In ﬂ uence of rotating magnetic ﬁ eld on Maxwell saturated ferro ﬂ uid ﬂ ow over a heated stretching sheet with heat generation/absorption

. This article is focused on Maxwell ferromagnetic ﬂ uid and heat transport characteristics under the impact of magnetic ﬁ eld generated due to dipole ﬁ eld. 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 magneto-thermo-mechanical (ferrohydrodynamic) interaction elaborates the ﬂ uid motion as compared to hydrodynamic case. Problem is modeled using continuity, momentum and heat transport equation. To implement the numerical procedure, ﬁ rstly 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 non-linear differentials equations is solved computationally with the aid of Runge – Kutta scheme with shooting algorithm using MATLAB. The ﬂ ow situation is carried out by considering the in ﬂ uence of pertinent parameters namely ferro-hydrodynamic interaction parameter, Maxwell parameter, suction/injection and viscous dissipation on ﬂ ow velocity ﬁ eld, temperature ﬁ eld, 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 ﬂ uid case ( g 1 =0) to check the validity of the solution. It is inferred that interaction of magneto-thermo-mechanical is to slow down the ﬂ uid motion. We also witnessed that by considering the Maxwell and ferrohydrodynamic parameter there is decrement in velocity ﬁ eld whereas opposite behavior is noted for temperature ﬁ eld.


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][3][4][5][6]. Furthermore, the industrial use of non-Newtonian 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 non-Newtonian liquids in industries. Numerous models of non-Newtonian 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 magneto-hydrodynamic (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 two-dimensional viscoelastic liquid flow over a moving surface. Majeed et al. [14] performed the heat transfer analysis of magneto-Eyring-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, non-magnetic carrier fluids are saturated with micron-sized 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, leak-proof seals, lithographic designing and much more [17][18][19][20][21]. Neuringer [22] demonstrated stagnation point flow of ferromagnetic liquid flow toward a cold wall along with surface temperature declines linearly. Sheikholeslami and Gorji-Bandpy [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 Gorji-Bandpy [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][30][31][32][33][34][35][36][37][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. Flow of magnetic fluid influenced due to dipole field (see Fig. 1) and c represents the scalar potential of dipole magnetic field, which are and components of magnetic field H are Magnetic body force and gradient of H are proportional to each other, we get: In component form, we have The representation of magnetization M is taken in the form of linear temperature as [39]

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 y-axis as shown Figure 1. A magnetic dipole is located at distance "a" with its center is on y-axis. Due to dipole, the direction of magnetic fluid in the +ve x-direction 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: with boundary conditions are Here, c > 0 signify the rate of stretching sheet, A is constant, v w represents the suction/injection velocity and l ¼ ffiffiffiffiffiffiffi n=c p is characteristic length.

Solution of the problem
We introduce the non-dimensional stream function ' (j, h) and temperature u (j, h) assumed by [39]. Here, Velocity components are: Putting equations (13)-(15) into equations (9) and (10), and obtain the expression up to the power of j 2 , we get: Correspondig boundary relations (11) and (12) takes the form The pertinent parameter appearing in the above equations (16)

Quantities of physical interest
Skin friction and heat transfer rate are stated as and Using the equations (13)- (15) we get where Re x ¼ rcx 2 m refer local Reynolds number. It is more attractive and appropriate to exchange the non-dimension Nusselt number termed as heat transfer rate as sheet.

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 b, Pr, S, g 1 , Q and l. 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 0 , z 3 = f 00 , z 4 = u 1 , z 5 = u 1 0 , z 6 = u 2 , z 7 = u 2 0 . By introducing these new constraints in the equations (16)-(22) we obtain: See equations (24) below.
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 x 1 , x 2 and x 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 h = h ∞ by z 3 (w 10 , w 20 , w 30 , h ∞ ), z 5 (w 10 , w 20 , w 30 , h ∞ ) and z 7 (w 10 , w 20 , w 30 , h ∞ ), respectively. Meanwhile y 3 , y 5 and y 7 are obviously a function of x 1 , x 2 and x 3 , they have opened in Taylor series around x 1 À w 10 , x 2 À w 20 and x 3 À 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 dx 1 = x 1 À w 10 , dx 2 = x 2 À w 20 and dx 3 = x 3 À w 30 , we obtain the new guess w 11 = w 10 + dw 10 , w 21 = w 20 + dw 20 and w 31 = w 30 + dw 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 x 1 , x 2 and x 3 until we fulfill the required boundary conditions. Lastly w 1n = w 1(nÀ1) + dw 1(nÀ1) , w 2n = w 2(nÀ1) + dw 2 (nÀ1) and w 3n = w 3(nÀ1) + dw 3(nÀ1) 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 À5 in order to obtain the best convergence.

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 Prð4z 2 z 6 À z 1 z 7 À Qz 6 Þ À 2z 1 z 6 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 ferro-hydrodynamic interaction parameter b 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 b leads to flattening the axial velocityf 0 (h). 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 b 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 (b = 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 g 1 . From graphs it is concluded that an increase in Maxwell parameter g 1 is to decline fluid velocity overhead the surface, and associated boundary layer thickness is decreased for a large value of g 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 l. Figure 9 simply demonstrates the influence of l on temperature profile. Result describes that temperature is decreased by enlarging the value of l. This happens due to extraordinary behavior of ferrofluid on boundary layer transport. On the other end, this is contradictory in the case of hydrodynamic (b = 0) whereas increasing in the value of l 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.

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 (g 1 ), ferromagnetic interaction parameter (b), viscous dissipa-tion parameter (l), 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 skin-friction coefficient.  Wall mass flux T c Curie temperature (K) Fig. 9. Impact of l on temperature profile.  Greek symbols r Density (kg m À3 ) c Scalar magnetic potential f Stream function (m 2 s À1 ) d Slip parameter e Curie temperature t w Shear stress l Viscous dissipation parameter l 1 Relaxation time a 1 Non-dimensional distance b Ferromagnetic interaction parameter x Strength of magnetic field (A/m) g 1 Maxwell parameter