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Suction/injection effects on an unsteady MHD Casson thin film flow with slip and uniform thickness over a stretching sheet along variable flow properties
- Sajid Rehman^{1}Email author,
- M. Idrees^{1},
- Rehan Ali Shah^{2} and
- Zeeshan Khan^{3}
- Received: 30 June 2018
- Accepted: 15 January 2019
- Published: 31 January 2019
Abstract
In this investigation the attention is given to a mathematical model of the non-Newtonian Casson liquid over an unsteady stretching sheet under the combined effects of different natural parameters with heat transfer in the presence of suction/injection phenomena. The movement of a laminar thin liquid film and associated heat transfer from a horizontal stretching surface is studied. Magnetic field is proposed perpendicular to the direction of flow, while surface tension is varied quadratically with temperature of the conducting fluid. Further, variable viscosity and thermal conductivity (linear function of temperature) of the flow are examined. The transformation allows to convert the boundary layer model to a system of nonlinear ODEs (ordinary differential equations). Analytical and numerical solutions of the resulting nonlinear ODEs are obtained by using HAM and BVP4C package. Thickness of the boundary layer is investigated by both methods for a classical selection of the unsteadiness parameter. A selection of the parameter ranges is studied for better solution of the problem. Present observation displays the joined effects of magnetic field, surface tension, suction/injection, and slippage at the boundary is to improve the thermal boundary layer thickness. Results for the heat flux (Nusselt number), skin friction coefficient, and free surface temperature are granted graphically and in a table form. Similarly, the effects of natural parameters on the velocity and temperature profiles are investigated.
Keywords
- Parameter range
- Viscosity parameter
- Suction/injection parameter
- Slip effect at the boundary
- Thermal conductivity parameter
1 Introduction
A large number of industrial processes, the effect of boundary layer flow with heat transfer over an unsteady stretching sheet with free surface flow have wide use in wire coating, drawing of plastic sheets, metal and polymer extrusion, foodstuff processing, daily life uses equipments, etc. Rate of heat transfer in the stretching sheet describes the good and bad quality of coating during manufacturing of the wire. Therefore, a number of experiments have been discussed under the primary effects of the boundary layer phenomena in a different frame of reference. Crane [1] started his work from the steady stretching sheet by considering Newtonian liquids under the assumption that velocity will vary in the direction of flow and that it must be linear to the distance from the specific point. Soon a researcher Winter [2] added a viscous dissipated term to the temperature equation that contributed a very important role to the energy distribution and determined the effects of energy during the heat transfer in the sheet. Meanwhile, Wang [3] took Newtonian fluid and discussed the time-dependent stretching sheet under the lead of Newtonian flow on the sheet. Navier–Stokes equations were converted to the sample equations of one independent variable called ordinary differential equations by the use of well form transformation, and solution was achieved by two methods, numerical and analytical. Similarly, Thompson and Troian [4] investigated the effects of slip velocity by assuming that slip velocity is directly dependent on the wall shear rate, i.e., slip length and shear rate (due to the diverging slip length). Dandapat et al. [5] presented the analysis of the heat transfer by considering the work of Wang [3]. Then they [6] extended his work by considering surface tension and constructed thermocapillary number in the work of Dandapat et al. [5]. They presented different physical effects on the heat transfer. The power-law model of the non-Newtonian liquid was discussed by assuming the unsteady liquid flow over a stretching sheet and a brief result on the heat transfer was presented; this was done by Chen [7]. Wang and Pop [8] took a step towards the solution of the problem presented by Chen [7] and solved it by HAM. Again Chen [9] discussed viscous dissipation effects to his work [7]. Abbas et al. [10] studied the effects of second grade liquid by introducing an unsteady stretching sheet of second grade liquid. Mahmoud and Megahed [11] developed a variation of thermal conductivity and viscosity in the thin liquid fluid over an unsteady stretching sheet in the presence of Hartmann number to the power-law liquid. Abel et al. [12] continued the concept of previous work and introduced viscous dissipation with magnetic effects to a plane stretching sheet over laminar fluid film.
