Issue 
Mechanics & Industry
Volume 20, Number 5, 2019



Article Number  502  
Number of page(s)  9  
DOI  https://doi.org/10.1051/meca/2019022  
Published online  15 July 2019 
Regular Article
Influence of rotating magnetic field on Maxwell saturated ferrofluid flow over a heated stretching sheet with heat generation/absorption
^{1}
Departmentof Mathematics Statistics, Bacha Khan University, 24420 Charsadda, KPK, Pakistan
^{2}
Department of Mathematics and Statistics, FBAS, IIUI, H10, Islamabad 44000, Pakistan
^{3}
Department of Informatics, Faculty of Mathematics and Natural Sciences, University of Oslo, Oslo, Norway
^{4}
Department of Electronic Engineering, University of Engineering and Technology, Taxila, Pakistan
^{*} email: aaqib@bkuc.edu.pk
Received:
29
January
2018
Accepted:
22
March
2019
This article is focused on Maxwell ferromagnetic fluid and heat transport characteristics under the impact of magnetic field generated due to dipole field. The viscous dissipation and heat generation/absorption are also taken into account. Flow here is instigated by linearly stretchable surface, which is assumed to be permeable. Also description of magnetothermomechanical (ferrohydrodynamic) interaction elaborates the fluid motion as compared to hydrodynamic case. Problem is modeled using continuity, momentum and heat transport equation. To implement the numerical procedure, firstly we transform the partial differential equations (PDEs) into ordinary differential equations (ODEs) by applying similarity approach, secondly resulting boundary value problem (BVP) is transformed into an initial value problem (IVP). Then resulting set of nonlinear differentials equations is solved computationally with the aid of Runge–Kutta scheme with shooting algorithm using MATLAB. The flow situation is carried out by considering the influence of pertinent parameters namely ferrohydrodynamic interaction parameter, Maxwell parameter, suction/injection and viscous dissipation on flow velocity field, temperature field, friction factor and heat transfer rate are deliberated via graphs. The present numerical values are associated with those available previously in the open literature for Newtonian fluid case (γ _{1} = 0) to check the validity of the solution. It is inferred that interaction of magnetothermomechanical is to slow down the fluid motion. We also witnessed that by considering the Maxwell and ferrohydrodynamic parameter there is decrement in velocity field whereas opposite behavior is noted for temperature field.
Key words: Maxwell ferrofluid / rotating magnetic field / heat generation / heat absorption / viscous dissipation / suction / injection
© AFM, EDP Sciences 2019
1 Introduction
Boundary layer theory by Prandtl confirmed to be greatly used in the flow of Newtonian fluids as the Navier–Stokes equations can be reduced to simpler equations. Such flow along with heat transfer analysis has acknowledged extensive attention in current decays because of their momentous contribution in industry. A few of applications are found like hot rolling, continuous casting of metals, glass blowing, cooling of an infinite metallic plate in a cooling bath, metal spinning and the aerodynamic extrusion of plastic sheets. The concept on boundary layer flow of over a continuously moving surface was initiated by Sakiadis [1]. The pioneer work of Sakiadis has been extended by many researchers for various physical aspect of the problem [2–6]. Furthermore, the industrial use of nonNewtonian fluids with heat transport analysis inspires the scientists and engineers in the research field. Industrial processes like in polymer depolarization, processing, composite, fermentation, bubble columns and absorption, boiling and lubrication which shows the consequence of nonNewtonian liquids in industries. Numerous models of nonNewtonian fluids describe different fluid behaviors exists. Maxwell fluid is one of such type fluid which is exclusive valuable for polymers for low molecular weight. Many mathematicians and scientists studied the behavior of Maxwell's liquid flow. Nadeem et al. [7] described numerical investigations of nanofluid over a stretching surface by utilizing Maxwell model. Majeed et al. [8] described magnetohydrodynamic (MHD) effect on stagnation point of Maxwell liquid flow due to stretched surface by applying homotopy analytical method (HAM). Hayat and Qasim [9] have examined chemical reaction and heat transport analysis of Maxwell ferromagnetic liquid flow toward a stretching sheet with the effects of Soret and suction. Aliakbar et al. [10] have investigated the radiation effect on MHD flow of Maxwellian liquids over a stretching surface. But these studies do not incorporate the heat transfer and heat absorption effects. Bataller [11] discussed radiation and heat source/sink impact on viscoelastic fluid flow and heat transport phenomena past a stretchable surface. Abel et al. [12] analyzed the combine impact of magnetic field and viscous dissipation on viscoelastic liquid flow due to stretched surface. Rashidi et al. [13] have obtained analytical approximate boundary layer solution of electrically conducting twodimensional viscoelastic liquid flow over a moving surface. Majeed et al. [14] performed the heat transfer analysis of magnetoEyring–Powell fluid flow over a stretchable surface with multiple slip effects and variable transverse magnetic field.
