### INTRODUCTION

- Top of page
- Abstract
- INTRODUCTION
- MATHEMATICAL FORMULATIONS
- NUMERICAL SOLUTIONS
- RESULTS AND DISCUSSION
- CONCLUSIONS
- NOMENCLATURE
- REFERENCES

The problem of boundary layer flow over a stretching surface has received great interest among researchers due to its wide range of applications in several engineering processes, for example materials manufactured by extrusion, glass-fibre and paper production. In industry, polymer sheets and filaments are manufactured by continuous extrusion of the polymer from a die to a wind-up roller, which is located at a finite distance away. In these cases, the final product of desired characteristics depends on the rate of cooling in the process. Sakiadis[1] introduced the study of boundary layer flow over a continuous solid surface moving with constant velocity. The dynamics of the laminar boundary layer flow of a Newtonian fluid caused by a flat elastic sheet whose velocity varies linearly with distance from the fixed point of the sheet was originated from the pioneering work of Crane.[2] Soundalgekar[3] investigated the Stokes problem for a viscoelastic fluid. A considerable amount of attention was drawn by Gupta and Gupta[4] in the stretching sheet problem with constant surface temperature. Rajagopal et al.[5] have analysed the flow of a viscoelastic fluid over a stretching sheet. Pillai et al.[6] investigated the effects of work done by deformation in viscoelastic fluid in porous media with uniform heat source. Cortell[7, 8] studied the flow and heat transfer of a viscoelastic fluid over a stretching surface considering both constant sheet temperature and prescribed sheet temperature. Hayat et al.[9] also investigated the effects of work done by deformation in second grade fluid with partial slip condition, in this no account of heat source has been taken into consideration.

The study of MHD flow of viscoelastic fluids over a continuously moving surface has a wide range of application in the production of synthetic sheet, of plastic sheet, cooling of metallic sheet, aerodynamic extrusion etc. Andersson[10] examined the flow of viscoelastic fluid over a stretching sheet under the influence of uniform magnetic field. Abel et al.[11] studied convective heat and mass transfer in a viscoelastic fluid flow through a porous medium over a stretching sheet. Makinde[12] analysed hydromagnetic boundary layer flow and mass transfer past a vertical plate in a porous medium with constant heat flux. Computational dynamics of hydromagnetic stagnation-point flow towards a stretching sheet was studied by Makinde and Charles.[13] Sibanda and Makinde[14] described MHD flow and heat transfer due to a rotating disk in a porous medium with Ohmic heating and viscous dissipation.

Cortell[15] investigated viscous fluid flow and heat transfer over a nonlinearly stretched sheet in the presence of thermal radiation and viscous dissipation in the case of prescribed wall heat flux, whereas Makinde and Ogulu[16] carried out an analysis to study the effects of variable viscosity and thermal radiation on heat and mass transfer past a vertical porous plate in the presence of a transverse magnetic field. A new dimension has been added to this investigation by Abo-Eldahab and El Aziz[17] to study the effects of space-dependent heat source/sink in addition to the temperature-dependent heat source/sink in their study of viscous flow. Study of viscoelastic fluid flow and heat transfer over a stretching sheet with variable viscosity was given by Abel et al.[18] Nandeppanavar et al.[19] analysed the heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet in the presence of thermal radiation and non-uniform heat source/sink.

In view of the above discussions, authors envisage to investigate the steady two-dimensional MHD mixed convection heat and mass transfer flow past a semi-infinite vertical porous plate embedded in a viscoelastic fluid-saturated porous medium in the presence of variable thermal conductivity, non-uniform heat source/sink parameter, chemical reaction, viscous and Ohmic dissipation since these parameters have significant contribution to convective transport process. The problem addressed here is a fundamental one that arises in many practical situations such as polymer extrusion process and combined effects of the physical parameters that will have a large impact on the heat and mass transfer characteristics. It is assumed that the stretching velocity, surface temperature and surface concentration vary linearly with the distance from the point. The nonlinearity of the basic equations and additional mathematical difficulties associated with it has led to the use of numerical method. The transformed dimensionless governing equations are solved numerically by using fifth-order Runge-Kutta-Fehlberg method (RKF45) with shooting technique.

