Abstract
This article studies the viscous flow and heat transfer over a plane horizontal surface stretched non-linearly in two lateral directions. Appropriate wall conditions characterizing the non-linear variation in the velocity and temperature of the sheet are employed for the first time. A new set of similarity variables is introduced to reduce the boundary layer equations into self-similar forms. The velocity and temperature distributions are determined by two methods, namely (i) optimal homotopy analysis method (OHAM) and (ii) fourth-fifth-order Runge-Kutta integration based shooting technique. The analytic and numerical solutions are compared and these are found in excellent agreement. Influences of embedded parameters on momentum and thermal boundary layers are sketched and discussed.
Introduction
The fundamental problem of two-dimensional flow due to stretching plane surface, initially discussed by Crane [1], is involved in various industrial processes such as metal and polymer extrusion, drawing of plastic films, paper production etc. Owing to such applications, the researchers have discussed this problem under various aspects including suction or injection, variable surface temperature, convective boundary condition, mass transfer, mixed convection etc. The three-dimensional flow due to plane bi-directional linearly stretching sheet was first discussed by Wang [2]. He found an exact similarity solution of the classical Navier-Stokes equations. Later, Lakshmisha et al. [3] numerically examined the unsteady three-dimensional flow with heat and mass transfer over an unsteady stretching sheet. In contrast to this problem, Takhar et al. [4] investigated the three-dimensional flow of an electrically conducting fluid due to an impulsive motion of the stretching sheet. Ariel [4] derived approximate analytic and numeric solutions for steady three-dimensional flow over a stretching sheet. Xu et al. [5] provided uniformly valid series solutions for three-dimensional unsteady flow caused by the impulsively stretching sheet. Liu and Andersson [6] considered the heat transfer in three-dimensional flow due to non-isothermal stretching sheet. The unsteady three-dimensional flow of elastico-viscous fluid and mass transfer due to unsteady stretching sheet with constant wall concentration was studied by Hayat et al. [7]. In another paper, Hayat et al. [8] described the three-dimensional flow of Jeffrey fluid due to stretching sheet. Liu et al. [9] firstly discussed the three-dimensional flow due to exponentially stretching sheet numerically. Steady flow of nanofluid past a linearly bi-directional stretching sheet through Buongiorno's model was examined by Junaid et al. [10]. Sheikholeslami and Ganji [11] discussed the flow and heat transfer of nanofluid between parallel sheets in the presence of Brownian motion and thermophoresis effects.
The literature cited above deals only with the case of either linearly or exponentially driven velocity of the sheet. Vajravelu [12] numerically discussed the viscous flow due to stretching sheet when the velocity of the sheet was assumed to obey the power-law distribution, i.e.
. He computed numerical solutions for various values of power-law index 
Cortell [13] extended this problem by considering viscous dissipation effects and variable surface temperature. The steady boundary layer flow of micropolar fluid over non-linearly stretching sheet was discussed by Bhargava et al. [14]. Radiation and viscous dissipation effects on the boundary layer flow above a non-linearly stretching sheet were explored by Cortell [15]. Homotopy analytic solutions for mixed convection flow of micropolar fluid past a non-linearly stretching vertical sheet were obtained by Hayat et al. [16]. Kechil and Hashim [17] derived analytic solutions for MHD flow over a non-linearly stretching sheet by Adomian decomposition method. Hayat et al. [18] used modified decomposition method for the series solutions of MHD flow of an electrically conducting fluid over a non-linearly stretching surface. The impact of chemical reaction on the flow over a non-linearly stretching sheet embedded in a porous medium was investigated by Ziabakhsh et al. [19]. Rana and Bhargava [20] computed numerical solutions for two-dimensional flow of nanofluid due to non-linearly stretching sheet by finite element method. Shahzad et al. [21] obtained closed form exact solutions for axisymmetric flow and heat transfer when the velocity of the stretching sheet was proportional to 
. Partial slip effects on the boundary layer flow past a non-linearly permeable stretching surface have been addressed by Mukhopadhyay [22]. In another paper, Mukhopadhyay [23] analyzed the flow and heat transfer of Casson fluid due to non-linearly stretching sheet. Rashidi et al. [24] derived homotopy based analytic solutions for flow over a non-isothermal stretching plate with transpiration.
