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absorption and convective condition
Alexandria Engineering Journal (2018) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/ absorption and convective condition G.K. Ramesh a, K. Ganesh Kumar b, B.J. Gireesha b, S.A. Shehzad c, F.M. Abbasi d a
Department of Mathematics, K.L.E’ S J.T. College, Gadag 82101, Karnataka, India Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta-577 451, Shimoga, Karnataka, India c Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan d Department of Mathematics, COMSATS University Islamabad Islamabad 44000, Pakistan b
Received 21 September 2017; revised 22 November 2017; accepted 2 December 2017
KEYWORDS Unsteadiness; MHD; Contracting cylinder; Nanoliquid; Uniform heat generation; Numerical solutions
Abstract In this article, the flow of an incompressible nanoliquid induced due to unsteady contracting cylinder is investigated. Analysis of heat transport phenomenon is reported under uniform heat absorption/generation and convective type boundary condition. A set of transformations is implemented to reduce PDEs to ODEs. The obtained coupled nonlinear equations are treated numerically through the help of RKF 45 technique. Physical evaluation of convective condition and heat absorption/generation on thermal analysis is discussed through graphs. Output shows that both the parameters of convective condition and heat absorption/generation give an increment in liquid temperature and its related thickness of boundary layer. Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Nanomaterials analysis can be recognized to the improvement of inventive technologies for production of nanometer size materials. This improvement has provided influential tools for the engineers and scientists to generate materials having extraordinary properties, larger than their complements. Flow of nanoliquid and energy transport towards stretched/shrinked E-mail addresses: [email protected] (G.K. Ramesh), ganikganesh @gmail.com (K. Ganesh Kumar), [email protected] (B.J. Gireesha), [email protected] (S.A. Shehzad), abbasisarkar @gmail.com (F.M. Abbasi) Peer review under responsibility of Faculty of Engineering, Alexandria University.
sheet has engrossed the attention of several scientists due to their numerous demands in manufacturing processes of industry like transportation, nuclear reactors, power generation, biomedicine, industrial and food applications. To the best of our knowledge Khan and Pop [1] firstly reported the flow characteristics of viscous nanomaterial due to linear stretching sheet. Nield and Kuznetsov [2] address the Cheng-Minkowycz problem on a flow of nanoliquid in porous medium. Makinde et al. [3] exhibits buoyancy effect in convectively heated stretched flow of viscous nanoliquid. Difference of single phase and two phase model in a wavy channel is addressed by Rashidi et al. [4]. Sheikholeslami et al. [5] computed the numerical solution of MHD nanoliquid under the influence of viscous dissipation. Ramesh [6] discussed the heat source/ sink phenomenon in a Jeffrey nanoliquid. Some recent works
https://doi.org/10.1016/j.aej.2017.12.009 1110-0168 Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
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Nomenclature aðtÞ a0 B Bi Cf C Cw C1 DB DT f Le k M Nux Nb Nt p Pr qw qm Q
radius of the cylinder positive constant time dependent uniform magnetic field biot number skin-friction coefficient concentration nanoparticle volume fraction at the wall nanoparticle volume fraction far from surface Brownian diffusion co-efficient thermophoetic diffusion co-efficient dimensionless stream function Lewis number thermal conductivity magnetic parameter local Nusselt number Brownian motion parameter thermophoresis parameter pressure Prandtl number heat flux at the wall mass flux at the wall uniform source/sink parameter
on two/three dimensional flow of nanoliquid can be seen [7–31]. All the above works analyses the flow and heat transport behavior on steady case only. Most of the engineering problems time dependent is important factors (see [32–36]). Flow due to stretching cylinder with time dependent was observed by Fang et al. [37]. Munawar et al. [38] address the unsteady behavior of vacillate stretching cylinder. Malvandi [39] considered nano-particles to report the time dependent factor in rotating sphere. Zaimi et al. [40] use the Buongiorno’s model to analysis the nanofluid flow in a contracting cylinder. Rohni et al. [41] explored the impact of suction on two dimensional nanoliquid flow of shrinking surface. Marinca and Ene [42] obtained the dual solutions of [38] by applying OHA method. Abbas et al. [43] utilized the slip condition for the analysis of unsteady stretching/shrinking cylinder. The idea of this article is to explore the behavior of uniform heat absorption/generation and various aspects of convective condition in magnetohydrodynamic nanoliquid past a contracting cylinder. Here the effect of heat source/sink acts as vital part in controlling the temperature and also technique of convective transport of heat has tremendous significance in the problems where relatively higher temperature is occurred. Some attempts have been done by the researchers [44–48] on heat source/sink and convective condition. Based on the above literatures no endeavor has been made on MHD nanoliquid over a contracting cylinder under the impact of heat source/sink and convective boundary condition. Numerical calculation is made to get the solution of coupled non-linear expressions. The outcome of dimensionless parameters has been visualized for different values of physical constraints.