### MATHEMATICAL FORMULATIONS

- Top of page
- Abstract
- INTRODUCTION
- MATHEMATICAL FORMULATIONS
- NUMERICAL SOLUTIONS
- RESULTS AND DISCUSSION
- CONCLUSIONS
- NOMENCLATURE
- REFERENCES

Consider steady two-dimensional MHD laminar convective heat and mass transfer flow in a viscoelastic fluid-saturated porous medium over a vertical stretching sheet in the presence of a uniform transverse magnetic field, non-uniform heat source sink, viscous-Ohmic dissipation and chemical reaction. The flow is generated due to the stretching of the sheet, caused by the simultaneous application of two equal and opposite forces along the *x*-axis. Keeping the origin fixed, the sheet is then stretched with a speed varying linearly with the distance from the slit. A uniform transverse magnetic field *B*_{0} is imposed along the *y*-axis as shown in Figure 1. The *x*-axis is taken in the direction of the main flow along the plate and the *y*-axis is normal to the plate with velocity components *u*, *v* in these directions. We adopt Walter's liquid *B*′ model for the constitutive equation of viscoelastic fluid as given by (in Cartesian tensor notation):

- (1)

where *τ*_{ij} is the stress tensor, *p* is the pressure, *e*_{ik} is the rate-of-strain tensor, *µ* is the coefficient of viscosity and *k*_{0} is elastic parameter. Under the usual boundary layer approximation, along with Walter's liquid *B*′ approximations, the governing equations describing the conservation of mass, momentum, energy and concentration are governed by the following equations[7, 8]:

(i) Equation of continuity:

- (2)

(ii) Equation of momentum:

- (3)

(iii) Equation of energy:

- (4)

(iv) Equation of momentum:

- (5)

where *ρ* is the density of the liquid, *T* and *C* are the temperature and concentration, respectively. *C*_{b} is the form of drag coefficient which is independent of viscosity and other properties of the fluid, but is dependent on the geometry of the medium, *k* is the permeability of the porous medium, *β*_{t} and *β*_{c} are the coefficients of thermal and concentration expansions, respectively, *c*_{p} is the specific heat at constant pressure, is the kinematic viscosity, *σ* is the electrical conductivity of the fluid, *B*_{0} is the externally imposed magnetic field strength in the *y*-direction, *D* is the molecular diffusivity, *κ* is the thermal conductivity, is the rate of internal heat generation/absorption coefficient. The other terms have their usual meanings as given in the nomenclature.

#### Similarity Solution for the Equation of Momentum

Using Equations (13) and (14), the boundary conditions for *f*_{0} and *f*_{1} are:

- (18)

- (19)

Now we find the zeroth-order stream function Equation (16) as a third order equation of *f*_{0}(*η*) for which three boundary conditions are prescribed by Equation (18). The first-order stream function Equation (17) is also third-order equation of *f*_{1}(*η*) for which three boundary conditions are prescribed by Equation (19). Since, the order of the differential equations (16) and (17) matches well with the number of boundary conditions prescribed by Equations (18) and (19), respectively, Equations (16) and (17) would produce unique solutions.

#### Similarity Solution of Energy and Mass-Diffusion Equations

The thermal conductivity of fluid *κ* is assumed to vary linearly with temperature and it is of the form:

- (20)

we consider non-isothermal temperature boundary condition as follows:

- (21)

We now define dimensionless temperature variable *θ* and concentration *φ* of the form:

- (22)

Using Equation (22) we have from Equations (4) and (5) as:

- (23)

- (24)

Equating *k*_{1}th and *x*^{2} order terms, we get:

- (27)

- (28)

#### Skin-Friction Coefficient, Nusselt Number and Sherwood Number

The most important physical quantities for the problem are skin-friction coefficient (*C*_{f}), local Nusselt number (*Nu*_{x}) and local Sherwood number (*Sh*_{x}) which are defined by the following relations:

- (30)

The skin-friction on the flat plate *τ*_{w}, rate of heat transfer *q*_{w} and rate of mass transfer *m*_{w} are given by:

- (31)

Substituting Equation (31) in Equation (30) thus we get:

- (32)

where is the local Reynolds number.