To our knowledge, the three-dimensional flow due to non-linearly stretching sheet has not been yet reported. It is obvious that three-dimensional flows can be appropriate in giving more clear physical insights of the real world problem when compared with the two-dimensional flows. The present work is therefore undertaken to fill such a void. The study also assumes that the temperature across the sheet is non-linearly distributed. Introducing a new set of similarity variables the boundary layer equations are first reduced into self-similar forms and then solved both analytically and numerically. It is pertinent to mention that computation of either approximate analytic or numerical solutions of the boundary layer equations governing the flow and heat transfer is often challenging (see [25]–[33] for details). Attention is focused on the physical interpretation of parameters including the power-law index 
Mathematical Modeling
Let us consider the three-dimensionalincompressible flow over a plane elastic sheet located at 
as shown in the Fig. 1. The flow is induced due to stretching of the sheet in two lateral directions. Let 
and 
be the velocities of the sheet along the 
and 
directions respectively with 
are constants (see Table 1). 
is the variable surface temperature where 
is constant and 
is the ambient fluid temperature. Under the usual boundary layer assumptions, the equations governing the three-dimensional flow and heat transfer in the absence of viscous dissipation and internal heat generation/absorption can be expressed as (see Liu et al. [9])
 |
(1) |
Figure 1. Physical configuration and coordinate system.
Table 1. List of symbols.
 Cartesian coordinate system |
 thermal conductivity |
 velocity components along the  directions |
 non-zero auxiliary parameter |
 velocity of the stretching sheet along  and  direction |
 1st order derivative with respect to 
|
 fluid temperature |
 2nd order derivative with respect to 
|
 wall temperature |
 3rd order derivative with respect to 
|
 ambient fluid temperature |
Greek symbols |
 positive constants |
 kinematic viscosity |
 Power-law index |
 thermal diffusivity |
 dimensionless stream function |
 dimensionless temperature |
 Prandtl number |
 similarity variable |
 skin friction coefficient along  and  direction |
 ratio of the stretching rates |
 local Nusselt number |
 wall shear stress along  and  direction |
 wall heat flux |
 density of the fluid |
 local Reynolds number along  and  direction |
 dynamic viscosity |
 |
(2) |
 |
(3) |
 |
(4) |
where 
and 
are the velocity components along the 
and 
directions respectively, 
is the kinematic viscosity, 
is the fluid temperature and 
is the thermal diffusivity (see Table 1). The boundary conditions are imposed as under:
 |
(5) |
We introduce the new similarity transformations as follows:
 |
(6) |
We have modified the similarity transformations used by Liu et al. [9] for the current problem. Using (6), Eq.(1) is identically satisfied and Eqs. (2)–(5) become
 |
(7) |
 |
(8) |
 |
(9) |
 |
(10) |
 |
where 
is the Prandtl number and 
is the ratio of stretching rate along the 
direction to the stretching rate along the 
direction (see Table 1). The above equations reduce to the case of two-dimensional flow when 
. Moreover, when 
, the equations governing the axisymmetric flow due to non-linearly stretching sheet are recovered. When 
the solution of 
is also a solution of 
. The quantities of practical interest are the skin friction coefficients and the local Nusselt number which are defined as below:
 |
(11) |
where 
and 
are the wall shear stresses and 
is the wall heat flux. These are given by
 |
(12) |
using Eqs. (6) and (12) in Eq. (11), one obtains
 |
(13) |
where 
and 
are the local Reynolds numbers along the 
and 
directions respectively (see Table 1). The vertical component of velocity at the far field boundary can be expressed as
 |
(14) |
Optimal homotopy analytic solutions
The non-linear differential equations (7)–(9) with the boundary conditions (10) have been solved by optimal homotopy analysis method (OHAM) [34], [35]. For this purpose, we first select the initial guesses 
and 
of 
and 
as under:
 |
(15) |
and the auxiliary linear operators are selected as below
 |
(16) |
If 
is the embedding parameter and 