S Shx t T Tw T1 u; w v
unsteadiness parameter local Sherwood number time temperature temperature at the wall temperature far from the wall velocity components along the r and z axes velocity vector
Greek symbols a thermal diffusivity b contraction/expansion strength g similarity variable h dimensionless temperature l viscosity q fluid density ðqcÞf heat capacity of the fluid ðqcÞ heat capacity of nanoparticle material / dimensionless nanoparticle volume fraction surface shear stress sw
2. Problem developments Here we elaborate unsteady laminar flow of nanoliquid past a cylinder in a contracting motion. The z-axis is selected along the axis of the cylinder and r -axis is perpendicular to it (see Fig. 1). We considered the diameter of the cylinder with pffiffiffiffiffiffiffiffiffiffiffiffiffi unsteady radius aðtÞ ¼ a0 1 bt; here b denote contraction strength, a0 > 0 and t be the time. We also deliberate about heat source/sink and convective condition. Flow analysis also consists of Brownian motion and thermophoresis. With the above assumptions equations governing the flow analysis are [22] 1 @ @w ðruÞ þ ¼ 0; r @r @z
Fig. 1
ð1Þ
Physical model and coordinate system.
Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder 2 @u @u @u 1 @p @ u 1 @u @ 2 u u rB2 ðtÞ þu þw ¼ þt þ u; þ 2 2 2 @t @r @z q @r @r r @r @z r q
3
2 d2 / d/ d/ dh Nt d h dh þ Le f Sg þ g 2þ ¼ 0: g 2þ dg dg dg dg Nb dg dg
ð2Þ 2 @w @w @w 1 @p @ w 1 @w @ 2 w rB2 ðtÞ þ þu þw ¼ þt þ w; 2 2 @t @r @z q @z @r r @r @z q ð3Þ @T @T @T @ 2 T 1 @T @ 2 T Þþ þu þw ¼ að 2 þ þ @t @r @z @r r @r @z2 " " ## 2 2 @C @T @C @T DT @T @T q þ 0 ðT T1 Þ; þ s DB þ þ qcp @r @r @z @z T1 @r @z ð4Þ 2 @C @C @C @ C 1 @C @ 2 C þu þw ¼ DB þ þ þ @t @r @z @r2 r @r @z2 2 2 DT @ T 1 @T @ T : þ þ T1 @r2 r @r @z2
ð5Þ
The boundary conditions for this analysis can be described as t < 0 : u ¼ w ¼ 0; T ¼ T1 ; C ¼ C1 ; t P 0 : u ¼ 0; w ¼
and the boundary condition (6) becomes fðgÞ ¼ 0;
df dh ¼ 1; ¼ Bið1 hðgÞÞ; /ðgÞ ¼ 1; at g ¼ 1 dg dg
df ! 0; hðgÞ ! 0; /ðgÞ ! 0 as g ! 1; dg
ð11Þ
The key parameters of the problem are Pr for Prandtl number, Le for Lewis number, S for unsteadiness parameter, Nb for Brownian motion parameter and Nt for thermophoresis parameter, M for magnetic parameter, Q for uniform source/ sink parameter, Bi for Biot number, which are defined as t t a2 b sDB ðCW C1 Þ Pr ¼ ; Le ¼ ; ; S ¼ 0 ; Nb ¼ a DB 4t t sDT ðTf T1 Þ Nt ¼ ; tT1 Q¼
q0 rB2 h ; M ¼ 2 0 ; Bi ¼ : cp q a0 q k
The term pressure can be obtained from Eq. (2) as Z P @u u 1 @u ¼ constant þ t þ u2 þ dr: q @r r 2 @t
1 4tz @T ; k ¼ hðTf Tw Þ; a2o 1 bt @r
C ¼ CW at r ¼ aðtÞ; w ! 0; T ! T1 ; C ! C1 as r ! 1;
ð10Þ
ð6Þ
here u; w; T and C represents the velocity, temperature and concentration components respectively, p the pressure, t the kinematic viscosity, q the liquid density, a the thermal diffusivity, s the ratio between effective heat capacity of nanoparticle materials and liquid heat capacity, cp the specific heat, B2 ðtÞ ¼ 4B2o tð1 btÞ1 the time dependent uniform magnetic field, DB the coefficient of Brownian diffusivity, h the coefficient of heat transport, DT the coefficient of thermophoretic diffusivity, Tw ¼ T1 þ bz½a0 tð1 btÞ1 the surface temperature, CW the constant surface nanoparticle volume fraction and T1 and C1 the constant temperature and nanoparticle volume fraction far from the surface of the cylinder respectively. Independent variables are defined as [37]: 9 0 ffiffiffiffiffiffiffi fðgÞ pffiffi ; w ¼ 12 4tz f ðgÞ; > u ¼ a12 p2t = ao 1bt o 1bt g ð7Þ 2 > ðTT1 Þ ðCC1 Þ r 1 ; ; / ¼ ; g ¼ : h ¼ ðT a0 1bt ðCw C1 Þ f T1 Þ Applying Eq. (7) into (1)–(6), noted that Eq. (1) is satisfied automatically and due to absence of longitudinal pressure gradient, Eqs. (2)–(5) reduces the following form 2 2 d3 f d2 f df df d f df df g 3þ 2þf S g 2þ M ¼ 0; dg dg dg dg dg dg dg ð8Þ 2 2 ! 1 d h dh dh dh d/ dh dh g 2þ þ f Sg þ g Nt þ Nb þ Qh ¼ 0; Pr dg dg dg dg dg dg dg ð9Þ
The coefficient of skin-friction Cf , the Nusselt number Nux and Sherwood number Shx can be written as Cf ¼