Ferrofluids (magnetic nanofluids) are liquids which are vastly magnetized under the impact of applied magnetic field. These fluids were first industrialized by Papell [15] at NASA in 1963. In ferrofluids, nonmagnetic carrier fluids are saturated with micronsized colloidal ferrous particles uniformly distributed [16]. The fluid shows remarkable applications other than in rocket fuel. These are used in shock absorbers, coolers of nuclear reactor, MEMS, microfluidic actuators, leakproof seals, lithographic designing and much more [17–21]. Neuringer [22] demonstrated stagnation point flow of ferromagnetic liquid flow toward a cold wall along with surface temperature declines linearly. Sheikholeslami and GorjiBandpy [23] reported the flow of ferrofluid in a heated cavity from beneath with the impact of magnetic field. Rehman et al. [24] investigated theoretically 2D boundary layer flow of Maxwell ferromagnetic fluid over a flat plate under the impact of two equal magnetic dipole and Cattaneo–Christov flux model. Hassan et al. [25] examined the nanoparticle shape effects on ferrofluids flow and heat transfer in the presence of low oscillating magnetic field. Feng et al. [26] examined experimental study for controlling heat transport characteristics of ferrofluid. Rashidi et al. [27] discussed the mixed convection flow of magnetic nanofluid flow in a channel with sinusoidal walls. Sheikholeslami and GorjiBandpy [28] investigated numerically ferrofluid flow in a cavity under the impact of applied magnetic field. Some of the literature related to ferromagnetic liquid flow because dipole field were executed by the scientist in the set [29–38].
Literature survey discloses that a lot of work has been done by several researchers to explore the heat transfer characteristic on boundary layer stretched flows. But the main theme of the present investigations is to discuss the heat generation or absorption effect on boundary layer Maxwell ferromagnetic liquid flow with the existence of point dipole field which has not been studied so far. The modelled equations of flow problem consisting of momentum and thermal transport are converted to a nonlinear ordinary differential equations (ODEs) using suitable similarity approach and then solve by Runge–Kutta (R–K) method. The influence of various constraints is discussed in detail through graphs.
2 Mathematical model
2.1 Magnetic field due to dipole
Flow of magnetic fluid influenced due to dipole field (see Fig. 1) and ψ represents the scalar potential of dipole magnetic field, which are(1)and components of magnetic field H are(2) (3)
Magnetic body force and gradient of H are proportional to each other, we get:(4)
In component form, we have(5) (6)
The representation of magnetization M is taken in the form of linear temperature as [39](7)
Fig. 1 Geometry of the problem. 
2.2 Flow analysis
Here, we consider 2D incompressible electrified Maxwell ferromagnetic liquid flow under the influence of line dipole over a linear permeable sheet. Sheet is stretched with u _{ w } and perpendicular to yaxis as shown Figure 1. A magnetic dipole is located at distance “a” with its center is on yaxis. Due to dipole, the direction of magnetic fluid in the +ve xdirection gives rise to magnetic field of appropriate strength to drench the ferromagnetic fluid. Here, wall temperature is T _{ w } and Curie temperature is T _{ c }, whereas the free stream temperature away from the wall is T _{∞} = T_{c} . Under Boussinesq's condition, the governing flow equations in the presence of magnetic dipole are:(8) (9) (10) with boundary conditions are (11) (12)
Here, c > 0 signify the rate of stretching sheet, A is constant, v _{ w } represents the suction/injection velocity and is characteristic length.
3 Solution of the problem
We introduce the nondimensional stream function φ (ξ, η) and temperature θ (ξ, η) assumed by [39].(13)
Here, , (ξ, η) are the nondimensional component of φ (14)
Putting equations (13)–(15) into equations (9) and (10), and obtain the expression up to the power of ξ^{2}, we get: (16) (17) (18)
Correspondig boundary relations (11) and (12) takes the form (19)
The pertinent parameter appearing in the above equations (16)–(18) are(20)
4 Quantities of physical interest
Skin friction and heat transfer rate are stated as (21)and(22)
