Study of differential transform technique for transient hydromagnetic Jeffrey fluid flow from a stretching sheet

Abstract This article investigates the time-dependent MHD heat transfer flow of Jeffrey fluid from a stretching sheet, the topic significance to non-Newtonian viscoelastic material processing. Using similarity transformations, the governing coupled non-linear PDE’s are remodel into ODE’s with suitable free stream and wall boundary conditions. The developed non-dimensional non-linear problem is revealed to be analysed by several key thermosphysical and rheological parameters, namely, Jeffrey fluid parameter (λ), Deborah number (β), Prandtl number (Pr), buoyancy parameter (ξ), magnetic parameter (M) and unsteadiness parameter (A). The semi-exact differential transform technique is applied to elucidate the coupled nonlinear governing equation of non-Newtonian Jeffrey fluid problem. Also, the solution is validated with numerical results attained via the MATLAB bvp4c function. Excellent accurateness is attained through the DTM approach. Further validation with available consequences from the existing literature is incorporated. The results indicate that fluid velocity and temperature are boosted with increasing Deborah number and stretching parameter however it shows a decreasing trend with Jeffrey fluid parameter and convection parameter. It also shows when augmenting the magnetic parameter which reduces the flow and increases the thickness of the boundary layer.


Introduction
The ow of non-Newtonian uids with heat transfer from stretching surfaces ensures frequent engineering applications. In point, several manufacturing processes consist of *Corresponding Author: Mahesh Kumar, Central University of Karnataka, Kalaburagi, 585367, India, E-mail: mkhmaths@gmail.com the fabrication of sheeting material which contains polymer sheets. Such kind of ows is particularly applicable in the aerodynamic extrusion of chemical compound sheets, hot rolling, petroleum reservoirs, bers spinning, arti cial bers, continuous lament extrusion from a die, manufacturing of elastic sheet, tinning and annealing of copper wires, polymers continuous casting, and glass blowing. Originally, the revolutionary work of boundary layer ow from a stretching sheet was premeditated by Sakiadis [1] and Erickson [2]. Later, Crane [3] extended the same [1,2] model work for 2-D ow for the stretching surface and reported the exact solution. Successively numerous investigators have further protracted the Sakiadis and Crane models for analysing suction or blowing problems [4], viscous dissipation e ects [5], stagnation ows [6] and variable viscosity [7]. Conversely, in several stretching ow problems, transient behaviour ascends owing to a swift elongating of the sheet. This characterizes some cases such as ow generated by a periodic variation of the temperature or the ow over an impulsive stretching sheet. Since instantaneous developed is occur as a result of sudden stretching of the sheet with de nite velocity. Interesting models related to the impulsive stretching sheet found in Ref. [8][9][10][11][12][13][14].
Though, the above-cited studies are circumscribed to viscous uid (Newtonian uids) and which are fails to explain the rheological (non-Newtonian) behaviour specifically, in polymer processing, extrusion of plastics, glass blowing, biochemical industries, coating protection, etc., The modeling studies compacts with non-Newtonian uids o er interesting challenges to the many researchers and which speci cally mimic several complex characteristics of real industrialized uids including couple stresses, viscoelasticity, viscoplasticity, spurt, etc. In spite of various physical constitutions of non-Newtonian uids, among all the Je rey uid is one which constituents characterizing the noticeable features of retardation and relaxation times [15][16][17][18][19]. The application of Je rey uid includes notably polymer solutions and multi-phase systems namely, emulsion, slurries, foams, etc. Also, it may consider as blood model [20][21][22]. Hamad et al. [23] con-sidered the heat transfer ow of a Je rey uid problem through a stretching sheet nearby stagnation point. Turkyilmazoglu and Pop [24] explored a similar type of problem with parallel external ow. The time-dependent ow of non-Newtonian Je rey uid for stretching sheet geometry was analysed by Hayat et al. [25]. Recently, Hayat et al. [26] examined the 3-D Je rey uid ow in a bidirectional stretching surface.