sw a0 ð1 btÞ1=2 qw a0 ð1 btÞ1=2 qm ; Nu ; Sh ; ¼ ¼ x x qw2w =2 2kðTf T1 Þ 2DB ðCW C1 Þ
where sw the surface shear stress, qw the surface heat flux and qm the surface mass flux. These quantities can be expressed as @w 1 8tlz ¼ f00 ð1Þ sw ¼ l @r r¼aðtÞ a3o ð1 btÞ3=2 , qw ¼ k
@T 2kðTW T1 Þ ¼ ½h0 ð1Þ; @r r¼aðtÞ a0 ð1 btÞ1=2
qm ¼ DB
@C 2DB ðCW C1 Þ ¼ ½/0 ð1Þ: @r r¼aðtÞ a0 ð1 btÞ1=2
Using similarity variables, we have Cf z=aðtÞ ¼ f00 ð1Þ; Nux ¼ h0 ð1Þ; Shx ¼ /0 ð1Þ: 3. Method of solutions Numerical calculations are carried out using Runge-KuttaFehlberg fourth-fifth order procedure by shooting technique to compute Eqs. (8)–(11). The equations f are in third order and h; / are in second order. These equations are reduced into seven simultaneous first order equations including fourteen unknowns. Shooting technique is applied to obtain the missing condition. Integration is performed via RKF-45 technique by considering fix values of g1 . The more accurate solutions
Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
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G.K. Ramesh et al.
can be achieved by using the proper step size h. Two various deformations of solutions are developed and examined at each step. Such deformations are only acceptable if both are close otherwise step size is reduced. Here we fixed step size dn ¼ 0:001; g1 ¼ 4; 6 and accuracy to fifth decimal place. The present work is good agreement with the existing work of Zaimi et al. [22], the comparison graph is displayed in Fig. 11. The flow chart of the present numerical procedure as follows 1. The boundary value problem is first converted into an initial value problem. 2. Initial value problem is solved by appropriately guessing the missing initial value using the shooting method for several sets of parameters. 3. The step size is h = 0.1 used for the computational purpose. 4. The solution process is repeated with another larger value of g1 until two successive values differ only after desired significant digit. 5. The last value g1 is taken as the finite value of the limit g1 for the particular set of physical parameters for determining velocity, temperature and concentration. 6. After getting all the initial conditions we solve this system of simultaneous equations using fourth order Runge–Kutta integration scheme.
Fig. 3
Temperature behavior on S.
4. Results and discussion Characteristics of Biot number, uniform heat absorption and generation in MHD nanoliquid flow through cylinder is analyses in this study. Figs. 2–4 are presented to describe the behavior of unsteadiness on velocity, temperature and concentration profiles. Fig. 2 shows the variation of f0 in response to modify in unsteadiness parameter S. Curves of f0 increases with an enchantment of S. But the opposite tend in nature can found for h and / which is displayed in Figs. 3 and 4. Physically it can analyze that S is directly proportional to a0 and S ¼ 0 represents the steady case. Important observation noted that the rate of cooling is much faster for negative values of unsteadiness parameter, whereas it may take longer time in steady flows.
Fig. 4
Fig. 5
Fig. 2
Velocity behavior on S.
Concentration behavior on S.
Velocity behavior on M.
Velocity field f0 for magnetic parameter M is sketched in Fig. 5. Higher values of M decay the velocity and its related momentum layer thickness. The physical impact is that ability
Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder
Fig. 6
Temperature behavior on Q.
Fig. 7
Temperature behavior on Bi.
5
Fig. 9 Variation of local Nusselt and Sherwood number with Nt for distinct values of Nb.
Fig. 10 Variation of local Nusselt and Sherwood number with Le for distinct values of Nb.
Fig. 8 M.
Variation of friction factor with S for distinct values of Fig. 11
Validation of the present work.
Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
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G.K. Ramesh et al. Table 1
Numerical values of f00 ð1Þ, /0 ð1Þ and h0 ð1Þ for different physical parameter.