Using the equations (13)–(15) we get(23) where refer local Reynolds number. It is more attractive and appropriate to exchange the nondimension Nusselt number with the distance free from ξ, obtained a ratio termed as heat transfer rate as sheet.
5 Numerical scheme
In this sections, we have solved equations (16)–(18) with corresponding boundary conditions (19). It is tough to find analytical solution, so to overcome this complexity, we solve it numerically. For this R–K based shooting method is used with the aid of Matlab package for numerous set of pertinent parameters β, Pr, S, γ _{1}, Q and λ. The set of coupled PDEs is transformed into differential equations for seven unknowns which are solved as an initial value problem (IVP). Let z _{1} = f, z _{2} = f ^{′}, z _{3} = f ^{′′}, z _{4} = θ _{1}, z _{5} = θ _{1} ^{′}, z _{6} = θ _{2}, z _{7} = θ _{2} ′. By introducing these new constraints in the equations (16)–(22) we obtain: (24)
Since z _{3}(0), z _{5}(0) and z _{7}(0) are not mentioned, so we establish initial guesses start with z _{3}(0) = w _{10}, z _{5}(0) = w _{20} and z _{7}(0) = w _{30}. Let χ _{1}, χ _{2} and χ _{3} be the correct estimation ofz _{3}(0),z _{5}(0) and z(0), respectively. The final form of ODEs is solved by utilizing R–K scheme and signifies the values of z _{3}, z _{5} and z _{7} at η = η _{∞} by z _{3}(w _{10}, w _{20}, w _{30}, η _{∞}), z _{5}(w _{10}, w _{20}, w _{30}, η _{∞}) and z _{7}(w _{10}, w _{20}, w _{30}, η _{∞}), respectively. Meanwhile y _{3}, y _{5} and y _{7} are obviously a function of χ _{1}, χ _{2} and χ _{3}, they have opened in Taylor series around χ _{1} − w _{10}, χ _{2} − w _{20} and χ _{3} − w _{30}, only linear terms survive. The treatment of difference quotients is made for the derivatives appeared in these Taylor series expansions. Thus, after resolving the set of Taylor series expansions for δχ _{1} = χ _{1} − w _{10}, δχ _{2} = χ _{2} − w _{20} and δχ _{3} = χ _{3} − w _{30}, we obtain the new guess w _{11} = w _{10} + δw _{10}, w _{21} = w _{20} + δw _{20} and w _{31} = w _{30} + δw _{30}. Following, process is repeated starting with z _{1}(0), z _{2}(0), w _{11}, z _{4}(0), w _{21}, y _{5}(0) and w _{31} is considered as initial values. The overall iteration procedure is recurring with most current values of χ _{1}, χ _{2} and χ _{3} until we fulfill the required boundary conditions.
Lastly w _{1n } = w _{1(n−1)} + δw _{1(n−1)}, w _{2n } = w _{2(n−1)} + δw _{2(n−1)} and w _{3n } = w _{3(n−1)} + δw _{3(n−1)} for n = 1, 2, 3,… are achieved which appeared to be the most preferred approximate initial values of z _{3}(0), z _{5}(0) and z _{7}(0). So, in this way, we obtained all initial conditions. Now it is possible to solve the set of ODEs by R–K method so that the required field for a definite set of convergence parameters can simply establish with error is chosen to be 10^{−5} in order to obtain the best convergence.
6 Graphical outcomes
In this segment, we have investigated the performance of physical parameters with the assistance of graphs and table. The graphical outcomes are displayed for dimensionless velocity profile, temperature profile, Nusselt number, and skin friction against MHD interaction parameter, Maxwell parameter, heat generation or absorption, viscous dissipation, Prandtl number and suction or injection parameter. For computational work, the value of the controlling parameters is considered as Pr = 7.0, β = 0.1, γ _{1} = 0.2, λ = 0.01, S = 0.1, Q = 0.1, ϵ = 2.0, α _{1} = 1.0. To validate and confirm the accuracy of the present numerical procedure, our current results of local Nusselt number −θ _{1} ^{′}(0) in case of Newtonian fluid (γ _{1} = 0) are compared with the existing data of Chen [6] and Abel et al. [12] for different values of Prandtl number are presented in Table 1, and found a superb accuracy with the publicized data.
Figure 2 is drawn for velocity field against heat generation/absorption parameter Q. From the outcomes, it is examined that energy boundary layer thickness rises for Q > 0 due to increasing in the thermal state of the fluid. From a physical characteristic heat source in thermal boundary layer produces energy, because of this fact temperature increases with the increase of Q. This increment in temperature provides an increase in flow field. Conversely, with the occurrence of heat absorption, there occurs decrement in the temperature field for Q < 0, so producing decrement in thermal boundary layer as seen in Figure 2.
Figures 3 and 4 represent the establishment of variation of sanction/injection parameter on the flow fields. It is noticed that when suction parameter is S > 0 fluid velocity reduces significantly whereas velocity of the fluid is increased for injection S < 0. Figure 4 points out that thermal boundary layer thickness diminishes by intensifying the suction parameter and increases the temperature profile due to injection parameter because temperature is overshoot for injection S < 0. This feature overcomes up to a specific height and afterward the procedure slows down and at distance away from the surface temperature vanishes.
