzation of ferrohydrodynamic interaction () on velocity field is seen in Figure 2. It is depicted that the velocity decreases due to the increasing values of ferrohydrodynamic interaction parameter (). This opposite behavior is due to the action of Lorentz forces which resist the flow and provide more resistance to transportation phenomena. From the graph, it is cleared that the magnetite ferrite offers more resistance due to which the velocity profile decreases in the case of magnetite ferrite . Figure 3 results an inclination in temperature with altering values of . It is detected that the temperature profile boosts up as soon as the ferrohydrodynamic interaction parameter () enhances. Reasoned is that Lorentz forces which produces under the action of magnetic field have the potential to produce resistance which in term produce heating among the layers of fluids hence thermal boundary layer thickness rises. It is also perceived that the magnetite ferrite offers more resistance due to which the rapid temperature enhancement occurred in the case of the magnetite ferrite . The sketch of velocity with augmenting values of Deborah number () is displayed in Figure 4. It is detected that the increasing Deborah number results in a decreasing velocity profile. As () precisely relate to relaxation time of fluid, immediately () increasing fluid relaxation time shot up and provide extra blocking to the fluid motion which results in thinning momentum boundary layer thickness. The maximum velocity is depicted in the case of nickel zinc ferrite . The graph of Deborah number () against the temperature field is seen in Figure 5. It is perceived that the augmented values of the Deborah number () amplifies the temperature field. This phenomena can be depicted as the enlarging Deborah number () tends to have a larger relaxation time of fluid which directly means to offer resistance and generates heat which enhances the temperature profile. The effects of the thermal relaxation parameter () upon the velocity profile is observed in Figure 6. It is clear from the figure that the increasing thermal relaxation parameter () is responsible for the increasing velocity profile. This fact is explained as by up surging thermal relaxation parameter () momentum of fluid particles also enhances which in term increase kinetic energy and velocity profile. Figure 7 enlightened the conduct of thermal relaxation time () on the temperature field. By expanding values of () temperature profile show reverse behavior, because enlarging thermal relaxation time implies materials particles demand extra pace to exchanging energy to their connecting particles, thus cause reduction in temperature profile. The impression of Prandtl number on velocity is realized in Figure 8. The increasing behavior of velocity is observed by the altering values of the Prandtl number, which is due to straight forward relation between the Prandtl number and momentum diffusivity. As soon as the Prandtl increases, the momentum diffusivity also increases; thus, in the result, fluid motion and momentum boundary layer thickness also boost up. Association between Prandtl number and temperature field clearly delineated in Figure 9, the contrary response of temperature towards Prandtl number, actually Prandtl number exhibit inverse relation to thermal diffusivity of fluid, So by enhancing Prandtl number directly means to lessen diffused heat betwixt fluid layers hence reduces temperature profile. The response of Schmidt number () towards the concentration field is characterized in Figure 10. The increase in Schmidt number () consequently increases the concentration field because the concentration field relates with a viscous diffusion rate to molecular diffusion rate, so when the Schmidt number () magnifies the viscous diffusion rate, it also magnifies which in turn enhances the concentration field. Figure 11 shows the connection between the strength of homogeneous reaction parameter and concentration field. It is observed by enhancing the strength of homogeneous parameterconcentration field that is going to shrink. This conduct is explained as reactants are consumed during the process of homogeneous reaction which depresses concentration profile. Also, the minimum concentration was observed in case of magnetite ferrite . The graph of concentration field against the altering values of strength of heterogeneous reaction parameter is displayed in Figure 12. An increase in causes a reduction in concentration field, while higher concentration is found in the case of nickel zinc ferrite . Figure 13 shows the impact of Pr number upon a heat transfer rate. As we increase the Prandtl number, the Nusselt number also increases because by increasing Prandtl number rate of momentum, the diffusivity also increases which gives rise to the kinetic energy and also the heat transfer through convection. The impact of strength of homogeneous parameter and Schmidt number () in the mass flux coefficient is carried out in Figure 14. It is seen that () shows a negative response, while displays an augmented amplitude against (). Figure 15 depicts the trend of () that relates with the strength of the heterogeneous parameter and Schmidt number (). As it is remarked because of the amplification in (), the mass flux coefficient () magnifies due to the accumulation of particles by the generation of the heterogeneous mixture. Also in the case of (), we observed that as soon as () increases, the Sherwood number () reduces by reducing the mass diffusivity.
5. Concluding Remarks
In this article, we have examined the response of double magnetic dipole in a chemically reactive viscoelastic fluid over a flat sheet by considering the two nanomagnetic ferrite particles. For heat transfer rate, we used the Cattaneo-Christov heat flux model which is the generalization of Fourier law by including thermal relaxation term. The thermal relaxation time converts the energy transport in form of thermal waves with finite speed. The modeled PDEs are converted into the system of ODEs and then solved numerically by shooting method. The results are depicted graphically with the impact of important parameters. The major outcomes of this study are as follows: (i)Due to the force generated by magnetic dipoles, the velocity of the fluid reduces, and the temperature increases by increasing the ferrohydrodynamic interaction parameter (ii)For the large value of Deborah number (), the velocity profile decreases(iii)The thermal relaxation parameter () shows a positive response towards velocity while the negative response towards temperature because particles take a surplus time to shifting energy(iv)The strength of homogeneous reaction parameter undermines concentration(v)The concentration field depresses due to the presence of strength of heterogeneous reaction parameter (vi)It has been observed that heat transfer rate increases in the presence of magnetite ferrite as compared to nickel zinc ferrite by increasing Pr(vii)From physical point of view, Pr is the proportion of momentum diffusivity to thermal diffusivity. The contrary response of temperature towards the Prandtl number is observed; actually, the Prandtl number exhibits an inverse relation to thermal diffusivity of the fluid, so by enhancing Prandtl number directly means to lessen the diffused heat between fluid layers and hence reduce the temperature profile
Nomenclature
| : | Velocity components |
| : | Length of plate |
| : | Surface temperature |
| : | Curie temperature |
| : | Magnetic permeability |
| : | Dynamic viscosity of nanofluid |
| : | Dynamic viscosity of base fluid |
| : | Thermal conductivity of nanofluid |
| : | Density of nanofluid |
| : | Heat capacitance of fluid |
| : | Heat capacitance of solid particle |
| : | Heat capacitance of nanofluid |
| : | Magnetic field intensity along direction |
| : | Magnetic field intensity along direction |
| M: | Magnetization |
| : | Kinematic viscosity |
| : | Wall shear stress |
| : | Relaxation time of fluid |
| : | Relaxation time of heat flux |
| : | Deborah number in case of double dipole |
| : | Thermal relaxation in case of double dipole |
| : | Schmidt number |
| : | Strength of homogeneous reaction |
| : | Strength of heterogeneous reaction |
| : | Tolerance rate |
| : | Ratio of mass diffusion coefficients |
| : | Magnetic scalar potential. |
Data Availability
Data are available within the article.
Conflicts of Interest
The authors declare no conflict of interest.
Acknowledgments
This study is supported by the Taif University Researchers Supporting Project number (TURSP-2020/31), Taif University, Taif, Saud Arabia.