the non-zero auxiliary parameter, then the generalized homotopic equations corresponding to (7)–(10) can be written as follows
 |
(17) |
 |
(18) |
 |
(19) |
 |
(20) |
 |
where the non-linear operators 
, 
and 
are
 |
(21) |
 |
(22) |
 |
(23) |
By Taylor's series expansion one obtains
 |
(24) |
 |
(25) |
 |
(26) |
Substituting 
in the above equations yields the final result. The functions 
and 
can be determined from the deformation of Eqs. (7)–(10). Explicitly the mth-order deformation equations corresponding to Eqs. (7)–(10) are as below
 |
(27) |
 |
(28) |
 |
(29) |
 |
(30) |
 |
 |
(31) |
 |
(32) |
 |
(33) |
 |
(34) |
In order to determine the optimal values of 
we define the squared residuals of the governing Eqs. (7)–(10), 
and 
as
 |
(35) |
 |
(36) |
 |
(37) |
Such kind of error has been considered in other works [36]–[41]. The smaller 
the more accurate the mth order approximation of the solution. The optimal values of 
can be obtained by minimizing the 
through the command Minimize of the software MATHEMATICA (see Liao [36] for details). Alternatively MATHEMATICA package bvph 2.0 can also be used to calculate such values (see [41] for details).
Numerical method
Eqs. (7)–(9) subject to the boundary conditions (10) have been solved numerically by shooting method with fifth order Runge-Kutta integration procedure. First, we reduce the original ODEs into a system of 1st order ODEs by substituting 
and 
which gives
 |
(38) |
and the corresponding initial conditions are
 |
(39) |
Suitable values of the unknown initial conditions 
and 
are guessed and then integration is carried out. The values of 
and 
are then iteratively computed through Newton's method such that the solutions satisfy the boundary conditions at infinity (given in Eq. (10)) with error less than 
.
Results and Discussion
This section contains the physical interpretations of the behavior of the interesting parameters entering into the problem. We compare the 15th-order OHAM solutions for temperature 
with the numerical ones for different values of 
Fig. 2 shows that data retrieved from both solution methods are identical, demonstrating the validation of our findings.
Figure 2. Comparison of analytical and numerical solutions for the temperature distribution.
Lines: 15th-order OHAM solutions, Circles: Numerical solution.
Figs. 3 and 4 show the variations in horizontal and vertical components of velocity with an increase in stretching rates ratio 
. It is clear that increase in 
corresponds to an increase in the stretching rate along the 
direction. Due to this reason the vertical component of velocity increases with an enhancement in 
while the velocity in the 
direction decreases correspondingly. The wall velocity gradients 
and entrainment velocity 
as functions of stretching rates ratio 
have been plotted in Fig. 5. Due to the bi-directional stretching sheet, there will be downward flow in the vertical direction. The vertical component at far field boundary is therefore expected to be negative in this situation. We notice that shear stresses at the wall increase when 
is increased. Further, the larger values of 
enhances the velocity of the cold fluid at the ambient. As a consequence, the entrainment velocity is an increasing function of 
.
Figure 3. Influence of stretching rates ratio 
on the 
component of velocity 
.
Figure 4. Influence of stretching rates ratio 
on the 
component of velocity 
.
Figure 5. Influence of stretching rates ratio 
on the skin friction coefficients 
and 
and entrainment velocity 
.
Fig. 6 indicates that temperature 
decreases with an increase in stretching rates ratio 
for unity Prandtl number. Physically, an increase in 
enhances the intensity of colder fluid at the ambient (as noticed in Fig. 6) towards the hot sheet which eventually corresponds to decrease the local fluid temperature. Fig. 7 perceives the behavior of Prandtl number 
on the temperature. A bigger Prandtl number fluid has less thermal diffusivity and hence it allows less thermal effect to penetrate deeper into the fluid. As a result, temperature decreases and the thermal boundary layer becomes thinner when 
is increased. This decrease in thickness of the thermal boundary layer is compensated with a larger wall slope of temperature function.