Mukhopadhay et al. [13] presented a two-dimensional problem of unsteady non-Newtonian liquid over a stretching surface with defined surface temperature. They used the Casson liquid model for the observation of non-Newtonian liquid with its physical behavior, and the result was obtained by the well-known method called shooting method. Moreover, an approximate solution was obtained for the steady momentum equation. Meanwhile, Nadeem et al. [14] discussed the effects of MHD Casson liquid flow with boundary layer momentum equation attached with the exponentially admitted shrinking sheet. They used Adomian decomposition method (ADM) for the analytical solution to the ODEs, and different graphs were constructed for arbitrary parameters to the velocity profiles and their possible conclusions. Mahdy [15] presented a numerical solution for the unsteady MHD Casson liquid over the stretching sheet with the presence of slip effect at the boundary, suction/blowing, and the defined surface temperature. Different physical behaviors were discussed for the local similar result of Casson model with the admission of robust computer algebra software MATLAB. Recently, Prasad et al. [16] discussed heat transfer of a Casson liquid flow having laminar boundary layer phenomena in a horizontal cylinder with heat and hydrodynamic slip boundary, where constant temperature was considered at the surface of the cylinder. The naturally parabolic boundary layer equation was normalized with the use of non-similar form, and then they used a scheme, which is efficient, well-tested, stable, and implicit, called Keller– box finite-difference scheme. A conducted viscoplastic fluid passing from a conducted channel was discussed for the MHD flow and heat transfer to their theoretical results by Akbar et al. [17]. They proposed a robust Casson model in the presence of magnetic field applied to the flow side along viscous dissipation term. Makinde and Rundora [18] studied heat decomposition in a time-dependent mixed convection reactive Casson liquid flow in a filled perpendicular channel existing saturated suction/injection medium. Walls of the channel were also porous with injection from the right wall and suction out from the left wall. Solution was discussed by using a finite difference method, a semi-discretized method, and along with the scheme of Runge–Kutta–Fehlberg integration of fourth order. Recently, a numerical solution for the flow of unsteady Casson thin film liquid over an unsteady sheet with the presence of variation in heat flux and viscous dissipation along the slip parameter at boundary condition were discussed by Megahed [19]. Results were obtained for the heat transfer with respect to different parameters by the help of shooting method, while the results of previous papers were compared. Furthermore, the influence of MHD on fluid flow in various geometries was studied in [20–22]. These are very resent works in the fluid mechanics having MHD flow over thin film with heat and mass transfers.
Motivated by the given investigation, the formulation of the constructed problem is to investigate the unsteady non-Newtonian Casson liquid over a stretching sheet under the combined effects of different natural parameters with Casson boundary layer flow and heat transfer. The movement of a laminar thin liquid film and associated heat transfer from a horizontal stretching surface will be studied. Magnetic field will be normal to the flow direction and surface tension will vary quadratically with temperature of the conducting fluid for viscous incompressible free surface flow. Further, the observation of variation of thermal conductivity and viscosity of the fluid flow will be observed for their linear functions of temperatures. The transformation will allow us to convert the boundary layer model to a system of nonlinear ODEs. Solutions of the resulting nonlinear ODEs will be obtained by using HAM and shooting method. Thickness of the boundary layer is investigated by both methods for a classical selection of the unsteadiness parameter. Present observation displays the joined effects of Casson liquid, suction/injection, and slip effect at the boundary, and magnetic field is to improve the thermal boundary layer thickness. Different parameters will be discussed for their physical importance such as Nusselt number, skin friction coefficient, free surface temperature, suction/injection, Casson parameter, slip parameter, Hartmann number, thermocapillary number, and Prandtl number.
2 Problem formulation
2.1 Governing equations
2.2 Similarity transformation
In view of the formulation structure, some limitations of the model problem are the following: The modeled problem is invalid for the flow of gases due to η mentioned in (20), because by the use of viscosity it becomes undefined. Similarly, for \(S=0\), the problem reduces from unsteady to steady state. Further, if \(\mathit{Ma}=0\), then there is no magnetic force, and we get a model having no magnetic force.
3 Solution approach
For solution purpose, we have the following subsections.
3.1 Skin friction coefficient and Nusselt number
The interesting physical parameters are the skin friction coefficient \(C_{fx}\) and the Nusselt number \(\mathrm{Nu}_{x}\) mentioned in [19]. The statements of these quantities are \(C_{fx}=\frac{\tau _{w}}{\rho U^{2}/2}\), \(\mathrm{Nu}_{x}=\frac{x q_{w}}{ \kappa (T_{\mathrm{ref}})}\), respectively. The notation \(\tau _{w}\) represents wall surface shearing stress, and \(q_{w}\) expresses the rate of heat transfer from an elastic sheet. The detailed expressions are as follows: \(\tau _{w}=(1+\frac{1}{\beta })\mu (\frac{\partial u}{\partial y})_{y=0}\), \(q_{w}=-\kappa (\frac{\partial T}{\partial y})_{y=0}\).
3.2 Optimal convergence control parameters
Optimal value of convergence control parameters versus different orders of approximation
Order of approximation | \(\hbar _{f}\) | \(\hbar _{\theta }\) | \(\varepsilon _{m}^{t}\) |
---|---|---|---|
2 | −0.126785 | −0.773882 | 4.23292 × 10^{−6} |
3 | −0.129292 | −0.727522 | 1.11854 × 10^{−5} |
4 | −0.130459 | −0.778901 | 5.97889 × 10^{−9} |
5 | −0.127917 | −0.787161 | 1.98668 × 10^{−9} |
6 | −0.130066 | −0.787120 | 1.91613 × 10^{−13} |
Individual averaged squared residual errors using optimal values of auxiliary parameters
m | \(\varepsilon _{m}^{f}\) | \(\varepsilon _{m}^{\theta }\) | CPU time |
---|---|---|---|
2 | 3.59250 × 10^{−5} | 2.69667 × 10^{−7} | 3.93754 seconds |
4 | 9.18224 × 10^{−10} | 7.82572 × 10^{−11} | 7.96887 seconds |
6 | 4.18373 × 10^{−13} | 1.99647 × 10^{−14} | 16.2815 seconds |
8 | 3.59431 × 10^{−16} | 6.63613 × 10^{−18} | 27.1254 seconds |
10 | 2.18166 × 10^{−19} | 2.56076 × 10^{−21} | 41.0319 seconds |
16 | 2.75479 × 10^{−29} | 2.04838 × 10^{−31} | 104.892 seconds |
20 | 4.73921 × 10^{−33} | 2.61926 × 10^{−34} | 169.253 seconds |
26 | 4.84321 × 10^{−33} | 2.31112 × 10^{−34} | 301.348 seconds |
30 | 4.84321 × 10^{−33} | 2.31112 × 10^{−34} | 423.631 seconds |
40 | 4.84321 × 10^{−33} | 2.31112 × 10^{−34} | 883.045 seconds |
Convergence of HAM on the basis of skin friction \(f''(0)\) and heat flux \(-\theta '(0)\) for selected values of \(\beta =0.2\), \(I_{0}=0\), \(\lambda =0.2\), \(A=0.3\), \(N=0.2\), \(S=0.4\), \(\mathit{Ma}=1\), \(M=1\), \(\varUpsilon =0.127013\), \(\mathit{Pr} =1\)
m | \(f''(0)\) (skin friction) | \(-\theta '(0)\) (heat flux) |
---|---|---|
1 | 0.87518974409683891741 | 0.07914447507774416643 |
5 | 0.87583582258625018543 | 0.08221669124276321393 |
10 | 0.87583587982339187017 | 0.08221658417434317797 |
15 | 0.87583587982465930304 | 0.08221658417178512573 |
20 | 0.87583587982465936218 | 0.08221658417178505904 |
25 | 0.87583587982465936218 | 0.08221658417178505903 |
30 | 0.87583587982465936218 | 0.08221658417178505903 |
35 | 0.87583587982465936218 | 0.08221658417178505903 |
40 | 0.87583587982465936218 | 0.08221658417178505903 |
3.3 Results and discussion
The modeled problem consists of a couple of ODEs shown in Eqs. (24) and (25) with the defined physically admitted boundary conditions mentioned in (26) and (27). They are solved by the well-known methods called HAM and shooting method for arbitrary values of the interesting non-dimensional parameters listed as Grashof number, unsteady parameter, skin friction, thermal conductivity parameter, heat flux, Hartmann parameter, variable viscosity parameter, film thickness, free surface temperature, suction/injection parameter, Prandtl number, Casson parameter, slip velocity parameter, and thermocapillary number.
Variation of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation when \(A=0.3\), \(I_{0}=0\), \(N=0.2\), \(S=0.4\), \(\mathit{Ma}=1\), \(M=1\), \(\mathit{Pr} =1\)
\(\hbar _{f}\) | \(\hbar _{\theta }\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) |
---|---|---|---|---|---|
β = 0.2, λ = 0.2 | |||||
−0.0265314 | −0.292062 | 2.82701 | −0.0156117 | 0.177729 | 2.19976 |
−0.0296361 | −0.315862 | 2.62701 | −0.0189389 | 0.211150 | 2.02976 |
−0.0333679 | −0.341349 | 2.42701 | −0.0229204 | 0.250369 | 1.85697 |
β = 1.0, λ = 0.2 | |||||
−0.330660 | −0.460604 | 1.62701 | −0.0915227 | 0.478447 | 1.133350 |
−0.343891 | −0.502384 | 1.42701 | −0.1097190 | 0.553971 | 0.943595 |
−0.352134 | −0.554164 | 1.22701 | −0.1299110 | 0.635164 | 0.753939 |
λ = 0.5, β = 0.2 | |||||
−0.0270687 | −0.290661 | 2.82701 | −0.00718384 | 0.177708 | 2.19878 |
−0.0301325 | −0.314454 | 2.62701 | −0.00872102 | 0.211112 | 2.02871 |
−0.0982505 | −0.315507 | 2.42701 | −0.01055900 | 0.250307 | 1.85587 |
λ = 1.0, β = 0.2 | |||||
−0.0584564 | −0.486120 | 1.62701 | −0.0114765 | 0.477491 | 1.129600 |
−0.1158530 | −0.494437 | 1.42701 | −0.0135242 | 0.552856 | 0.940594 |
−0.1173810 | −0.549646 | 1.22701 | −0.0157667 | 0.633955 | 0.751751 |
Comparison of the values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation and shooting method for the case \(\lambda =0.2\), \(I_{0}=0\), \(A=0.3\), \(N=0.2\), \(S=0.4\), \(\mathit{Ma}=1\), \(M=1\), \(\mathit{Pr} =1\) and several values of β
β | HAM | BVP4C | ||||||
---|---|---|---|---|---|---|---|---|
β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | |
0.2 | 2.927010 | −0.0141535 | 0.162948 | 2.28385 | 2.927010 | −0.0141539 | 0.162955 | 2.28392 |
0.5 | 2.627010 | −0.0281333 | 0.211237 | 2.03197 | 2.627010 | −0.0281332 | 0.211238 | 2.03197 |
1.0 | 2.327010 | −0.0454676 | 0.272640 | 1.77295 | 2.327010 | −0.0454676 | 0.272640 | 1.77294 |
1.5 | 2.162701 | −0.0575470 | 0.312715 | 1.62763 | 2.162701 | −0.0575470 | 0.312714 | 1.62762 |
2.0 | 1.827010 | −0.0843351 | 0.410243 | 1.32158 | 1.827010 | −0.0843277 | 0.410241 | 1.32160 |
2.5 | 1.527010 | −0.1159610 | 0.515644 | 1.03948 | 1.527010 | −0.1159575 | 0.515643 | 1.03948 |
Comparison of the values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation and shooting method for the case \(\beta =0.2\), \(I_{0}=0\), \(A=0.3\), \(N=0.2\), \(S=0.4\), \(\mathit{Ma}=1\), \(M=1\), \(\mathit{Pr} =1\) and several values of λ
λ | HAM | BVP4C | ||||||
---|---|---|---|---|---|---|---|---|
β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | |
0.0 | 2.927010 | −0.0651851 | 0.163076 | 2.29018 | 2.927010 | −0.0650983 | 0.163074 | 2.29021 |
0.2 | 2.627010 | −0.0189389 | 0.211150 | 2.02976 | 2.627010 | −0.0189335 | 0.211149 | 2.02977 |
0.5 | 2.327010 | −0.0116064 | 0.272305 | 1.76827 | 2.327010 | −0.0116024 | 0.272307 | 1.76833 |
1.0 | 2.162701 | −0.0071275 | 0.312168 | 1.62200 | 2.162701 | −0.0071262 | 0.312169 | 1.62204 |
1.5 | 1.827010 | −0.0065553 | 0.409284 | 1.31620 | 1.827010 | −0.0065549 | 0.409284 | 1.31622 |
2.0 | 1.527010 | −0.0064081 | 0.514285 | 1.03508 | 1.527010 | −0.0064080 | 0.514284 | 1.03508 |
Comparison of the values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation and BVP4C for the case \(\beta =0.2\), \(I_{0}=0\), \(\lambda =0.2\), \(A=0.3\), \(N=0.2\), \(S=0.4\), \(\mathit{Ma}=1\), \(M=1\) and several values of Pr
Pr | HAM | BVP4C | ||||||
---|---|---|---|---|---|---|---|---|
β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | |
0.7 | 2.927010 | −0.0215291 | 0.245797 | 1.87652 | 2.927010 | −0.0215291 | 0.245798 | 1.87652 |
1.0 | 2.627010 | −0.0189333 | 0.211146 | 2.02976 | 2.627010 | −0.0189335 | 0.211149 | 2.02977 |
2.0 | 2.327010 | −0.0108576 | 0.118714 | 2.58480 | 2.327010 | −0.0108139 | 0.118543 | 2.58522 |
3.0 | 2.162701 | −0.0074303 | 0.080081 | 2.95425 | 2.162701 | −0.0073173 | 0.079346 | 2.95632 |
4.0 | 1.827010 | −0.0082316 | 0.086707 | 2.87997 | 1.827010 | −0.0081283 | 0.086067 | 2.88159 |
5.0 | 1.527010 | −0.0103011 | 0.106688 | 2.68538 | 1.527010 | −0.0102314 | 0.106327 | 2.68614 |
Comparison of the values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation and shooting method for the case \(\beta =0.2\), \(I_{0}=0\), \(\lambda =0.2\), \(A=0.3\), \(N=0.2\), \(S=0.4\), \(\mathit{Pr} =1\), \(M=1\) and several values of Ma
Ma | HAM | BVP4C | ||||||
---|---|---|---|---|---|---|---|---|
β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | |
0.0 | 1.927011 | −0.037135 | 0.378277 | 1.40976 | 1.927011 | −0.037132 | 0.378276 | 1.40978 |
1.0 | 1.627012 | −0.047255 | 0.47781 | 1.13090 | 1.627012 | −0.047254 | 0.477809 | 1.13091 |
3.0 | 1.327013 | −0.059117 | 0.593312 | 0.84702 | 1.327013 | −0.059116 | 0.593311 | 0.84703 |
5.0 | 1.162704 | −0.066144 | 0.661408 | 0.69259 | 1.162704 | −0.066144 | 0.661407 | 0.69259 |
7.0 | 0.827015 | −0.082756 | 0.801535 | 0.39413 | 0.827015 | −0.082755 | 0.801534 | 0.39413 |
10 | 0.527016 | −0.096483 | 0.910891 | 0.17387 | 0.527016 | −0.096483 | 0.910891 | 0.17387 |
Comparison of the values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation and shooting method for the case \(\beta =0.2\), \(I_{0}=0\), \(\lambda =0.2\), \(A=0.3\), \(N=0.2\), \(S=0.4\), \(\mathit{Pr} =1\), \(\mathit{Ma}=1\) and several values of M
M | HAM | BVP4C | ||||||
---|---|---|---|---|---|---|---|---|
β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | |
0.0 | 1.927011 | 0.00000000 | 0.380657 | 1.42266 | 1.927011 | 0.00000000 | 0.380657 | 1.42265 |
0.3 | 1.627012 | −0.0142617 | 0.48024 | 1.13867 | 1.627012 | −0.0142609 | 0.48023 | 1.13868 |
0.5 | 1.327013 | −0.0302575 | 0.59549 | 0.85101 | 1.327013 | −0.0302571 | 0.595489 | 0.85102 |
0.7 | 1.162704 | −0.0477753 | 0.662731 | 0.69433 | 1.162704 | −0.0477751 | 0.662730 | 0.69433 |
1.0 | 0.827015 | −0.0845684 | 0.801455 | 0.39407 | 0.827015 | −0.0845683 | 0.801454 | 0.39408 |
1.2 | 0.527016 | −0.117232 | 0.910344 | 0.17375 | 0.527016 | −0.1172314 | 0.910343 | 0.17375 |
Comparison of the values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation and shooting method for the case \(\beta =0.2\), \(I_{0}=0\), \(\lambda =0.2\), \(A=0.3\), \(N=0.2\), \(M=1\), \(\mathit{Pr} =1\), \(\mathit{Ma}=1\) and several values of S
S | HAM | BVP4C | ||||||
---|---|---|---|---|---|---|---|---|
β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) | |
0.4 | 1.927011 | −0.036380 | 0.378317 | 1.41000 | 1.927011 | −0.036377 | 0.378316 | 1.41002 |
0.6 | 1.627012 | −0.034292 | 0.35886 | 1.47003 | 1.627012 | −0.034288 | 0.358859 | 1.47006 |
0.8 | 1.327013 | −0.038051 | 0.393797 | 1.36371 | 1.327013 | −0.038048 | 0.393796 | 1.36373 |
1.0 | 1.162704 | −0.039363 | 0.405902 | 1.32835 | 1.162704 | −0.039360 | 0.405900 | 1.32837 |
1.2 | 0.827015 | −0.055462 | 0.551115 | 0.94681 | 0.827015 | −0.055462 | 0.551114 | 0.94681 |
1.4 | 0.527016 | −0.076787 | 0.735821 | 0.53102 | 0.527016 | −0.076787 | 0.735820 | 0.53102 |
Different values of \(\beta =\varUpsilon ^{\frac{1}{2}}\), \(f''(0)\), \(\theta (1)\), and \(-\theta '(0)\) using 20th-order HAM via Mathematica package BVPh2.0 approximation for the case \(S=0.4\), \(\beta =0.2\), \(\lambda =0.2\), \(A=0.3\), \(N=0.2\), \(M=1\), \(\mathit{Pr} =1\), \(\mathit{Ma}=1\) and several values of \(I_{0}\)
\(I_{0}>0\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) |
---|---|---|---|---|
0.20 | 1.927011 | −0.212293 | 0.793334 | 0.402894 |
0.25 | 1.627012 | −0.249894 | 0.833711 | 0.320317 |
0.30 | 1.327013 | −0.287485 | 0.873305 | 0.240635 |
0.35 | 1.162704 | −0.323504 | 0.900353 | 0.186803 |
0.40 | 0.827015 | −0.360966 | 0.937495 | 0.114535 |
0.45 | 0.527016 | −0.397485 | 0.966743 | 0.059133 |
\(I_{0}<0\) | β | \(f''(0)\) | θ(1) | \(-\theta '(0)\) |
---|---|---|---|---|
−0.20 | 1.927011 | 0.057075 | 0.715539 | 0.553116 |
−0.25 | 1.627012 | 0.086068 | 0.744323 | 0.492561 |
−0.30 | 1.327013 | 0.114487 | 0.777895 | 0.424047 |
−0.35 | 1.162704 | 0.144611 | 0.797593 | 0.384115 |
−0.40 | 0.827015 | 0.171291 | 0.845510 | 0.290587 |
−0.45 | 0.527016 | 0.197796 | 0.894847 | 0.196312 |
Suction case \(I_{0} > 0\):
Based on Table 11, by increasing the value of suction parameter \(I_{0}\) and reducing the film thickness \(\beta ^{2}\), heat flux and local skin friction coefficient of suction side are decreased, while free surface temperature is increased for fixed values of the remaining parameters. Physically, due to the low pressure at the inlet face, the fluid enters through medium, and due to the higher pressure at the outlet face, the fluid outs at the discharge side.
Injection case \(I_{0} < 0\):
Similarly, based on Table 11, by decreasing the value of injection parameter \(I_{0}\) and reducing the thickness of the sheet \(\beta ^{2}\), free temperature and skin friction coefficient of suction side are increased, while heat flux is decreased for fixed values of the remaining parameters. Usually, it happens that due to the higher pressure at the injection face, the fluid is forced out through medium, and due to the lower pressure at the suction face, the fluid enters at the discharge side. Excellent agreements are observed between both the techniques, shooting method and HAM, for a variety of physical parameter values as displayed in Table 5.
4 Concluding remarks
- 1.
As β is increased, the corresponding velocity increases and the corresponding temperature remains unchanged.
- 2.
As λ is increased, the corresponding flow speed reduces but the corresponding temperature is unchanged.
- 3.
The increase of A causes the flow velocity slightly rise and then decelerate; therefore, the corresponding temperature of the flow rises slightly.
- 4.
When N is increased, the corresponding flow velocity decreases while the corresponding temperature rises.
- 5.
Increasing of S increases the velocity profile while temperature decreases.
- 6.
The increasing of \(\beta ^{2}\) decreases velocity and increases temperature.
- 7.
Increased M leads to decrease in flow velocity while temperature is slightly decreased.
- 8.
Increased value of Ma results in the increased corresponding velocity while temperature is unchanged.
- 9.
Increasing the value of Pr increases the flow velocity while decreasing the temperature.
- 10.
The increase in \(I_{0}\) (in both cases, suction and injection) the corresponding azimuthal velocity increases while velocity decreases and the rise of temperature is seen.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their constructive comments and valuable suggestions to improve the quality of the paper.
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Funding
This paper is self-supported by authors in respect of funding and technically supported by Islamia College University, Khyber Pakhtunkhwa, Peshawar, Pakistan.
Authors’ contributions
All authors participated in the analysis of the results and manuscript coordination. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no conflict of interests.
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