Magnetohydrodynamic (MHD) ow of an electrically conducting uid as a result of the sheet geometry is of signi cant interest in contemporary metallurgic and metalworking processes, condensation process and also in polymer industries. It has several other applications including induction ow meter, chemical factories, in lubrication of machine components, in coolers and heaters of mechanical and electrical devices, etc. The hydromagnetic boundary layer ow of Je rey uid has been premeditated by several authors [27][28][29][30][31][32][33]. Recently, Khan et al. [34] analysed the MHD e ects of Je rey uid problem from inclined stretching sheet. Qayyum et al. [35] analysed the MHD ows of Jeffrey nano uid past a non-linear stretching surface. Zokri et al. [36] studied Je rey uid with MHD e ects over the sheet with the impact of dissipation. Similarly, the MHD stagnation point ow of Je rey uid from a stretched surface was analysed by Hayat et al. [37].
Most of the uid mechanics problems are basically in nonlinear form and are referred to nonlinear ordinary or partial di erential equations. The analytical methods could not evaluate these types of nonlinear equations accurately. Further methods, which are numerical/semianalytical in nature, therefore need to be employed and to handle these type of problem the authors prefer certain standard methods of decomposition technique, perturbation technique, Iteration technique, homotopy analysis and optimal asymptotic techniques and similarly, di erential transform technique. These semianalytical/numerical methods are recurrently applied on account of their greater exactness and ease in producing results. In the current article, the Di erential Transform Method (DTM) [38] is hired. It is a vigorous semiexact method, which is not constructed on the subsistence of large or small factors. This method has essential bene ts over the other estimated semi-exact methods. It is useful to solve various di erential equations viz. di erence equation, eigenvalue problems, nonlinear integral, high index, di erential-algebraic equations fractional, pantograph equation, nonlinear oscillators and integro-di erential equation without restrictive assumptions, linearization, perturbation, discretization or roundo error. Many researchers applied DTM to solve various ow-eld problems. Hatami et al. [39] applied the DTM to solve the non-Newtonian uid ow problem for plate geometry. Recently, Sobamowo [40] applied DTM to analyse the 2D nano uid problem from the porous channel. Usman et al. [41] studied the DTM to examine the transient nano uid ow problem. Other recent work on DTM can be found in ref. [42][43][44]. A close review of the earlier literature survey speci es that a very less number of analysis have been done for the DTM study on nonlinear transient ow of Je rey uid over a sheet. Therefore, the existing study aspires to discuss the DTM to the convective heat transfer ow of Je rey uid over a stretching sheet problem.

Mathematical modelling
Transient, 2-D, incompressible, laminar thermal convective electrically conducting Je rey uid ow over stretching sheet geometry is deliberated and is described in Figure 1. In the beginning i.e., t = 0, the considered sheet is precipitately stretched by velocity Uw (x, t) along the direction of x-axis, by xing origin with ambient uid temperature T∞ and sheet temperature Tw (x, t) is presumed to vary linearly from the x-coordinate. The stationary Cartesian coordinate system has its xed origin placed at the forefront of the sheet in which x-axis is taken alongside the sheet in an upward direction and the y-axis positioned normal direction to the sheet. Following by all these suppositions (with the approximations of Boussinesq and boundary layer), the mathematical governing non-linear equations for hydromagnetic ow of Je rey uid with heat transfer [25,26] are: The appropriate boundary conditions on the ow are given by: The assumptions of sheet velocity and temperature namely, Uw(x, t)and Tw(x, t) are given by and Where the constants a and c (with a > and c ≤ , where ct < ) are measured in terms of frequency units i.e. (time) − , while the constant b is measured in unit of temperature/length, with b < and b > relate to the buoyancy opposing ow and buoyancy assisting ow, respectively. Furthermore b = corresponds to forced convection ow. It is noteworthy that at the outset, i.e. t = , the governing