Figure 6. Influence of stretching rates ratio 
on the temperature 
.
Figure 7. Influence of Prandtl number 
on the temperature 
.
Fig. 8 plots the wall temperature gradient against 
with the variation in stretching rates ratio 
. The wall heat transfer rate approaches the zero value for vanishing Prandtl number 
, a fact that is clear from the energy equation (9). Moreover, this Fig. compliments the results of Fig. 4. In bigger Prandtl number fluids the convection is effective in transferring energy from the stretching sheet compared to pure conduction. Due to this reason the wall heat transfer rate is an increasing function of 
. The reduction in thermal boundary layer thickness with an increase in 
meets with the bigger magnitude of local Nusselt number. In other words the enhanced intensity of cold fluid at the ambient towards the hot fluid closer to the sheet results in larger heat transfer rate at the sheet.
Figure 8. Influence of Prandtl number 
and stretching rates ratio 
on the wall temperature gradient 
.
Tables 2 and 3 provide the numerical values of skin friction coefficients and local Nusselt number for different values of parameters by employing shooting method. The results are compared with the MATLAB built in function bvp5c and found in excellent agreement. We notice that wall shear stresses increase with an increase in 
more rapidly at 
when compared with 
. The thinner thermal boundary layer accounted for larger 
accompanies with larger temperature gradient along the sheet. The magnitude of increase in wall temperature gradient 
with an increase in 
increases when 
is increased.
Table 2. Numerical values of 
and 
for different values of 
and 
.
|
|
|
|
||
| shooting | bvp5c | shooting | bvp5c | ||
| 1 | 0 | −1 | −1 | 0 | 0 |
| 0.5 | −1.224745 | −1.224742 | −0.612372 | −0.612371 | |
| 1 | −1.414214 | −1.414214 | −1.414214 | −1.414214 | |
| 3 | 0 | −1.624356 | −1.624356 | 0 | 0 |
| 0.5 | −1.989422 | −1.989422 | −0.994711 | −0.994711 | |
| 1 | −2.297186 | −2.297182 | −2.297186 | −2.297182 | |
Table 3. Numerical values of local Nusselt number 
for various values of 
and 
.
|
|
|
|
|
| shooting | bvp5c | |||
| 1 | 0.7 | 0 | 0.793668 | 0.793668 |
| 0.5 | 0.972033 | 0.972029 | ||
| 1 | 1.122406 | 1.122321 | ||
| 1 | 0 | 1.000000 | 0.999990 | |
| 0.5 | 1.224745 | 1.224742 | ||
| 1 | 1.414214 | 1.414214 | ||
| 7 | 0 | 3.072250 | 3.072251 | |
| 0.5 | 3.762723 | 3.762724 | ||
| 1 | 4.344818 | 4.344779 | ||
| 3 | 0.7 | 0 | 1.292193 | 1.292194 |
| 0.5 | 1.582607 | 1.582607 | ||
| 1 | 1.827437 | 1.827427 | ||
| 1 | 0 | 1.624356 | 1.624356 | |
| 0.5 | 1.989422 | 1.989422 | ||
| 1 | 2.297186 | 2.297182 | ||
| 7 | 0 | 4.968777 | 4.968777 | |
| 0.5 | 6.085484 | 6.085485 | ||
| 1 | 7.026912 | 7.026913 | ||
Conclusions
For the first time, the flow and heat transfer over a plane surface stretched non-linearly in two lateral directions have been investigated. The simulation in this study assumes that the temperature across the sheet is non-linearly distributed. Both analytic and numerical solutions are obtained and found in excellent agreement. Following are the major results of this study.
I. It is seen that shear stress at the wall increase when the stretching rates ratio is increased. The entrainment velocity is negative, representing a downward flow in the vertical direction, which is a consequence of the bi-directional stretching sheet.
II. The increased intensity of the cold fluid at the ambient towards the stretching sheet with an increase in stretching rates ratio
decreases the fluid temperature.III. The temperature decreases and thermal boundary layer thins when the power-law index
is increased.IV. The results for the case of two-dimensional and axisymmetric flows can be obtained as special cases of present study when
and 
respectively.