The impact of ferromagnetic on problem is considered by taking ferrohydrodynamic interaction parameter β and dimensionless distance between the center of the magnetic dipole and origin. From physical point of view there is an interaction between the fluid motion and the external magnetic field due to dipole. Consequently enhancement in ferromagnetic parameter β leads to flattening the axial velocityf ^{′}(η). In fact, ferromagnetic fluid fundamentally has a transporter liquid flow with tinny size ferrite particles which improve the viscosity of the liquid, and therefore velocity profile reduces for higher value of β transfer of heat is also improved via decay motion. It is more fascinating to perceive that due to impact of magnetic dipole, fluid velocity remains lower as compared to MHD case (β = 0) as shown in Figure 5. Since there is an interference among the motion of fluid and the stroke of point dipole. This type of intervention slows down the fluid velocity and increases the frictional heating involving inside the fluid layers which are responsible for the improvement in the energy transport as cleared in Figure 6.
Figures 7 and 8 are outlined for velocity and temperature profile versus Maxwell parameter γ _{1}. From graphs it is concluded that an increase in Maxwell parameter γ _{1} is to decline fluid velocity overhead the surface, and associated boundary layer thickness is decreased for a large value of γ _{1}. From physical view point, when shear stress is detached fluid will come to rest. These sorts of philosophy are revealed in many cases of polymeric fluids which are not clear in the Newtonian model. For higher value Maxwell parameter will produce a retarding force between two adjacent layers which develop a suppression in the fluid velocity and momentum boundary layer thickness as shown in Figure 7. Also, perceived that temperature profile increases by the large value of Maxwell parameter, since thermal boundary layer thickness grows into broadened when we enhance Maxwell parameter as illustrated in Figure 8. Therefore, the cooling of a heated sheet can be enhanced by picking a coolant having a small Maxwell parameter. The consequence of frictional heating because of viscosity and magnetic dipole on temperature field is signified by dissipation parameter λ. Figure 9 simply demonstrates the influence of λ on temperature profile. Result describes that temperature is decreased by enlarging the value of λ. This happens due to extraordinary behavior of ferrofluid on boundary layer transport. On the other end, this is contradictory in the case of hydrodynamic (β = 0) whereas increasing in the value of λ shows an increment in temperature profile near the boundary layer region.
Figures 10 and 11 are plotted to see the impact of suction parameter on local Nusselt and skin friction coefficient versus ferromagnetic interaction parameter. Given result shows skin friction enhancement for numerous values of ferromagnetic interaction parameter and suction parameter. This behavior appears because of external magnetic field which produced a drag force known as Lorentz force. Due to this force, fluid velocity is depressed and hence increasing skin friction coefficient, but inverse trend is noted for Nusselt number as shown in Figure 11.
Calculation of Nusselt number −θ _{1} ^{′}(0), when β = λ = γ _{1} = S = Q = 0.
Fig. 2 Impact of Q on temperature profile. 
Fig. 3 Impact of S on velocity distribution. 
Fig. 4 Impact of S temperature profile. 
Fig. 5 Impact of β on velocity profile. 
Fig. 6 Impact of β on temperature profile. 
Fig. 7 Impact of γ _{1} on velocity profile. 
Fig. 8 Impact of γ _{1} on temperature profile. 
Fig. 9 Impact of λ on temperature profile. 
Fig. 10 Impact of S and β on skin friction coefficient. 
Fig. 11 Impact of S and β on Nusselt number. 
7 Closing remarks
In the present study, impact of heat generation or absorption on Maxwell ferromagnetic liquid flow due to stretched surface affected by magnetic dipole field with suction or injection parameter. Mathematical PDEs of momentum and energy are first converted into ODEs, then solved by adopting R–K based shooting procedure using MATLAB package. Some effective governing parameters on the flow problem like Maxwell parameter (γ _{1}), ferromagnetic interaction parameter (β), viscous dissipation parameter (λ), heat generation or absorption parameter (Q), sanction or injection parameter (S) on velocity, temperature, skin friction and heat transfer rate are explained graphically and discussed. Some of the major observations of the current flow problem are summarized as below:

Impact of both suction and ferromagnetic parameter is to suppress the fluid velocity which causes the improvement in skinfriction coefficient.

Nusselt number is reduced by rising the ferromagnetic interaction parameter.

The influence of Maxwell parameter is to decrease the velocity profile and enhance the temperature profile in the region of boundary layer.
Nomenclature
c _{ p } : Specific heat transfer (J kg^{−1} K^{−1})
k : Thermal conductivity (W m^{−1} K^{−1})
K ^{*} : Pyromagnetic coefficient
Q : Heat generation/absorption
S : Suction/injection parameter
(u, v): Velocity components (m s^{−1})
(x, y): Cartesian coordinates (m)
μ _{0} : Magnetic permeability
T _{ c } : Curie temperature (K)
Greek symbols
ϕ : Stream function (m^{2} s^{−1})
λ : Viscous dissipation parameter
α _{1} : Nondimensional distance
β : Ferromagnetic interaction parameter
χ : Strength of magnetic field (A/m)
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Cite this article as: A. Majeed, A. Zeeshan, F.M. Noori, U. Masud, Influence of rotating magnetic field on Maxwell saturated ferrofluid flow over a heated stretching sheet with heat generation/absorption, Mechanics & Industry 20, 502 (2019)
All Tables
All Figures
Fig. 1 Geometry of the problem. 

In the text 
Fig. 2 Impact of Q on temperature profile. 

In the text 
Fig. 3 Impact of S on velocity distribution. 

In the text 
Fig. 4 Impact of S temperature profile. 

In the text 
Fig. 5 Impact of β on velocity profile. 

In the text 
Fig. 6 Impact of β on temperature profile. 

In the text 
Fig. 7 Impact of γ _{1} on velocity profile. 

In the text 
Fig. 8 Impact of γ _{1} on temperature profile. 

In the text 
Fig. 9 Impact of λ on temperature profile. 

In the text 
Fig. 10 Impact of S and β on skin friction coefficient. 

In the text 
Fig. 11 Impact of S and β on Nusselt number. 

In the text 
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