1 Introduction

The study of magnetohydrodynamic (MHD) boundary layer flow due to a stretching surface in a stationary fluid is an important aspect of various industrial applications. For example,in the manufacturing of plastic film and artificial fibers,the extruded material exits a slit formed by two solid blocks into a region containing fluid at rest that cools the sheet and alters its mechanical properties^[1]. These properties depend on the rate of cooling,which is strongly influenced by whether the fluid is Newtonian or non-Newtonian. Other practical applications of the flow in consideration include the drawing of copper wires,glass blowing,and the extrusion of material that travels between feed and wind-up roller.

The study of two-dimensional boundary layer flow over a moving surface in a fluid at rest was initiated by Sakiadis^[2]. Crane^[3] extended the work of Sakiadis by obtaining an exact analytical solution for the case of a linearly stretching sheet. Since then several researchers have studied various extensions of these influential works that consider a special class of non- Newtonian fluids known as viscoelastic fluids,i.e.,viscoelastic fluids. For example Rajagopal et al.^[4] derived numerical solutions to the flow of a second-order fluid over a stretching sheet. Siddappa and Abel^[5] presented a similar analysis for the flow of Walters’ liquid B fluid over a stretching sheet. Dandapat and Gupta^[6] provided an exact solution to the flow of Walters’ liquid B and an analysis of the heat transfer.

In this paper,we perform an analysis of viscoelastic boundary layer flow and heat transfer over a stretching sheet with nonuniform heat source/sink subject to a uniform transverse mag- netic field that was recently studied by Abel and Nandeppanavar^[7]. First,we uncover an error in ^[7],in which analytical solutions of the temperature field are given in terms of Kummer’s function that,as will be demonstrated,do not converge at the boundary. Then,we present analytical solutions obtained using the homotopy analysis method (HAM)^{[8, 9, 10, 11, 12]} that converge for all η∈[0,∞) under all conditions considered.

In light of these results,we believe that the HAM should be given serious consideration by researchers who seek analytical solutions to problems involving viscoelastic boundary layer flow and heat transfer. The HAM is a nonperturbative analytical method for obtaining series solutions to various types of nonlinear equations that has two main advantages over similar analytical methods. First,it is nonperturbative,i.e.,it does not rely on the existence of a small/large parameter,and thus,has a larger region of convergence with respect to the model parameters. Second,it provides the ability to adjust and control the convergence of a solution via the so-called convergence-control parameter. 2 Boundary value problem

The fluid flow in consideration involves an incompressible and electrically conducting vis- coelastic fluid subject to a transverse uniform magnetic field in which a two-dimensional flow field is induced by the motion of a linearly stretching sheet. The corresponding heat transfer analysis includes the effects of a nonuniform heat source/sink. For a complete mathematical formulation of the nonlinear partial differential equations governing the fluid flow and heat transfer,we can refer to ^[7].

After an appropriate similarity transformation,the nonlinear ordinary differential equation governing the fluid flow is given by

with boundary conditions where f(η) is the dimensionless stream function,η is the dimensionless similarity variable,k₁ is the viscoelastic parameter,and M is the magnetic parameter.

An appropriate similarity transformation of the governing thermal boundary layer equation in the presence of a nonuniform internal heat source/sink yields

where θ(η) is the dimensionless temperature,Pr is the Prandtl number,A is the spatial heat source/sink coefficient,and B is the temporal heat source/sink coefficient. We consider two general types of non-isothermal boundary conditions,i.e.,prescribed surface temperature (PST) and prescribed heat flux (PHF),which are given by for the PST case,and for the PHF case,where g(η) is used to denote the solution to (3) for the PHF case. 3 HAM

In this section,we use the HAM to solve the nonlinear boundary value problem in (1)−(4) for f(η) and θ(η) in the PST case. The procedure is the same as the PHF case,in which (4) is replaced by (5). We choose the linear operators to be

and the nonlinear operators,Nf and Nθ,to match (1) and (3),respectively. To satisfy the boundary conditions in (2) and (4),we choose the initial guesses to be

The zeroth-order deformation equations are then

where the boundary conditions are given in (2) and (4). h_f and h_θ are the convergence-control parameters and q ∈^{[0, 1]} is an embedding parameter such that