Data Availability
The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper.
Funding Statement
The author(s) received no specific funding for this work.
References
- 1.Crane LJ (1970). Flow past a stretching plate. Z Angew Math Phys, 21(4), 645–647.
- 2.Wang CY (1984). The three-dimensional flow due to a stretching flat surface. Phys Fluids, 27(8), 1915–1917.
- 3.Lakshmisha KN, Venkateswaran S, Nath G (1988). Three-dimensional unsteady flow with heat and mass transfer over a continuous stretching surface. J Heat Trans, 110(3), 590–595.
- 4.Takhar HS, Chamkha AJ, Nath G (2001). Unsteady three-dimensional MHD-boundary-layer flow due to the impulsive motion of a stretching surface. Acta Mech, 146(1–2), 59–71.
- 5.Xu H, Liao SJ, Pop I (2007). Series solutions of unsteady three-dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate. Eur J Mech-B/Fluids, 26(1), 15–27.
- 6.Liu I, Andersson HI (2008). Heat transfer over a bidirectional stretching sheet with variable thermal conditions. Int J Heat Mass Transf, 51(15), 4018–4024.
- 7.Hayat T, Mustafa M, Hendi AA (2011). Time-dependent three-dimensional flow and mass transfer of elastico-viscous fluid over unsteady stretching sheet. Appl Math Mech, 32(2), 167–178.
- 8.Hayat T, Awais M, Obaidat S (2012). Three-dimensional flow of a Jeffery fluid over a linearly stretching sheet. Commun Nonlinear Sci Numer Simul, 17(2), 699–707.
- 9.Liu IC, Wang HH, Peng YF (2013). Flow and heat transfer for three-dimensional flow over an exponentially stretching surface. Chem Eng Commun, 200(2), 253–268.
- 10.Khan JA, Mustafa M, Hayat T, Farooq MA, Alsaedi A, et al. (2014). On model for three-dimensional flow of nanofluid: An application to solar energy. J Mol Liq, 194, 41–47.
- 11.Sheikholeslami M, Ganji DD (2014). Numerical investigation for two phase modeling of nanofluid in a rotating system with permeable sheet. J Mol Liq, 194, 13–19.
- 12.Vajravelu K (2001). Viscous flow over a nonlinearly stretching sheet. Appl Math Comput, 124(3), 281–288.
- 13.Cortell R (2007). Viscous flow and heat transfer over a nonlinearly stretching sheet. Appl Math Comput, 184(2), 864–873.
- 14.Bhargava R, Sharma S, Takhar HS, Bég OA, Bhargava P (2007). Numerical solutions for micropolar transport phenomena over a nonlinear stretching sheet. Nonlinear Anal.-Model Control, 12, 45–63.
- 15.Cortell R (2008). Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Phys Lett A, 372(5), 631–636.
- 16.Hayat T, Abbas Z, Javed T (2008). Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet. Phys Lett A, 372(5), 637–647.
- 17.Kechil S, Hashim I (2008). Series solution of flow over nonlinearly stretching sheet with chemical reaction and magnetic field. Phys Lett A, 372(13), 2258–2263.
- 18.Hayat T, Hussain Q, Javed T (2009). The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet. Nonlinear Anal-Real World Appl, 10(2), 966–973.
- 19.Ziabakhsh Z, Domairry G, Bararnia H, Babazadeh H (2010). Analytical solution of flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. J Taiwan Inst Chem Eng, 41(1), 22–28.
- 20.Rana P, Bhargava R (2012). Flow and heat transfer of a nanofluid over a nonlinearly stretching sheet: a numerical study. Commun Nonlinear Sci Numer Simul, 17(1), 212–226.
- 21.Shahzad A, Ali R, Khan M (2012). On the exact solution for axisymmetric flow and heat transfer over a nonlinear radially stretching sheet. Chinese Phys Lett, 29(8), 084705.