system of Eqs. (1)-(3) reduces to the steady ow situation. These speci c forms of Uw (x, t) and Tw (x, t) have been desired so as to extract a novel similarity variables, which modi es the governing non-linear PDE's (Eqs. (1) -(3)) into a set of ODE's. Succeeding with the scrutiny, let us consider the subsequent non-dimensional functions f and θ, and similarity variable ηas follows [8,9]: Here ψ(x, y, t) denotes stream function and is well-de ned in standard way as (u, v) = ∂ψ ∂y , − ∂ψ ∂x which satis es the equation (1) in the exact manner. Now, Substituting Eqn. (6) into (2) and (3) we get: Pr Here prime represents di erentiation pertaining to η, β = aλ /υ ρ( − ct) is the Deborah number, λ denotes the Jeffrey uid parameter, A = c/a is the stretching ratio parameter. Also, ξ is the buoyancy parameter which is de ned as ξ = Gr/Re where Gr = gβ T (Tw − T∞) x /υ signi es local Grashof number, Pr = υ/α is the Prandtl number and Re = Uw x/υ represents the local Reynolds number. Further, ξ is non-dimensional constant with ξ < implies buoyancy opposing ow and ξ + > related buoyancy assisting ow respectively, ξ = signi es the forced convection ow. The associate boundary conditions are: The essential physical parameters of wall gradient features such as friction C f , shear stress at the sheet surface τw, heat transfer rate Nu and surface heat ux qw are wellde ned as follows:

Di erential transform method solutions
The nonlinear dimensionless well-de ned ordinary di erential Eqs. (7) -(9) of the boundary value problem is elucidated with DTM. Zhou proposed the idea of DTM for the rst time in 1986 [38] and it was developed to evaluate the non-linear and linear forms of IVP's in the analysis of the electric circuit. This DTM composes an analytical result in the polynomial form and it is completely based on the standard Taylor-series method. The foremost advantage of DTM is used for elucidating nonlinear mathematical equations without necessitating linearization, and discretization. Hence, the method does not pretend by errors related to discretization. It also eases the size of the computational part and it is pertinent to numerous mathematical problems easily. DTM has been implemented successfully in several uid dynamics, multi-physical mechanics, and heat transfer models in the current centuries. These include Burgers and coupled Burgers equations [45], application to nonlinear oscillators [46], plane Couette uid ow problem [47], free vibration analysis [48], micropolar uid ows [49,50], non-Newtonian nano uids ow analysis [51], solving nonlinear dynamic problems [52], Acoustic and Wave propagation problems [53] and viscoelastic Winkler foundation [54]. Di erential Transform Method has been revealed to be a very e ective way in this article. Though convergence can be hastened with alterations of this method e.g. Padé estimations, it is not compulsory. The complete procedure of the method to the present problem is now de ned. The kth derivative of di erential transform of the function f (η)is well-de ned as: where f (η) represents original function and F (p) signi es transformed function. The inverse di erential transformation is as follows: For the practical models, the function f (η) is expressed by a truncated series and hence the above Eqn. (12) can be written as: Eqn. (13) implies that ∞ p=m+ F(p)(η − η ) p is trivially small, where m denotes the series size in this problem. Few essential of the assets of DTM are revealed in Table 1. These assets are obtained from Eqs. (12) and (13). Now, implementing DTM and taking di erential transform of non-linear governing conservation Eqs. (7) and (8) along with corresponding boundary conditions (9) we get subsequent recurrence relations as follow: Corresponding boundary conditions: where F (p) , G[p] are the di erential transform of f (η) , θ (η) and n , n , n are unknowns which can be calculated from boundary conditions i.e. Eqn. (9). Following relations can be obtained through transformed boundary conditions (16a) and (16b):  Original function Transformed function G [ ] = Pr n n + Pr n − Pr ( + A) G [ ] = Pr n n − Pr (n + A n ) Pr n ( + A ) − Pr n (n + A n ) − Pr n n + Pr n − Pr ( + A) The above solution procedure is goes on. By substituting Eqs. (17) and (18) in Eqn. (13) based on DTM, the series solutions can be formed as: θ (η) = + n η + Pr ( + A) η + Pr (n + A n ) η + Pr n n + Pr n − Pr ( + A) From the above