Note that as q increases from 0 to 1,F(η,q) and Θ(η,q) vary from the initial guesses given in (8)−(9) to the desired solutions f(η) and θ(η). Now,we consider Maclaurin series expansions of F(η,q) and Θ(η,q) with respect to q of the form

where If the auxiliary linear operators,the initial guesses,and the convergence-control parameters are properly chosen so that these Maclaurin series all converge at q = 1,we obtain the homotopy- series solutions

Differentiating (10) and (11) m times with respect to the embedding parameter q,dividing by m!,and then setting q = 0,we construct the mth-order deformation equations

subject to the boundary conditions for the PST case,and for the PHF case,where

Note that (20)−(21) are linear nonhomogeneous differential equations. Starting with the ini- tial guesses in (8)−(9),fm(η) and θm(η) for m > 1 are obtained iteratively by solving (20)−(23) with the symbolic computational softwareMATHEMATICA. This procedure is terminated after a fixed number of iterations N to yield the approximate analytical solutions

to the nonlinear boundary value problem of interest. 4 Results and discussion

The convergence of the HAM solutions given in (28)−(29) depends on the convergence- control parameters h_f and h_θ. To determine optimal values for the PST case (the PHF case can be done similarly),we minimize the discrete squared residuals defined by^[12]

with respect to h_f and h_θ,where ηj = 0.1j andM = 100. The optimal values of the convergence- control parameters for all cases considered are obtained by first minimizing (30) since it only depends on h_f and then substituting the optimal value of h_f into (31) to find the optimal value of h_θ. We plot the discrete squared residuals in Figs. 1−3 for illustrative purposes.

To demonstrate convergence of the HAM solutions,we present values of the physically rel- evant boundary derivatives for different orders of approximation in Table 1 and note that the convergence is clearly obtained. In Tables 2 and 3,we compare the values of θ′(0) and g(0) computed with the HAM solutions to the values in ^[7] that were obtained using Kummer’s function. It is clear that the values of θ′(0) and g(0) obtained with the two methods in con- sideration are different,especially when the internal heat source is higher. These discrepancies require further analysis,and so,we define the residual function

where f(η) and θ(η) are approximate solutions to (1) and (3),respectively.

For both the PST and PHF cases,we substitute the HAM solution given in (28)−(29) and the solution in terms of Kummer’s function given in ^[7] into (32) and plot the results in Figs. 4−7. Note that the residuals of the solutions given in terms of Kummer’s function have a maximum at η = 0 that is roughly 0.06 for the case of a low internal heat source and roughly 1 for the case of a much higher heat source,and that the residual of the HAM solution is negligible in comparison. These figures provide ample explanation for the discrepancies in values of θ′(0) and g(0) noted above,and we claim that the correct values are given by HAM, and furthermore,solutions in terms of Kummer’s function are only valid in the absence of a nonuniform internal heat source/sink as evidenced in Table 4. These results have significant implications on previously reported research,in which Kummer’s function was used to represent solutions to the temperature field in the presence of a nonuniform heat source/sink.

In Figs. 8−11,we plot the temperature profiles for the two different thermal boundary conditions,PST and PHF,for various values of the relevant parameters. For the PST case, it is clear that an increase in the magnetic parameter increases the thermal boundary layer thickness. It is worth noting that although Table 1 in ^[7] reflects the same qualitative behavior, Table 2 reflects the opposite behavior in the same article. It is also evident that an internal heat source increases the thermal boundary layer thickness whereas an internal heat sink has the opposite effect as one would expect. Similar conclusions can be made for the PHF case.

5 Conclusions

In this study,the nonlinear differential equations governing viscoelastic boundary layer flow and heat transfer over a linearly stretching surface subject to a transverse uniform magnetic field are solved using the HAM. We provide a graphical and numerical demonstration of convergence of the HAM solutions. We tabulate the physically relevant boundary derivatives and compare our results with those given in ^[7]. Our comparison demonstrates that solutions of the energy equation given in ^[7] are not valid when a nonuniform internal heat source/sink is considered.

These results demonstrate that HAM is a very effective analytical method for solving nonlin- ear problems in science and engineering,and moreover,it should be given serious consideration by researchers who seek analytical solutions to problems involving viscoelastic boundary flow and heat transfer.