Bi
Le
M
Nb
Nt
Pr
Q
S
f00 ð1Þ
/0 ð1Þ
h0 ð1Þ
0.5 0.6 0.7 0.5
1
4
0.5
0.1
6.8
0.5
0.9
1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.6466 1.8954 2.1189 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.9445 1.9194 1.8954
0.3074 0.3167 0.3260 0.3074 0.2519 0.2069 0.2923 0.3074 0.3191 0.3074 0.2070 0.1483 0.3074 0.6064 0.9551 0.3638 0.3388 0.3198 0.3074 0.2626 0.2244 0.1680 0.2385 0.3074
0.2568 0.3085 0.3602 0.2568 0.2553 0.2541 0.2308 0.2568 0.2774 0.2568 0.1288 0.0371 0.2568 0.2630 0.2862 0.4789 0.3721 0.2999 0.2568 0.1444 0.0573 -0.0713 0.0963 0.2568
0.5
1 2 3 1
0.5
1
3 4 5 4
0.5
1
4
0.5 0.6 0.7 0.5
0.5
1
4
0.5
0.1 0.2 0.3 0.1
0.5
1
4
0.5
0.1
4 5 6 6.8
0.5
1
4
0.5
0.1
6.8
of the force magnet which is normal to electrical-conducting fluid gives the drag force named as Lorentz force which acts opposite direction to the flow. Fig. 6 is elaborated to see the influence of Q on temperature field h. It reveals that presence of heat source (positive values) in the thermal layer gives the energy this causes the enhancement in the temperature and its related boundary layer thickness. As noted that negative values of Q decay the temperature filed and Q ¼ 0 correspond to the absence of heat source/sink. Fig. 7 elucidate the effect of Biot number Bi on h. This Fig. gives the clear insight that for large values of Q raise the temperature rapidly near the boundary. The central reason is convective heat exchange at the surface plate leading to an increase in thermal boundary layer thickness. The parameter Bi at any location x is directly proportional to the heat transfer coefficient associated with the hot fluid hf . Thus as Bi increases, the hot fluid side convection resistance decreases. It is also observed that in the absence of M; Q and Bi our Eqs. (8)–(11) reduces the problem of Zaimi et al. [22] (see Fig. 11). Figs. 8–10 are displayed to study the engineering interest of friction factor f00 ð1Þ, local Nusselt number h0 ð1Þ and Sherwood number /0 ð1Þ on certain physical parameters. Fig. 8 is demonstrated unsteadiness S and magnetic parameter M on f00 ð1Þ. As increasing values of S and M gives the increase in f00 ð1Þ profile. Figs. 9 and 10 provide the analysis of Nb and Nt; Le versus h0 ð1Þ and /0 ð1Þ. It is analyzed from these figures that temperature and thermal layer are enhanced for larger Nb but opposite trend for concentration. Physically higher values of Nb relate to much random motion of the particles which results in an enhancement of temperature and reduction on concentration. The Brownian motion of nanoparticles can enhance thermal conduction through one of two systems-either a direct effect owing to nanoparticles
0.5 0.6 0.7 0.5
0.6 0.7 0.8
that transport heat or alternatively through an indirect contribution due to micro-convection of fluid surrounding individual nanoparticles. For small particles, Brownian motion is strong, and the parameter Nb will have high values, the converse is the case for large particles. 5. Conclusions We examined the behavior of MHD, heat absorption/generation and convective type boundary condition on time dependent flow of Buongiorno’s model nanoliquid over a contracting cylinder. The new set of similarity variables is applied to get the ordinary non-linear differential equations and RKF45 method is adopted to get the solution of the mathematical problem. It is noted from Table 1 that f00 ð1Þ is higher for M and lower for S and there is no variations in Q; Nb; Nt; Le and Bi which can observe from Eqs. (8)– (10). Here h0 ð1Þ is a higher value for Bi; M; Nt; S and lower values for Le; Nb; Pr; Q. The same behavior can also found for /0 ð1Þ. Significant observation is found that cooling rate is much higher for negative values of unsteadiness parameter and heat sink parameter. Acknowledgments The authors are very much thankful to the editor and reviewers for their encouraging comments and suggestions to improve the presentation of this manuscript. References [1] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483.
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8 [41] A.M. Rohni, S. Ahmad, A.I.Md Ismail, I. Pop, Flow and heat transfer over an unsteady shrinking sheet with suction in a nanofluid using Buongiorno’s model, Int. Commun. Heat Mass Transfer 43 (2013) 75–80. [42] V. Marinca, R.D. Ene, Dual approximate solutions of the unsteady viscous flow over a shrinking cylinder with optimal homotopy asymptotic method, Adv. Math. Phys. 2014 (2014) 417643. [43] Z. Abbas, S. Rasool, M.M. Rashidi, Heat transfer analysis due to an unsteady stretching/shrinking cylinder with partial slip condition and suction, Ain Shams Eng. J. 6 (2015) 939–945. [44] G.K. Ramesh, B.J. Gireesha, R.S.R. Gorla, Boundary layer flow past a stretching sheet with fluid-particle suspension and convective boundary condition, Heat Mass Transfer 51 (2015) 1061–1066. [45] T. Hayat, S.A. Shehzad, A. Alsaedi, MHD three-dimensional flow of Maxwell fluid with variable thermal conductivity and
G.K. Ramesh et al. heat source/sink, Int. J. Numer. Methods Heat Fluid Flow 24 (2014) 1073–1085. [46] T. Hayat, T. Muhammad, 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, Int. J. Numer. Methods Heat Fluid Flow 25 (2015) 762–781. [47] A. Malvandi, S.A. Moshizi, D.D. Ganji, Nanofluids flow in microchannels in presence of heat source/sink and asymmetric heating, J. Thermophys. Heat Transfer 30 (2016) 111–119. [48] T. Hayat, M. Waqas, S.A. Shehzad, A. Alsaedi, Mixed convection flow of viscoelastic nanofluid by a cylinder with variable thermal conductivity and heat source/sink, Int. J. Numer. Methods Heat Fluid Flow 26 (2016) 214–234.