- 22.Mukhopadhyay S (2013). Analysis of boundary layer flow over a porous nonlinearly stretching sheet with partial slip at the boundary. Alexandria Eng J, 52(4), 563–569.
- 23.Mukhopadhyay S (2013). Casson fluid flow and heat transfer over a nonlinearly stretching surface. Chinese Phys B, 22(7), 074701.
- 24.Rashidi MM, Freidoonimehr N, Hosseini A, Bég OA, Hung TK (2014). Homotopy simulation of nanofluid dynamics from a non-linearly stretching isothermal permeable sheet with transpiration. Meccanica, 49(2), 469–482.
- 25.Rashidi MM, Keimanesh M (2010). Using differential transform method and padé approximant for solving MHD flow in a laminar liquid film from a horizontal stretching surface. Math Prob Eng, 2010 Article ID 491319.
- 26.Rashidi MM, Pour SM (2010). Analytic approximate solutions for unsteady boundary-layer flow and heat transfer due to a stretching sheet by homotopy analysis method. Nonlinear Anal.-Model Control, 15(1), 83–95.
- 27.Rashidi MM, Erfani E (2011). The modified differential transform method for investigating nano boundary-layers over stretching surfaces. Int J Numer Methods Heat Fluid Flow, 21(7), 864–883.
- 28.Sheikholeslami M, Ganji DD (2014). Magnetohydrodynamic flow in a permeable channel filled with nanofluid. Scientia Iranica, 21(1), 203–212.
- 29.Sheikholeslami M, Gorji-Bandpy M (2014). Free convection of ferrofluid in a cavity heated from below in the presence of an external magnetic field. Powder Technol, 256, 490–498.
- 30.Sheikholeslami M, Gorji-Bandpy M, Ganji DD, Soleimani S (2014). Thermal management for free convection of nanofluid using two phase model. J Mol Liq, 194, 179–187.
- 31.Sheikholeslami M, Gorji-Bandpy M, Ganji DD, Rana P, Soleimani S (2014). Magnetohydrodynamic free convection of Al2O3-water nanofluid considering Thermophoresis and Brownian motion effects. Comput Fluids, 94, 147–160.
- 32.Sheikholeslami M, Gorji-Bandpy M, Ganji DD (2014). Lattice Boltzmann method for MHD natural convection heat transfer using nanofluid. Powder Technol, 254, 82–93.
- 33.Sheikholeslami M, Ganji DD, Gorji-Bandpy M, Soleimani S (2014). Magnetic field effect on nanofluid flow and heat transfer using KKL model. J Taiwan Inst Chem Eng, 45(3), 795–807.
- 34.Marinca V, Herişanu N (2008). Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int Commun Heat Mass, 35(6), 710–715.
- 35.Niu Z, Wang C (2010). A one-step optimal homotopy analysis method for nonlinear differential equations. Commun Nonlinear Sci Numer Simul, 15(8), 2026–2036.
- 36.Liao S (2010). An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun Nonlinear Sci Numer Simul, 15(8), 2003–2016.
- 37.Abbasbandy S, Shivanian E, Vajravelu K (2011). Mathematical properties of ħ-curve in the frame work of the homotopy analysis method. Commun Nonlinear Sci Numer Simul, 16(11), 4268–4275.
- 38.Mushtaq A, Mustafa M, Hayat T, Rahi M, Alsaedi A (2013). Exponentially Stretching Sheet in a Powell–Eyring Fluid: Numerical and Series Solutions. Z Naturforsch, 68a(12), 791–798.
- 39.Mustafa M, Farooq MA, Hayat T, Alsaedi A (2013). Numerical and series solutions for stagnation-point flow of nanofluid over an exponentially stretching sheet. PLoS ONE, 8(5), e61859. [DOI] [PMC free article] [PubMed]
- 40.Farooq U, Xu H (2014). Free convection nanofluid flow in the stagnation-point region of a three-dimensional body. The Scientific World Journal Article ID 158269. [DOI] [PMC free article] [PubMed]
- 41.Zhao YL, Liao SJ (2014). Advances in the homotopy analysis method, Chapter 9, World Scientific.
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper.