Framing the features of MHD boundary layer flow past an unsteady stretching cylinder in presence of non-uniform heat source

Journal of Molecular Liquids 225 (2017) 418–425

Contents lists available at ScienceDirect

Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq

Framing the features of MHD boundary layer flow past an unsteady stretching cylinder in presence of non-uniform heat source Nilankush Acharya a,⁎, Kalidas Das b, Prabir Kumar Kundu a a b

Dept. of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India Dept. of Mathematics, A.B.N. Seal College, Cooch Behar PIN-736101, West Bengal, India

a r t i c l e

i n f o

Article history: Received 30 July 2016 Received in revised form 18 October 2016 Accepted 23 November 2016 Available online 24 November 2016 Keywords: Stretching cylinder MHD Non uniform heat source

a b s t r a c t In this article main effort has been devoted towards the study of MHD boundary layer flow analysis past an unsteady continuously moving stretching cylinder invoking the partial slip mechanism. Furthermore we have analysed our investigation including the presence of non uniform heat source in the flow field. The emerging foremost flow related non-linear equations have been solved numerically via RK-4 approach which includes shooting technique. The impact of pertinent parameters on velocity and temperature profile has been deliberated with physical justification through tables and graphs. Furthermore statistical technique has been employed to reveal the correlation of parameters with the physical properties of the flow system. Our investigation explores that the temperature escalates due to the improvisation of curvature parameter. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Magneto-hydrodynamics i.e. MHD allow us to define it as the coupling of the dynamics of fluid flow features and electro-magnetism characteristics. More precisely it is the noble way to capture activities of fluid flow which is electrically conducted in presence of magnetic. The flow and heat transfer mechanism for a viscous fluid in presence of magnetic field has massive submission in many engineering and technological fields such as MHD power generators, industries such as in petroleum, significant performance in nuclear reactors cooling, studies in the field of plasma, extractions of energy in geothermal field, orientation the configuration of boundary layer structure. Several artificial methods have been developed and carried out in order to control the boundary layer structure, but out of that, the code of applying MHD is an important method for affecting the flow field in the required route by varying the configuration of the boundary layer. Therefore understanding the theory of flow and heat transfer phenomenon has seized the fancy of MHD among scientists and engineers over the last half century. Recently fluid flow dynamics over flat or stretching sheet has turned to be the ground of keen interest owing to its several applicative natures. Crane [1] explores the stretching flow passed over a flat surface. Extension of the work [1] was communicated [2] introducing heat and mass transfer scrutiny under several physical circumstances. More or ⁎ Corresponding author. E-mail addresses: [email protected] (N. Acharya), [email protected] (K. Das), [email protected] (P.K. Kundu).

http://dx.doi.org/10.1016/j.molliq.2016.11.085 0167-7322/© 2016 Elsevier B.V. All rights reserved.

less same investigation in this context can be found in [3–4]. Impression of irregular heat source on fluid flow which is visco-elastic in nature was reported in [5]. Recently, a variety of such problem has been addressed by many researchers [6–10]. Impact of cross diffusion on radiative flow over a stretching sheet demonstrated in [11]. Hayat et al. [12] disclose the effect of soret and dufour effects on three dimensional flow over exponential stretched sheet. More literatures can be encountered in [13– 15]. Fluid flow repealing over cylinders is recognised to be two-dimensional when the comparative study uplifts the body radius as compared to the thickness of the boundary layer. On the other hand, the flow obtains its axi-symmetric definition instead of two-dimensional [16–17] if the cylindrical radius together with boundary layer thickness found to be of the same order for a thin cylinder. Steady flow outside of a cylinder in an ambient fluid at rest has been introduced by Wang [18]. The outcome of Hartmann number on flow over a cylindrical stretching surface was communicated by Ishak et al. [19]. Ishak and Nazar [20] investigated the flow features over a stretching cylinder and obtained similarity solutions assuming that the stretching velocity is linear with axial direction and declared that their study may be considered as the extension of the investigation reported by Grubka and Bobba [21] and Ali [22]. Studies related to the flow characteristics over a cylinder in presence of magnetic field or prescribed heat flux can be found in [23–29]. Most of the above mentioned studies are limited to no-slip conditions. But when the fluid runs over a stretching surface, then it is not always valid to neglect the slip factor. Navier [30] was the first to introduce the velocity slip phenomenon in the flow field. Recently MHD slip flow mechanism past a stretching cylinder can be traced in

N. Acharya et al. / Journal of Molecular Liquids 225 (2017) 418–425

419

such type of flows are as follows: ∂ ∂ ðruÞ þ ðrvÞ ¼ 0 ∂x ∂r ∂u ∂u ∂u þu þv ¼ ∂t ∂x ∂r

ð1Þ

    μf 1 ∂ ∂u σB2 u r − ρ ρ r ∂r ∂r

  ∂T ∂T ∂T α ∂ ∂T 1 ∂q  þu þv ¼ r þ q‴ −  r ∂r ∂t ∂x ∂r ∂r ρC p f ∂r

ð2Þ

ð3Þ

Where u and v are the velocity components of the fluid in the directions x and rrespectively, ρ is the density of the fluid, μf is the fluid dyκ

Fig. 1. Sketch of the physical model.

[31–32]. Numerical and ANN modelling for the flow features over a permeable stretching cylinder with chemical reaction was addressed by Reddy and Das [33]. Existence of heat source or sink frequently performs a key role in the analytic discussion of boundary layer. Quality and fabrication of the final product depends on these factors because the orientation of temperature inside the boundary layer depends on heat source/sink mechanism. More relevant studies unwrapping the effects of non-uniform heat source are in [34–37]. To the authors knowledge no studies developed in this article have far been explored or communicated. So motivated by the above investigations in this paper we are on the way to capture the impact of magnetic field on the viscous fluid flow over an unsteady cylindrically stretched sheet considering non-uniform heat source. Governing partial differential equations with coupled nonlinearity have been condensed to ordinary ones by submitting similarity renovation. After that solving those equations via RK-4 method, result and discussion section have been made highly enriched via graphical and tabular approach coupled with a statistical technique.

2. Formulation of the problem

namic viscosity, σ represents the electrical conductivity, α ¼ ðρC pf Þ

f

denotes the thermal diffusivity, νf is the kinematic viscosity, Cp denotes the specific heat of the fluid, q is the radiative heat flux, non-uniform 0   w heat source is represented by [34] in as q‴ ¼ αU xν f ½A ðT w −T ∞ Þf þ B ðT− ⁎ ⁎ T ∞ Þ, temperature of the fluid is denoted by T, A and B are temperature and space dependent heat source respectively. Now the radiative heat flux for radiation can be simplified by employing the Rosseland approximation as q¼−

4σ  ∂T 4 : 3κ  ∂y

ð4Þ

where σ⁎ is the Stefan Boltzmann constant, k⁎ is the mean absorption coefficient and expanding T4 in Taylor series about T∞ and neglecting higher order we get T 4 ¼ 4T:T 3∞ −3T 3∞

ð5Þ

Now surrogating Eqs. (4) and (5) in Eq. (3) we have    ∂T ∂T ∂T α ∂ ∂T αU w   0 A ðT w −T ∞ Þf þ B ðT−T ∞ Þ þu þv ¼ r þ r ∂r xν f ∂t ∂x ∂r ∂r 2 16σT ∞ 3 ∂ T  þ  ð6Þ 3κ ρC p ∂r 2

2.1. Scrutiny of the flow regime Consider a two dimensional axisymmetric laminar flow of viscous incompressible fluid along an impermeable unsteady continuously moving and stretched cylindrical sheet. The coordinate system has been selected in such a way such that x-axis runs along the axis of the cylinder and r-axis is measured along the radial direction as shown in Fig. 1. The sheet is being stretched with velocity U w ðx; tÞ ¼ ax ð1−λtÞ along

x-axis where a is the stretching rate,λ is the positive con-

stant with the property λt b 1 and dimension of λ is (time)− 1. Also, it is assumed that the temperature of the surface Tw (x, t) varies in terms of x and time t. Let T ∞ be the temperature of the fluid far ! away from the sheet. Magnetic field having strength B = (0, B, 0) is supplied to the fluid flow. Magnetic Reynolds number is assumed to be small in comparison with applied magnetic field in order to avoid the induced magnetic field. So the Lorentz force after simplifiB0 cation can be rewritten as − σB2u where B ¼ pffiffiffiffiffiffiffiffiffi and B0 is the initial 1−λt

intensity of the magnetic field acting along the radial direction. Before continuing further we restrict ourselves in assuming that all body forces are neglected. Also the viscous and joule dissipation, Hall effects are neglected. Considering the above mentioned circumstances the governing equations unfolding the continuity, momentum and energy profile for

2.2. Relevant boundary conditions Now the appropriate boundary conditions for the present problem are as follows: 9 = ∂u ; v ¼ 0; T ¼ T w at r ¼ R ∂r ; as r→∞

u ¼ U w þ B1 ν f u→0; T→T ∞

ð7Þ

where B1 is the velocity slip, R is the radius of cylindrical sheet and the bx surface temperature Tw is of the form T w ðx; tÞ ¼ T ∞ þ ð1−λtÞ with b as constant. Table 1 Comparison of −θ′(0) for various values of Pr. Pr

Wang [18]

Khan and Pop [38]

Present work

0.2 0.7 2.0 7.0 20.0 70.0

0.1691 0.4539 0.9114 1.8954 3.3539 6.4622

0.1691 0.4539 0.9114 1.8954 3.3539 6.4622

0.16918852 0.45392176 0.91142210 1.89541203 3.35397421 6.46221004

420

N. Acharya et al. / Journal of Molecular Liquids 225 (2017) 418–425

Table 2 Values of Cfr for different values of pertinent parameters. β

A 1.00 2.00 3.00 4.00 1.00

ξ

0.25

0.50

0.00 0.25 0.50 0.75 0.25

0.00 0.50 1.00 1.50

Table 4 Values of correlation coefficient. r

Cfr M = 0.0

M = 2.0

‐0.454219 ‐0.577781 ‐0.627299 ‐0.674809 ‐0.526318 ‐0.544010 ‐0.559651 ‐0.577027 ‐1.633010 ‐0.856747 ‐0.590975 ‐0.453123

‐0.493118 ‐0.593295 ‐0.646389 ‐0.631042 ‐0.595202 ‐0.615754 ‐0.633270 ‐0.648564 ‐1.921296 ‐0.946037 ‐0.636319 ‐0.480876

A β ξ

Cfr M = 0.0

M = 2.0

Nur M = 0.0

M = 2.0

‐0.970804 ‐0.992654 0.963847

0.977623 ‐0.974811 0.903924

0.998514 ‐0.991723 ‐0.939974

0.997584 ‐0.984217 ‐0.951624

  4N ″ η

0 0 θ þ 2βθ0 þ Pr Q  f þ Qθ þ f θ0 −2 f θ−A 2θ þ θ0 ð1 þ 2ηβÞ 1 þ ¼0 3 2

ð10Þ

Here β ¼

2.3. Similarity renovation

qffiffiffiffiffiffi νf

aR2

is the curvature parameter, A ¼ λa is the unsteadiness 2

Now the necessary similarity transformations to convert the nonlinear partial differential Eqs. (2) and (6) into ordinary ones are as follows: ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi aν f r 2 −R2 a T−T ∞ ψ¼ xRf ðηÞ; η ¼ ; θðηÞ ¼ ν f ð1−λt Þ 1−λt 2R T w −T ∞

ð8Þ

3

T∞ parameter, M ¼ σBρa0 denotes the magnetic parameter, N ¼ 4σ κ f κ  repre-

sents the radiation parameter, Pr ¼ 

A ν f ðρC p Þ f

νf α

is the usual Prandtl number, Q  ¼



B and Q ¼ aν f ðρC are renamed as space and time dependent heat pÞ f

source parameters respectively. It is to be noted that β = 0.0 corresponds to the flat surface i.e. when we employ β = 0.0(i . e . R → ∞) we have flow features over a stretching flat surface as communicated by Ali [22]. 2.5. Changed boundary conditions

where f(η), θ(η) are the dimensionless functions with ψ as the free stream function and it satisfies the continuity Eq. (1) on behalf of its outand v ¼ − 1r ∂ψ . ward appearance as u ¼ 1r ∂ψ ∂r ∂x

Boundary condition as depicted in Eq. (7) acquires its shape in the following format 0

2.4. Changed profile of leading equations Introducing the similarity transformation as prescribed in Eq. (8) the governing Eqs. (2) and (6) captures its new changed dimensionless form as η ″

‴ ″ 02 0 0 ð1 þ 2ηβÞf ðηÞ þ f ð f þ 2βÞ−f −A f þ f −M 2 f ¼ 0 2

ð9Þ

where ξ ¼ B1

Table 3 Values of Nur for different values of pertinent parameters.

1.00 2.00 3.00 4.00 1.00

β 0.25

0.00 0.25 0.50 0.75 0.25

ξ 0.50

0.00 0.50 1.00 1.50 0.50

Q⁎ 0.30

0.00 1.00 2.00 3.00 0.30

Q 0.30

0.00 1.00 2.00 3.00

Nur M = 0.0

M = 2.0

2.989711 3.904637 4.651901 5.299441 2.066496 1.993282 1.929965 1.895658 1.489562 1.267900 1.157313 1.086237 1.951952 1.732083 1.512214 1.440114 1.755053 1.657135 1.552502 1.439397

2.670819 3.723217 4.529015 5.218078 1.103182 1.047325 1.001717 0.985594 1.418349 1.121012 1.016594 0.906591 1.746037 1.564897 1.316029 1.146337 1.672027 1.568926 1.457931 1.336787

 ð11Þ

pffiffiffiffiffiffiffiffi aν f is the velocity slip parameter.

2.6. Physical quantities of engineering interest The physical quantities which provide significant practical implication are the Skin friction coefficient and Nusselt number. They are defined as Cf ¼

A

″

f ð0Þ ¼ 1 þ ξf ð0Þ; f ð0Þ ¼ 0; θð0Þ ¼ 1 at η ¼ 0 0 f →0; θ→0 as η→∞

2τ w ρU w

; Nu ¼ 2

) xqw κ f ðT w −T ∞ Þ

ð12Þ

Þ ; q ¼ −κ f ð∂T Þ . where τ w ¼ μð∂u ∂r r¼R w ∂r r¼R Now with the help of similarity transformation one can achieve the reduced skin friction coefficient, reduced Nusselt number respectively as follows: ″

1

C fr ¼ Rex 2 C f ¼ f ð0Þ

ð13Þ

Nur ¼ Rex −2 Nu ¼ −θ0 ð0Þ

ð14Þ

1

w where Rex ¼ xU ν f is the local Reynolds number.

Table 5 Values of P.E(r). P . E(r) A β ξ

Cfr M = 0.0

M = 2.0

Nur M = 0.0

M = 2.0

0.019405241 0.004447839 0.021444786

0.013362542 0.015011247 0.055190524

0.000904258 0.004886531 0.023707541

0.001506344 0.009456632 0.029066411

N. Acharya et al. / Journal of Molecular Liquids 225 (2017) 418–425

421

and denoted by r. The process is of estimating covariance and variance on a sample. Suppose we have two dataset {x1,x2, ......,xk} and {y1,y2, ......,yk}

Table 6 r Values of P:EðrÞ . r P:EðrÞ

Cfr M = 0.0

A β ξ

49.847977 223.164703 44.944972

k

M = 2.0

Nur M = 0.0

M = 2.0

73.162700 64.939047 16.377818

1104.220256 198.875771 40.490151

662.201096 104.075461 32.739283

3. Numerical experiment 3.1. Numerical methodology Since the governing Eqs. (9) and (10) are nonlinear in nature, hence to solve them we approach towards the Runga-Kutta-Fehlberg method. In the numerical procedure scheme we choose MAPLE-17 software which satisfies our desired RK-4 methodology in conjunction with shooting criteria. The inner iteration is executed with convergence criteria of 10‐6 in all cases taking step size h = 0.01. 3.2. Code of authentication To arrest the precision of our present analysis we have computed the values of −θ′(0)for various values of Pr by letting the values of parameter as β = 0.0 , A = 0.0 , M = 0.0 , N = 0.0 , Q⁎ = 0.0 , Q = 0.0 , ξ = 0.0. We have enlisted those values in Table 1 and compared those values with Wang [18] and Khan and Pop [38]. It confirms our desired accuracy. Thus the code of verification is justified. 4. Statistical sketch out Now we are progressively heading towards the discussions of the various consequences of pertinent parameters on flow profile. But before proceeding further we have to understand the underlying relation between the physical parameters and to achieve the same we have listed the values of reduced skin friction coefficient, reduced Nusselt number in Table 2 and Table 3 respectively. After that we have estimated the values of the correlation coefficients between them and presented them in Table 4. (See Table 1.)

∑ðxi −xÞðyi −yÞ i¼1 ffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x; y then the formula stands as r ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k

∑ ðxi −xÞ2 i¼1

4

4

i¼1

i¼1

here x ¼ 2:5; y ¼ −0:9386 and ∑ ðxi −xÞ2 ¼ 5; ∑ ðyi −yÞ2 ¼ 0:5313 . Proceeding similarly and calculating the rest of the values as in the formula of r as mentioned above we get the values of r as ‐0.970804 in Table 4. Then we have estimated the value of Probable error in Table 5 as P .E(r) =0.019405241 adapting the procedure as mentioned in (15) where r = −0.970804 and n =4. After that we have calculated the r modulus of the values of P:EðrÞ in Table 6. One can easily check that numerical values are all greater than 6. Hence we can surely declare following Eq. (16) that all the correlation coefficients are significant and parameters are highly related to the physical characteristics.

5. Results and discussions This section provides the graphical and tabular outlook on the effect of various flow related parameters on the velocity and temperature profile. Here our investigation lies on the comparative study of flow characteristics considering the presence and absence of magnetic field. Moreover we also captured the effect of magnetic field on velocity and temperature profile by introducing a comparative discussion between flat surface and cylindrical surface. The whole demonstration has been achieved by taking the values of parameter as Pr = 6.2, M = 2.0, β = 0.25, A = 0.5, N = 0.5, Q⁎ = 0.3, Q = 0.3, ξ = 0.5 unless otherwise specified.

Probable error is one of the most classical and straightforward methodology to authenticate observed correlation coefficient. Fisher [39] provides the pioneer contribution to suggest it. The Probable error is defined by the following relation 

1−r 2 pffiffiffi n

 ð15Þ

where r is the correlation coefficient and n is the number of observation or number of data we have received in each case of calculating Cfr and Nur whatever the presence and absence of magnetic field it may be and the significance of the factor 0.6745 is that μ ± 0.6745σ covers 50% of total region in a normal distribution. Here μ is the mean and σ is the standard deviation of the sample. Now the rule by which the final judgment has to be made is that: rbP:Eðr Þ ⇒correlation is not significant rN6:P:Eðr Þ⇒correlation is significant

 ð16Þ

4.2. Statistical declaration about parameters In evaluating correlation coefficient we have made utilize of the Pearson's correlation coefficient which is mostly classified for a sample

i¼1

are the mean of the sample. For simplification let us introduce the way to calculate it. Now from Table 2 we have acquired the value of Cfr in absence of M for the parameter A. If we take the values of A as the first set of data i.e. {x1,x2, ......,xk} then the corresponding value of Cfr in the column 4th column (i.e. M = 0 column) will be treated as the 2nd dataset i.e. {y1,y2, ......,yk}. Clearly here we count four observations i.e. the value of k is 4. Now clearly

4.1. On probable error

P:Eðr Þ ¼ 0:6745:

k

∑ ðyi −yÞ2

Fig. 2. Effect of A on velocity.

422

N. Acharya et al. / Journal of Molecular Liquids 225 (2017) 418–425

Enhancement of A allows transferring less heat from the sheet. An important observation is that the rate of cooling is much faster for higher values of unsteadiness parameter, whereas it may take longer time for cooling during steady flows. Comparatively presence of magnetic field aids the fluid to gain the temperature high inside the boundary layer region. Since the magnetic field confirms the presence of Lorentz force, hence this resistive force generates some frictional heat at the time of interaction of the fluid with the cylindrical surface. 5.2. Effect of curvature parameterβ

Fig. 3. Effect of A on temperature.

5.1. Effect of unsteadiness parameter A

The impact of curvature parameter β on velocity has been demonstrated in Fig. 4. Dual characteristic for velocity has been captured. It is noticed that within the range 0.0≤ η ≤ 1.0 (not accurately determined) velocity reduces but the totally reverse effect is perceived for η ≥ 1.0. The parametric definition of β ensures us that the kinematic viscosity νf increases when β amplifies. Since νf is known to be the resistance against the fluid's velocity, hence initially velocity reduces. But to keep the mass flow rate conservative a decrease in the fluid velocity will be compensated by the increasing fluid velocity. That's why at η = 1.2 (not accurately determined) we found a point of separation and backflow takes place. Fluid acquires high velocity in absence of M owing to same logic in section 5.1. Moreover temperature is also significantly enhanced inside the boundary layer region. Enhancement is seemed to be more pronounced in presence of magnetic field. This occurrence has been characterized in Fig. 5. Physics behind this occurrence informs us that, as νf increases due to β, so the resistance between fluid and surface will generate some frictional heat and this will be more effective in presence of magnetic field.

Fig. 2 reflects on the fact that how unsteadiness parameter behaves with fluid velocity in presence and absence of magnetic field. Significant reduction in the fluid velocity is perceived. Clear and distinct effect is noticed near the surface. Within the region 0.5 ≤ η ≤ 6.0 (not precisely determined) the effect is more prominent and after that it satisfies far field boundary condition asymptotically. Absence of the parameter M uplifts the velocity because M corresponds to resistive Lorentz strength. Temperature profile due to the unsteadiness parameter has been sketched out in Fig. 3. Like velocity profile, temperature of the fluid drops off. Distinct and clear consequences have been traced out inside the region 0.3≤η ≤ 1.7. Simultaneously thermal boundary layer reduces.

The effect of ξ on f′(η) has been sketched in Fig. 6. Graphical view confirms the reduction of fluid velocity inside the boundary layer region both in presence and absence of magnetic field. Thus the thickness of velocity boundary layer drops off. This reduction outlook has been found more effective and significantly distinct within 0.0 ≤ η ≤ 2.5. The basis of this trend is same as we have discussed previously in Section 5.2.

Fig. 4. Effect of β on velocity.

Fig. 5. Effect of β on temperature.

5.3. Effect of slip parameterξ

N. Acharya et al. / Journal of Molecular Liquids 225 (2017) 418–425

423

Fig. 6. Effect of ξ on velocity. Fig. 8. Effect of Q⁎ on temperature.

Fig. 7 depicts that the temperature of the fluid rises with the increase of the parameterξ. Effect has been traced out considerable enhancement within 0.5 ≤ η ≤ 6.0. After that such effect is smeared out and the curve converges asymptotically. This occurred because of same reason as we mentioned in Section 5.2, temperature is high when we consider the presence of magnetic field.

both cases. Because the existence of heat source parameters within the flow province generate more heat. Generally presence of magnetic field enriches the heat transport mechanism significantly in both cases as compared to absence of magnetic field.

5.5. Effect of magnetic field parameter M 5.4. Effect of non-uniform heat source parameter Figs.s 8 and 9 enlighten on the ramification of space dependent heat source parameter (Q⁎) and time dependent heat source parameter (Q) on temperature. Temperature is found to be increasing in nature for

It is to be noted that β = 0.0 corresponds to the flat surface. We are interested to watch out the differences of velocity and temperature profile between flat surface and cylindrical surface in presence of magnetic field. Those differences have been pictured in Figs.s 10 and 11. It is

Fig. 7. Effect of ξ on temperature.

Fig. 9. Effect of Q on temperature.

424

N. Acharya et al. / Journal of Molecular Liquids 225 (2017) 418–425 Table 7 Values of Cfr and Nur for different values of M. M 0.00 1.50 3.00 4.50

Fig. 10. Effect of M on velocity.

observed from Fig. 10 that presence of magnetic field behaves as a retarding factor for velocity of the fluid. The effect is very prominent near the surface. It is interesting to note that velocity for flat surface lies in higher position as compared to cylindrical surface inside 0.0 ≤ η ≤ 1.7 (not accurately determined). But more prominent reverse effect snatches our attention within 1.7 ≤ η ≤ 4.2. Fig. 11 conveys that temperature amplifies in both cases following the same fashion. But comparatively temperature for flat surface is on the lower side. 5.6. Tabular profile of physical quantities It is observed from Table 2 that for both presence and absence of magnetic field the reduced skin friction coefficient reduces for the parameters A and β, but opposite situation comes into view for ξ. The

Cfr β = 0.25

β = 0.0

Nur β = 0.25

β = 0.0

‐0.508913 ‐0.535634 ‐0.591956 ‐0.648385

‐0.484952 ‐0.513023 ‐0.573044 ‐0.633643

1.986917 1.752843 1.564985 1.477323

2.044462 1.793475 1.588131 1.493249

rate of reduction has been found 14.47% and 3.12% for A and β respectively in absence of magnetic field. It has been tabulated 10.54% and 2.90% in presence of magnetic field. Hence presence of M behaves as a delaying factor for Cfr. For ξ the percentage figure in terms of enhancing skin friction stands approximately 33.96% and 35.97% for absence and presence of magnetic field respectively. Table 3 performs as evidence that reduced Nusselt number reduces for the parameters β , ξ , Q , Q⁎. If we consider absence of magnetic field then reduction phenomenon has been traced out as 2.83%, 9.91%, 9.57%, 6.39% for β , ξ , Q , Q⁎ respectively. Consequently the numerical values convey 3.67%, 13.70%, 13.05% and 7.18% in case of presence of magnetic field. Thus the rate of reduction amplifies due to presence of magnetic field. But totally opposite behaviour may be authenticated for A from Table 3. Comparatively for the unsteadiness parameter A reduced Nusselt number increases approximately 4.2% in presence of magnetic field. Additionally Table 7 exhibits that presence of magnetic field aids the Nusselt number and skin friction to reduce for both flat surface and cylindrical surface. The reduction is slightly high near about 3.745% for cylindrical surface as compared to flat surface. But for Nusselt number comparatively 1.8% reduction is noticed for flat surface. 6. Final remarks In this article we have captured the MHD boundary flow analysis past an unsteady stretching cylinder considering the presence of nonuniform heat source. The nonlinear momentum and energy equations have been modified into its dimensionless forms by means of similarity transformation and then solved numerically. Ramification of various pertinent parameters on the flow system has been demonstrated by means of graphs, tables together with a statistical approach. Based on the whole study some major conclusions may be pointed out as follows: ➢ Velocity of the fluid reduces due to A , ξ, but a dual characteristic of velocity has been noticed for β within a certain province of the flow system. ➢ Temperature profile has been found to boost due to the positive impact of ξ ,Q ,Q⁎ where as opposite situation is perceived for A. ➢ Comparative study between flat surface and cylindrical surface in presence of magnetic field reveals that temperature for flat surface trims down. ➢ The truth for physical parameters of being highly correlated has been established via statistical approaches. ➢ Significant enhancement in reduced skin friction coefficient has been encountered due to the positive improvisation of ξ. Acknowledgement The authors wish to express their cordial thanks to reviewers for valuable suggestions and comments to improve the presentation of this article. References

Fig. 11. Effect of M on temperature.

[1] L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970) 645–670. [2] P.S. Gupta, A.S. Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (1977) 744–760.

N. Acharya et al. / Journal of Molecular Liquids 225 (2017) 418–425 [3] B.K. Datta, P. Roy, A.S. Gupta, Temperature field in the flow over a stretching sheet with uniform heat flux, Int Commun Heat Mass Transfer. 12 (1985) 89–94. [4] C.K. Chen, M.I. Char, Heat transfer of a continuous stretching surface with suction or blowing, J. Math. Anal. Appl. 135 (1988) 568–580. [5] M.S. Abel, N. Mahesha, Heat transfer in viscoelastic fluid over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Appl. Math. Model. 32 (10) (2008) 1965–1983. [6] A. Ishak, R. Nazar, I. Pop, Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder, Energy Convers. Manag. 49 (2008) 3265–3269. [7] T. Hayat, M. Qasim, S. Mesloub, MHD flow and heat transfer over permeable stretching sheet with slip conditions, Int. J. Numer. Methods Fluids 66 (8) (2011) 963–975. [8] I.C. Mandal, S. Mukhopadhya, Heat transfer analysis for fluid flow over an exponentially stretching porous sheet with surface heat flux in porous medium, Ain Shams Engineering Journal. 4 (1) (2013) 103–110. [9] R. Cortell, Fluid flow and radiative nonlinear heat transfer over a stretching sheet, Journal of King Saud University-Science. 26 (2) (2014) 161–167. [10] F. Mabood, W. A. Khan, A.I.Md. Ismail, MHD flow over exponentially radiating stretching sheet using homotopy analysis method, Journal of King Saud UniversityEng. Sci. doi:10.1016/j.jksues.2014.06.001. [11] C.S.K. Raju, N. Sandeep, C. Sulochana, V. Sugunamma, M.J. Babu, Radiation, inclined magnetic field and cross-diffusion effects on flow over a stretching surface, Journal of the Nigerian Mathematical Society. 34 (2) (2015) 169–180. [12] T. Hayat, T. Muhamad, S.A. Shehzad, A. Alsaedi, Soret and dufour effects in three dimensional flow over an exponentially stretching surface with porous medium, chemical reaction and heat source/sink, International Journal of Numerical Methods For Heat and Fluid Flow 25 (2015) 724–748. [13] T. Hayat, M.I. Khan, M. Farooq, N. Gull, A. Alsaedi, Unsteady three dimensional mixed convection flow with variable viscosity and thermal conductivity, J. Mol. Liq. 223 (2016) 1297–1310. [14] T. Hayat, M.I. Khan, M. Farooq, T. Yasmeen, A. Alsaedi, Stagnation point flow with cattaneo-christov heat flux and homogeneous heterogeneous reactions, J. Mol. Liq. 220 (2016) 49–55. [15] T. Hayat, M.I. Khan, M. Farooq, A. Alsaedi, M. Waqas, T. Yasmeen, Impact of cattaneochristov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface, Int. J. Heat Mass Transf. 99 (2016) 702–710. [16] E.M.E. Elbarbary, N.S. Elgazery, Flow and heat transfer of a micropolar fluid in an axisymmetric stagnation flow on a cylinder with variable properties and suction (numerical study), Acta Mech. 176 (2005) 213–229. [17] P. Datta, D. Anilkumar, S. Roy, N.C. Mahanti, Effect of nonuniform slot injection (suction) on a forced flow over a slender cylinder, Int. J. Heat Mass Transf. 49 (2006) 2366–2371. [18] C.Y. Wang, Fluid flow due to a stretching cylinder, Phys. Fluids 31 (1988) 466–468. [19] A. Ishak, R. Nazar, I. Pop, Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder, Energy Convers. Manag. 49 (2008) 3265–3269. [20] A. Ishak, R. Nazar, Laminar boundary layer flow along a stretching cylinder, Eur. J. Sci. Res. 36 (1) (2009) 22–29. [21] L.G. Grubka, K.M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, ASME J Heat Transfer. 107 (1985) 248–250.

425

[22] M.E. Ali, Heat transfer characteristics of a continuous stretching surface, Heat Mass Transf. 29 (1994) 227–234. [23] N. Bachok, A. Ishak, Flow and heat transfer over a stretching cylinder with prescribed surface heat flux Malaysian J, Math. Sci. 4 (2010) 159–169. [24] W.K. Hussam, M.C. Thompson, G.J. Sheard, Dynamics and heat transfer in a quasi two dimensional MHD flow past a circular cylinder in a duct at high Hartmann number, Int. J. Heat Mass Transf. 54 (2011) 1091–1100. [25] T. Hayat, M. Farooq, A. Alsaedi, Inclined magnetic field effect in stratified stagnation point flow over an inclined cylinder, Zeitschrift Fur Naturforschung 70 (2015) 317–324. [26] S. Mukhopadhyay, Mixed convection boundary layer flow along a stretching cylinder in porous medium, J. Pet. Sci. Eng. 96 (2012) 73–78. [27] K. Vajravelu, K.V. Prasad, S.R. Santhi, Axisymmetric magnetohydrodynamic flow and heat transfer at a non-isothermal stretching cylinder, Appl. Math. Comput. 219 (2012) 3393–4005. [28] S. Munawar, A. Mehmood, A. Ali, Time dependent flow and heat transfer over stretching cylinder, Chin. J. Phys. 50 (5) (2012) 828–848. [29] K. Vajravelu, K.V. Prasad, S.R. Santhi, V. Umesh, Fluid flow and heat transfer over a permeable stretching cylinder, J. Appl. Fluid Mech. 7 (1) (2014) 111–120. [30] C.L.M.H. Navier, Memoire sur les lois du mouvement des fluids, Mem. Acad. R. Sci. Inst. Fr. 6 (1823) 389–440. [31] S. Mukhopadhyay, MHD boundary layer slip flow along a stretching cylinder, Ain Shams Eng. J. 4 (2013) 317–324. [32] S. Mukhopadhyay, Chemically reactive solute transfer in a boundary layer slip flow along a stretching cylinder, Font Chem. Sci. Eng. 5 (3) (2011) 385–391. [33] P. B. A. Reddy, R. Das, Estimation of MHD boundary layer slip flow over a permeable stretching cylinder in the presence of chemical reaction through numerical and artificial neural network modelling, Engineering Science and Technology, an International Journal. doi:10.1016/j.jestch.2015.12.013. [34] E.M. Abo-Eldahab, M.A. El Aziz, Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation/absorption, Int. J. Thermal. Sci. 43 (7) (2004) 709–719. [35] A.M. Subhas, N. Mahesha, Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Appl. Math. Model. 32 (2008) 1965–1983. [36] M.S. Abel, J. Tawade, M.M. Nandeppanavarb, Effect of nonuniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet, Int. J. Nonlinear Mechanics. 44 (2009) 990–998. [37] F. Mabood, S.M. Ibrahim, M.M. Rashidi, M.S. Shadloo, G. Lorenzini, Non-uniform heat source/sink and soret effects on MHD non-Darcian convective flow past a stretching sheet in a micropolar fluid with radiation, Int. J. Heat Mass Transf. 93 (2016) 674–682. [38] W.A. Khan, I. Pop, Boundary-layer flow of nanofluid past a stretching sheet, Int. J. Heat Mass Transf. 53 (2010) 2477–2483. [39] R.A. Fisher, On the “probable error” of a coefficient of correlation deduced from a small sample, Metron 1 (1921) 03–32.