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H O S T E D BY
Alexandria University
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ORIGINAL ARTICLE
Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/ absorption and convective condition G.K. Ramesh a, K. Ganesh Kumar b, B.J. Gireesha b, S.A. Shehzad c, F.M. Abbasi d a
Department of Mathematics, K.L.E’ S J.T. College, Gadag 82101, Karnataka, India Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta-577 451, Shimoga, Karnataka, India c Department of Mathematics, COMSATS University Islamabad, Sahiwal 57000, Pakistan d Department of Mathematics, COMSATS University Islamabad Islamabad 44000, Pakistan b
Received 21 September 2017; revised 22 November 2017; accepted 2 December 2017
KEYWORDS Unsteadiness; MHD; Contracting cylinder; Nanoliquid; Uniform heat generation; Numerical solutions
Abstract In this article, the flow of an incompressible nanoliquid induced due to unsteady contracting cylinder is investigated. Analysis of heat transport phenomenon is reported under uniform heat absorption/generation and convective type boundary condition. A set of transformations is implemented to reduce PDEs to ODEs. The obtained coupled nonlinear equations are treated numerically through the help of RKF 45 technique. Physical evaluation of convective condition and heat absorption/generation on thermal analysis is discussed through graphs. Output shows that both the parameters of convective condition and heat absorption/generation give an increment in liquid temperature and its related thickness of boundary layer. Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Nanomaterials analysis can be recognized to the improvement of inventive technologies for production of nanometer size materials. This improvement has provided influential tools for the engineers and scientists to generate materials having extraordinary properties, larger than their complements. Flow of nanoliquid and energy transport towards stretched/shrinked E-mail addresses: [email protected] (G.K. Ramesh), ganikganesh @gmail.com (K. Ganesh Kumar), [email protected] (B.J. Gireesha), [email protected] (S.A. Shehzad), abbasisarkar @gmail.com (F.M. Abbasi) Peer review under responsibility of Faculty of Engineering, Alexandria University.
sheet has engrossed the attention of several scientists due to their numerous demands in manufacturing processes of industry like transportation, nuclear reactors, power generation, biomedicine, industrial and food applications. To the best of our knowledge Khan and Pop [1] firstly reported the flow characteristics of viscous nanomaterial due to linear stretching sheet. Nield and Kuznetsov [2] address the Cheng-Minkowycz problem on a flow of nanoliquid in porous medium. Makinde et al. [3] exhibits buoyancy effect in convectively heated stretched flow of viscous nanoliquid. Difference of single phase and two phase model in a wavy channel is addressed by Rashidi et al. [4]. Sheikholeslami et al. [5] computed the numerical solution of MHD nanoliquid under the influence of viscous dissipation. Ramesh [6] discussed the heat source/ sink phenomenon in a Jeffrey nanoliquid. Some recent works
https://doi.org/10.1016/j.aej.2017.12.009 1110-0168 Ó 2018 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
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Nomenclature aðtÞ a0 B Bi Cf C Cw C1 DB DT f Le k M Nux Nb Nt p Pr qw qm Q
radius of the cylinder positive constant time dependent uniform magnetic field biot number skin-friction coefficient concentration nanoparticle volume fraction at the wall nanoparticle volume fraction far from surface Brownian diffusion co-efficient thermophoetic diffusion co-efficient dimensionless stream function Lewis number thermal conductivity magnetic parameter local Nusselt number Brownian motion parameter thermophoresis parameter pressure Prandtl number heat flux at the wall mass flux at the wall uniform source/sink parameter
on two/three dimensional flow of nanoliquid can be seen [7–31]. All the above works analyses the flow and heat transport behavior on steady case only. Most of the engineering problems time dependent is important factors (see [32–36]). Flow due to stretching cylinder with time dependent was observed by Fang et al. [37]. Munawar et al. [38] address the unsteady behavior of vacillate stretching cylinder. Malvandi [39] considered nano-particles to report the time dependent factor in rotating sphere. Zaimi et al. [40] use the Buongiorno’s model to analysis the nanofluid flow in a contracting cylinder. Rohni et al. [41] explored the impact of suction on two dimensional nanoliquid flow of shrinking surface. Marinca and Ene [42] obtained the dual solutions of [38] by applying OHA method. Abbas et al. [43] utilized the slip condition for the analysis of unsteady stretching/shrinking cylinder. The idea of this article is to explore the behavior of uniform heat absorption/generation and various aspects of convective condition in magnetohydrodynamic nanoliquid past a contracting cylinder. Here the effect of heat source/sink acts as vital part in controlling the temperature and also technique of convective transport of heat has tremendous significance in the problems where relatively higher temperature is occurred. Some attempts have been done by the researchers [44–48] on heat source/sink and convective condition. Based on the above literatures no endeavor has been made on MHD nanoliquid over a contracting cylinder under the impact of heat source/sink and convective boundary condition. Numerical calculation is made to get the solution of coupled non-linear expressions. The outcome of dimensionless parameters has been visualized for different values of physical constraints.
S Shx t T Tw T1 u; w v
unsteadiness parameter local Sherwood number time temperature temperature at the wall temperature far from the wall velocity components along the r and z axes velocity vector
Greek symbols a thermal diffusivity b contraction/expansion strength g similarity variable h dimensionless temperature l viscosity q fluid density ðqcÞf heat capacity of the fluid ðqcÞ heat capacity of nanoparticle material / dimensionless nanoparticle volume fraction surface shear stress sw
2. Problem developments Here we elaborate unsteady laminar flow of nanoliquid past a cylinder in a contracting motion. The z-axis is selected along the axis of the cylinder and r -axis is perpendicular to it (see Fig. 1). We considered the diameter of the cylinder with pffiffiffiffiffiffiffiffiffiffiffiffiffi unsteady radius aðtÞ ¼ a0 1 bt; here b denote contraction strength, a0 > 0 and t be the time. We also deliberate about heat source/sink and convective condition. Flow analysis also consists of Brownian motion and thermophoresis. With the above assumptions equations governing the flow analysis are [22] 1 @ @w ðruÞ þ ¼ 0; r @r @z
Fig. 1
ð1Þ
Physical model and coordinate system.
Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder 2 @u @u @u 1 @p @ u 1 @u @ 2 u u rB2 ðtÞ þu þw ¼ þt þ u; þ 2 2 2 @t @r @z q @r @r r @r @z r q
3
2 d2 / d/ d/ dh Nt d h dh þ Le f Sg þ g 2þ ¼ 0: g 2þ dg dg dg dg Nb dg dg
ð2Þ 2 @w @w @w 1 @p @ w 1 @w @ 2 w rB2 ðtÞ þ þu þw ¼ þt þ w; 2 2 @t @r @z q @z @r r @r @z q ð3Þ @T @T @T @ 2 T 1 @T @ 2 T Þþ þu þw ¼ að 2 þ þ @t @r @z @r r @r @z2 " " ## 2 2 @C @T @C @T DT @T @T q þ 0 ðT T1 Þ; þ s DB þ þ qcp @r @r @z @z T1 @r @z ð4Þ 2 @C @C @C @ C 1 @C @ 2 C þu þw ¼ DB þ þ þ @t @r @z @r2 r @r @z2 2 2 DT @ T 1 @T @ T : þ þ T1 @r2 r @r @z2
ð5Þ
The boundary conditions for this analysis can be described as t < 0 : u ¼ w ¼ 0; T ¼ T1 ; C ¼ C1 ; t P 0 : u ¼ 0; w ¼
and the boundary condition (6) becomes fðgÞ ¼ 0;
df dh ¼ 1; ¼ Bið1 hðgÞÞ; /ðgÞ ¼ 1; at g ¼ 1 dg dg
df ! 0; hðgÞ ! 0; /ðgÞ ! 0 as g ! 1; dg
ð11Þ
The key parameters of the problem are Pr for Prandtl number, Le for Lewis number, S for unsteadiness parameter, Nb for Brownian motion parameter and Nt for thermophoresis parameter, M for magnetic parameter, Q for uniform source/ sink parameter, Bi for Biot number, which are defined as t t a2 b sDB ðCW C1 Þ Pr ¼ ; Le ¼ ; ; S ¼ 0 ; Nb ¼ a DB 4t t sDT ðTf T1 Þ Nt ¼ ; tT1 Q¼
q0 rB2 h ; M ¼ 2 0 ; Bi ¼ : cp q a0 q k
The term pressure can be obtained from Eq. (2) as Z P @u u 1 @u ¼ constant þ t þ u2 þ dr: q @r r 2 @t
1 4tz @T ; k ¼ hðTf Tw Þ; a2o 1 bt @r
C ¼ CW at r ¼ aðtÞ; w ! 0; T ! T1 ; C ! C1 as r ! 1;
ð10Þ
ð6Þ
here u; w; T and C represents the velocity, temperature and concentration components respectively, p the pressure, t the kinematic viscosity, q the liquid density, a the thermal diffusivity, s the ratio between effective heat capacity of nanoparticle materials and liquid heat capacity, cp the specific heat, B2 ðtÞ ¼ 4B2o tð1 btÞ1 the time dependent uniform magnetic field, DB the coefficient of Brownian diffusivity, h the coefficient of heat transport, DT the coefficient of thermophoretic diffusivity, Tw ¼ T1 þ bz½a0 tð1 btÞ1 the surface temperature, CW the constant surface nanoparticle volume fraction and T1 and C1 the constant temperature and nanoparticle volume fraction far from the surface of the cylinder respectively. Independent variables are defined as [37]: 9 0 ffiffiffiffiffiffiffi fðgÞ pffiffi ; w ¼ 12 4tz f ðgÞ; > u ¼ a12 p2t = ao 1bt o 1bt g ð7Þ 2 > ðTT1 Þ ðCC1 Þ r 1 ; ; / ¼ ; g ¼ : h ¼ ðT a0 1bt ðCw C1 Þ f T1 Þ Applying Eq. (7) into (1)–(6), noted that Eq. (1) is satisfied automatically and due to absence of longitudinal pressure gradient, Eqs. (2)–(5) reduces the following form 2 2 d3 f d2 f df df d f df df g 3þ 2þf S g 2þ M ¼ 0; dg dg dg dg dg dg dg ð8Þ 2 2 ! 1 d h dh dh dh d/ dh dh g 2þ þ f Sg þ g Nt þ Nb þ Qh ¼ 0; Pr dg dg dg dg dg dg dg ð9Þ
The coefficient of skin-friction Cf , the Nusselt number Nux and Sherwood number Shx can be written as Cf ¼
sw a0 ð1 btÞ1=2 qw a0 ð1 btÞ1=2 qm ; Nu ; Sh ; ¼ ¼ x x qw2w =2 2kðTf T1 Þ 2DB ðCW C1 Þ
where sw the surface shear stress, qw the surface heat flux and qm the surface mass flux. These quantities can be expressed as @w 1 8tlz ¼ f00 ð1Þ sw ¼ l @r r¼aðtÞ a3o ð1 btÞ3=2 , qw ¼ k
@T 2kðTW T1 Þ ¼ ½h0 ð1Þ; @r r¼aðtÞ a0 ð1 btÞ1=2
qm ¼ DB
@C 2DB ðCW C1 Þ ¼ ½/0 ð1Þ: @r r¼aðtÞ a0 ð1 btÞ1=2
Using similarity variables, we have Cf z=aðtÞ ¼ f00 ð1Þ; Nux ¼ h0 ð1Þ; Shx ¼ /0 ð1Þ: 3. Method of solutions Numerical calculations are carried out using Runge-KuttaFehlberg fourth-fifth order procedure by shooting technique to compute Eqs. (8)–(11). The equations f are in third order and h; / are in second order. These equations are reduced into seven simultaneous first order equations including fourteen unknowns. Shooting technique is applied to obtain the missing condition. Integration is performed via RKF-45 technique by considering fix values of g1 . The more accurate solutions
Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
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G.K. Ramesh et al.
can be achieved by using the proper step size h. Two various deformations of solutions are developed and examined at each step. Such deformations are only acceptable if both are close otherwise step size is reduced. Here we fixed step size dn ¼ 0:001; g1 ¼ 4; 6 and accuracy to fifth decimal place. The present work is good agreement with the existing work of Zaimi et al. [22], the comparison graph is displayed in Fig. 11. The flow chart of the present numerical procedure as follows 1. The boundary value problem is first converted into an initial value problem. 2. Initial value problem is solved by appropriately guessing the missing initial value using the shooting method for several sets of parameters. 3. The step size is h = 0.1 used for the computational purpose. 4. The solution process is repeated with another larger value of g1 until two successive values differ only after desired significant digit. 5. The last value g1 is taken as the finite value of the limit g1 for the particular set of physical parameters for determining velocity, temperature and concentration. 6. After getting all the initial conditions we solve this system of simultaneous equations using fourth order Runge–Kutta integration scheme.
Fig. 3
Temperature behavior on S.
4. Results and discussion Characteristics of Biot number, uniform heat absorption and generation in MHD nanoliquid flow through cylinder is analyses in this study. Figs. 2–4 are presented to describe the behavior of unsteadiness on velocity, temperature and concentration profiles. Fig. 2 shows the variation of f0 in response to modify in unsteadiness parameter S. Curves of f0 increases with an enchantment of S. But the opposite tend in nature can found for h and / which is displayed in Figs. 3 and 4. Physically it can analyze that S is directly proportional to a0 and S ¼ 0 represents the steady case. Important observation noted that the rate of cooling is much faster for negative values of unsteadiness parameter, whereas it may take longer time in steady flows.
Fig. 4
Fig. 5
Fig. 2
Velocity behavior on S.
Concentration behavior on S.
Velocity behavior on M.
Velocity field f0 for magnetic parameter M is sketched in Fig. 5. Higher values of M decay the velocity and its related momentum layer thickness. The physical impact is that ability
Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009
Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder
Fig. 6
Temperature behavior on Q.
Fig. 7
Temperature behavior on Bi.
5
Fig. 9 Variation of local Nusselt and Sherwood number with Nt for distinct values of Nb.
Fig. 10 Variation of local Nusselt and Sherwood number with Le for distinct values of Nb.
Fig. 8 M.
Variation of friction factor with S for distinct values of Fig. 11
Validation of the present work.
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G.K. Ramesh et al. Table 1
Numerical values of f00 ð1Þ, /0 ð1Þ and h0 ð1Þ for different physical parameter.
Bi
Le
M
Nb
Nt
Pr
Q
S
f00 ð1Þ
/0 ð1Þ
h0 ð1Þ
0.5 0.6 0.7 0.5
1
4
0.5
0.1
6.8
0.5
0.9
1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.6466 1.8954 2.1189 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.8954 1.9445 1.9194 1.8954
0.3074 0.3167 0.3260 0.3074 0.2519 0.2069 0.2923 0.3074 0.3191 0.3074 0.2070 0.1483 0.3074 0.6064 0.9551 0.3638 0.3388 0.3198 0.3074 0.2626 0.2244 0.1680 0.2385 0.3074
0.2568 0.3085 0.3602 0.2568 0.2553 0.2541 0.2308 0.2568 0.2774 0.2568 0.1288 0.0371 0.2568 0.2630 0.2862 0.4789 0.3721 0.2999 0.2568 0.1444 0.0573 -0.0713 0.0963 0.2568
0.5
1 2 3 1
0.5
1
3 4 5 4
0.5
1
4
0.5 0.6 0.7 0.5
0.5
1
4
0.5
0.1 0.2 0.3 0.1
0.5
1
4
0.5
0.1
4 5 6 6.8
0.5
1
4
0.5
0.1
6.8
of the force magnet which is normal to electrical-conducting fluid gives the drag force named as Lorentz force which acts opposite direction to the flow. Fig. 6 is elaborated to see the influence of Q on temperature field h. It reveals that presence of heat source (positive values) in the thermal layer gives the energy this causes the enhancement in the temperature and its related boundary layer thickness. As noted that negative values of Q decay the temperature filed and Q ¼ 0 correspond to the absence of heat source/sink. Fig. 7 elucidate the effect of Biot number Bi on h. This Fig. gives the clear insight that for large values of Q raise the temperature rapidly near the boundary. The central reason is convective heat exchange at the surface plate leading to an increase in thermal boundary layer thickness. The parameter Bi at any location x is directly proportional to the heat transfer coefficient associated with the hot fluid hf . Thus as Bi increases, the hot fluid side convection resistance decreases. It is also observed that in the absence of M; Q and Bi our Eqs. (8)–(11) reduces the problem of Zaimi et al. [22] (see Fig. 11). Figs. 8–10 are displayed to study the engineering interest of friction factor f00 ð1Þ, local Nusselt number h0 ð1Þ and Sherwood number /0 ð1Þ on certain physical parameters. Fig. 8 is demonstrated unsteadiness S and magnetic parameter M on f00 ð1Þ. As increasing values of S and M gives the increase in f00 ð1Þ profile. Figs. 9 and 10 provide the analysis of Nb and Nt; Le versus h0 ð1Þ and /0 ð1Þ. It is analyzed from these figures that temperature and thermal layer are enhanced for larger Nb but opposite trend for concentration. Physically higher values of Nb relate to much random motion of the particles which results in an enhancement of temperature and reduction on concentration. The Brownian motion of nanoparticles can enhance thermal conduction through one of two systems-either a direct effect owing to nanoparticles
0.5 0.6 0.7 0.5
0.6 0.7 0.8
that transport heat or alternatively through an indirect contribution due to micro-convection of fluid surrounding individual nanoparticles. For small particles, Brownian motion is strong, and the parameter Nb will have high values, the converse is the case for large particles. 5. Conclusions We examined the behavior of MHD, heat absorption/generation and convective type boundary condition on time dependent flow of Buongiorno’s model nanoliquid over a contracting cylinder. The new set of similarity variables is applied to get the ordinary non-linear differential equations and RKF45 method is adopted to get the solution of the mathematical problem. It is noted from Table 1 that f00 ð1Þ is higher for M and lower for S and there is no variations in Q; Nb; Nt; Le and Bi which can observe from Eqs. (8)– (10). Here h0 ð1Þ is a higher value for Bi; M; Nt; S and lower values for Le; Nb; Pr; Q. The same behavior can also found for /0 ð1Þ. Significant observation is found that cooling rate is much higher for negative values of unsteadiness parameter and heat sink parameter. Acknowledgments The authors are very much thankful to the editor and reviewers for their encouraging comments and suggestions to improve the presentation of this manuscript. References [1] W.A. Khan, I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, Int. J. Heat Mass Transfer 53 (2010) 2477–2483.
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Please cite this article in press as: G.K. Ramesh et al., Magnetohydrodynamic nanoliquid due to unsteady contracting cylinder with uniform heat generation/absorption and convective condition, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2017.12.009