### NUMERICAL SOLUTIONS

- Top of page
- Abstract
- INTRODUCTION
- MATHEMATICAL FORMULATIONS
- NUMERICAL SOLUTIONS
- RESULTS AND DISCUSSION
- CONCLUSIONS
- NOMENCLATURE
- REFERENCES

The coupled ordinary differential Equations (16), (17) and (25)-(28) are third order in *f*_{0}, *f*_{1}, second order in *θ*_{0}(*η*), *θ*_{1}(*η*) and *φ*_{0}(*η*), *φ*_{1}(*η*) which have been reduced to a system of 14 simultaneous equations of first-order for 14 unknowns. In order to solve this system of equations numerically we require 14 initial conditions but two initial conditions on *f*_{0} and *f*_{1}, one initial condition each on *θ*_{0}, *θ*_{1} and *φ*_{0}, *φ*_{1} are known. However, the values of *θ*_{0}, *θ*_{1} and *φ*, *φ*_{1} are known as *η* ∞. These six end conditions are utilised to produce six unknown initial conditions at *η* = 0 by using shooting technique. The most crucial factor of this scheme is to choose the appropriate finite value of *η*_{∞}. In order to estimate the value of *η*_{∞}, we start with an initial guess value and solve the boundary value problem consisting of Equations (16), (17) and (25)-(28) to obtain . The solution process is repeated with another large value of *η*_{∞} until two successive values of differ only after desired significant digit. The last value of *η*_{∞} is taken as the final value of *η*_{∞} for a particular set of physical parameters for determining velocity *f*(*η*), temperature *θ*(*η*) and concentration *φ*(*η*) in the boundary layer. After knowing all the seven initial conditions, we solve this system of simultaneous equations using fifth-order Runge-Kutta-Fehlberg integration scheme with automatic grid generation scheme which ensures convergence at a faster rate. The results are provided in several tables and graphs.

### RESULTS AND DISCUSSION

- Top of page
- Abstract
- INTRODUCTION
- MATHEMATICAL FORMULATIONS
- NUMERICAL SOLUTIONS
- RESULTS AND DISCUSSION
- CONCLUSIONS
- NOMENCLATURE
- REFERENCES

The present nonlinear boundary value problem cannot be solved in closed-form, so numerical method is adopted in order to describe the physics of the problem well. The resulting nonlinear ordinary differential equations are integrated by fourth-order Runge-Kutta-Fehlberg method with shooting technique. In order to assess the accuracy of the numerical results, the validity of the numerical code has been checked under some limiting cases. Comparison of skin friction coefficient *f*″(0) for various values of *k*_{1} are made with Bhattacharyya et al.[21] and Mahapatra et al.[20] in the absence of viscous dissipation, buoyancy force, porous parameter, magnetic parameter and Eckert number which shows a very good agreement as seen from Table 1.

Table 1. Comparison of skin friction coefficient *f*″(0), *Ha* = 0, *Sc* = 0.0 and various values of *k*_{1} with Bhattacharyya et al.[21] and Mahapatra et al.[20]*k*_{1} | Bhattacharyya et al.[21] | Mahapatra et al.[20] | Present results |
---|

0.001 | 1.0005 | 0.9964 | 1.0002 |

0.005 | 1.0025 | 0.9984 | 1.0022 |

0.01 | 1.0050 | 1.0009 | 1.0050 |

Figure 2 shows the behaviour of velocity profiles for different values of viscoelastic parameter *k*_{1} = 0.001, 0.005, 0.01. It is seen from this figure that the transverse component velocity increases with increasing values of *k*_{1}. Figure 3 shows the behaviour of velocity profiles for different values of Hartmann number *Ha* with viscoelastic parameter *k*_{1} = 0.0, 0.01. It is observed from this figure that fluid velocity is higher in viscous fluid than viscoelastic fluid. The Hartmann number represents the importance of the magnetic field on the flow field. It is observed from this figure that the effect of the magnetic field decreases the velocity profiles, due to the fact that the application of transverse magnetic field normal to the flow direction has a tendency to give rise to a resistive type force called the Lorentz force and hence results in retarding the velocity profile. Thus, the increase in the Hartmann number results in the increase in the Lorentz force due to which the velocity profile decreases. So the effect of the magnetic field is to retard the velocity of the fluid. Figure 4 depicts that the horizontal velocity profile increases with the increase in the value of the thermal Grashof number *Gr*_{t} and the peak is observed near the stretching sheet which exponential decreases away from the stretching surface.

Figure 5 shows the velocity profiles for various values of solutal buoyancy force whose effect is taken into account by the parameter *Gr*_{c} (solutal Grashof number). It is noticed that the effect of increasing *Gr*_{c} is to increase the velocity profile. Thus what actually is observed is that the velocity overshoots (i.e. the velocity at a certain value of *η* which exceeds the velocity at the edge of the boundary layer) in the boundary layer region and buoyancy force acts like a favourable pressure gradient which accelerates the fluid velocity within the boundary layer. Figure 6 displays the influence of the inverse Darcy number *Da*^{−1} on the velocity profile. It is seen that the increasing the value of inverse Darcy number is to decrease the velocity profiles which is due to the presence of porous medium that causes higher retardation to the fluid velocity. Figure 7 represents the effect of ∈ and *Ha* on velocity profiles. It is observed from this figure that the effect of magnetic field is to decrease the velocity profile and similar effect of ∈ is seen on velocity profiles.

Figure 13 shows the plot of skin-friction coefficient with Schmidt number *Sc* for various values of Hartmann number. It is examined from this figure that there is significant effect of Hartmann number on *f*″(0), that is skin-friction coefficient decreases with increase in the value of Hartmann number for all values of *Sc* whereas no significant effect of *Sc* is seen on the skin-friction coefficient. The effect of Hartmann number and Schmidt number *Sc* is demonstrated in Figure 14. We observe by analysing the graph that the effect of increasing the value of Hartmann number *Ha* is to decrease the rate of heat transfer in the flow region, which is responsible for decreasing Nusselt number in the thermal boundary layer. Whereas the increase in Schmidt number *Sc* is to decrease the Nusselt number in the boundary layer. Figure 15 shows the graph of Sherwood number for different values of Schmidt number *Sc* and *Ha*. Increase of Schmidt number means decrease of molecular diffusivity *D*, which results in decrease in the Schmidt number in the solutal boundary layer. Figure 16 depicts the variation of chemical reaction parameter on concentration profile. It is seen that *R*_{1} produces a decrease in concentration profile since chemical reaction decelerates the concentration of species.

### CONCLUSIONS

- Top of page
- Abstract
- INTRODUCTION
- MATHEMATICAL FORMULATIONS
- NUMERICAL SOLUTIONS
- RESULTS AND DISCUSSION
- CONCLUSIONS
- NOMENCLATURE
- REFERENCES

The problem of two-dimensional mixed convection flow due to a viscoelastic fluid-saturated porous medium in the presence of inverse Darcy number, Prandtl number, Schmidt number, non-uniform heat source/sink parameter, magnetic parameter, chemical reaction parameter, local thermal Grashof number and local solutal Grashof number effects was investigated. The resulting partial differential equations, describing the problem, are transformed into ordinary differential equations by using similarity transformations. These equations are more conveniently solved numerically using Runge-Kutta Fehlberg method with shooting technique for the computation of the flow, heat and mass transfer characteristics for various values of physical parameters. The numerical results obtained and compared with previously reported cases available in the open literature and they are found to be in very good agreement. From the present investigation, the following conclusions are drawn:

- Increase in the viscoelastic parameter
*k*_{1} is to increase the velocity in the momentum boundary layer. - Increase in the inverse Darcy number
*Da*^{−1} is to decrease the velocity profile. - Increasing thermal buoyancy parameter leads to increase the value of velocity profiles.
- Velocity profiles are strongly influenced by the magnetic field in the momentum boundary layer, which decreases with the increase in the Hartmann number.
- Prandtl number influences the temperature profiles in the thermal boundary layer as a result the temperature decreases in the thermal boundary layer.
- Chemical reaction parameter deteriorates the concentration profile.
- Skin-friction and local Nusselt number decrease with the increase in Hartmann number whereas local Sherwood number increases with Hartmann number.
- Local Nusselt number and local Sherwood number decrease with the increase in the Schmidt number.