Eqs. (19) and (20) the unidenti ed n , n , n values are evaluated by using boundary conditions stated in eqn. (9). Afterwards, substituting the n , n , n into the Eqs. (19) and (20) for selected case of control parameters (ξ = . , λ = . , A = . , Pr = . , β = . , M = . ), the expressions of f (η) and θ (η) can be written as follows: In order to corroborate the present existing results of the considered problem and endorse accuracy, the obtained DTM results are equated with existing available numerical results as described in Table 2 for various values of physical parameter, from which one can see that results of DTM computations are in good agreement. The numerical result of the present model is performed with the help of MATLAB built in solver. Additionally, the current DTM results are also compared in Table 3 with those obtainable results for the case of Newtonian uid (β = , λ = , M = ) of Ishak et al. [7] for absence of the buoyancy term ξ θ in Eqn. (7) and Pr = 1.0 and A = 0 (steady-state ow). Yet again, an excellent correspondence is achieved which endorses the validity of the DTM computations. The consequences of Deborah number β on velocity f (η) and temperature θ (η) pro le are presented in Figures 2a and 2b. As β increases, the velocity and the boundary layer thickness is increases. This is due to the existence  of rheological features of viscoelastic ow and impermeability of the wall upsurge the ow velocity and thickness of the boundary layer. It has also been remarked that the in uence of β is to augments the temperature and thermal boundary layer thickness become lesser. Figures 3a-3b portray the in uence of λ on velocity and temperature curves through the boundary layer. From Figure 3a, the velocity declines for augmenting values of λ with the augmenting distance (η) of the sheet. Physically as λ being the viscoelastic parameter displays it shows both elastic and viscous characteristics or more we can say it augments non-Newtonian behaviour and as a consequence it boosts boundary layer thickness. Therefore, the uid motion is slow down and velocity reduces. The temperature pro le in Figure 3b shows the similar trend as that of velocity. As λ become higher, the temperature decreases negatively along the sheet.

Results and discussion
The impact of unsteadiness parameter A on velocity and temperature distribution is demonstrated in Figures  4a and 4b, respectively. Scrutiny of Figure 4a reveals that velocity shows a increasing tendency with ampli ed values of A. Since A is reciprocal of positive constant coecient a. Thus, boost in the A values upturns the stretching ratio. This results the increment in velocity. Also, the temperature pro le displays analogous trend for variation of A values when < η < . . Conversely, the reverse trend is exhibited for η > . as illustrated in Figure 4b.
Figures 5a and 5b respectively elucidate the e ect of convection parameter ξ on ow eld pro les. The variation of cumulative values of ξ is to decrease the velocity. Substantially ξ > signi es assisting ow, ξ < reveals opposing ow and ξ = implies forced convection ow. It is examined that ow velocity is reduces increasing ξ values. Since greater values of ξ relates to the extreme buoyancy pressure force greater than viscous force and which Finally, Figures 7a and 7b display the variation of MHD parameter M on distribution of velocity and temperature curves. The velocity and temperature decreasing with increase M values. The applied magnetic eld develops the resistive force so-called Lorentz force. Due to this uid movement tend to decelerate, which results to the reduction in velocity and temperature boundary layer.

Conclusions
In this work, a semi-analytical approach of di erential transform method for resolving nonlinear coupled govern- ing equations is presented to reveal the validity and ease of this method. For elucidating the correctness of described method, a numerical method via the MATLAB symbolic code 'bvp4c' was also used to solve the problems. Further comparison of present solutions with available existing literature is incorporated. The present calculations reveal that: • The ow is boosted with increasing Deborah number whereas it is slow down with Je rey uid parameter. • Temperature is augments with Deborah number while it is depleted with Je rey uid parameter. • Increasing stretching parameter, the velocity increases and temperature reduces.