Analysis of Melting Heat Transport in a Cross Flow Direction: A Comparative Study

Commun. Theor. Phys. 70 (2018) 777 784 Vol. 70, No. 6, December 1, 2018 Analysis of Melting Heat Transport in a Cross Flow Direction: A Comparative Study G. K. Ramesh 1, and S. A. Shehzad 2 1 Department of Mathematics, K.L.E S J.T. College, Gadag-582101, Karnataka, India 2 Department of Mathematics, COMSATS Institute of Information Technology, Sahiwal-57000, Pakistan (Receive July 4, 2018; revised manuscript received August 10, 2018) Abstract An incompressible three-dimensional laminar flow in a cross flow direction is described in this work. The term of melting and viscous dissipation is incorporated in the mathematical modeling of present flow problem. The flow expressions are converted into dimensionless equations, which are solved with help of Runge-Kutta scheme. Impact of the emerging parameters on the non-dimensional velocities and temperature and friction-factors and local Nusselt number are examined. The convergence analysis is found for ϵ < 0 and 0 < ϵ 2. Comparative analysis is made between the obtained results and published data for limiting case. It is explored at the surface that the melting parameter retards the liquid temperature while it enhances the fluid velocity. DOI: 10.1088/0253-6102/70/6/777 Key words: melting effect, cross flow, moving plate, viscous heating, numerical solution 1 Introduction From the learning of dynamic theory of issue, each solid melts if open to a high temperature. Nowadays, an enthusiasm for supportable, clean and low cost essentialness sources is the main problem for engineers and researchers. Elective sources like sun based imperativeness, wind control imperativeness, and joined heat and power plants offer a considerable measure of essentialness yet expecting almost no exertion. Everything considered heat essentialness is generally secured by three procedures: slow heat vitality, sensible warmth criticalness, and fast heat hugeness. Among these glow storing procedures, the inactive warmth limit is a connecting with procedure because of its ability to offer a high-essentialness amassing. Warm essentialness is secured in a material through inactive warmth by the dissolving system. Starting late, frames concerning softening warmth trade inside non-newtonian fluids have shown to have a wide combination of uses in inventive warm working, for outline, magma solidifying, welding shapes, sit still warm imperativeness accumulating, perfect use of essentialness, warm confirmation dissolving of the permafrost, status of semiconductors, warm assurance, geothermal imperativeness recuperation, and so on. The gathering heat phenomena of ice set into a hot stream of air in solid state were investigated for the essential experienced by Robert. [1] Epstein and Cho [2] studied the surface melting heat exchange for laminar flow past a static level plate. Cheng and Lin [3] described the effect of melting on convective heat transport induced by moving plate. Ishak et al. [4] discussed the point of confinement flow layer and heat transport from laminar, warm liquid stream to moving and melting surfaces. Hardly, any current examinations that arrange the Corresponding author, E-mail: gkrmaths@gmail.com c 2018 Chinese Physical Society and IOP Publishing Ltd melting heat exchange investigation are given in Refs. [5 10]. Recently, much importance is devoted to the examination of boundary-layer flow behavior and heat transport mechanism of Newtonian material induced due to movement of plate by virtue of its expansive implications in planning systems. The boundary-layer conditions accept central part in various parts of fluid mechanics since they depict the development of thick fluid closer to surface. Forced convective flows have been inspected comprehensively from both practical and theoretical view point throughout late decades. Early examinations generally possessed with finding the closeness features within boundary-layer frame-work. The literature on boundarylayer witnessed that stream past flat surface with uniform free-stream is generally mulled over. The progression of boundary-layer velocity on flat surface was firstly analyzed by Blasius. [11] Cortell [12] presented the numerical examination on the Blasius flat surface problem. Introducing a composite speed where is the speed of free stream and is speed of plate, Blasius and Sakiadis issues adequately merged and got a singular game plan of conditions by Afzal et al. [13] Abdelhafez [14] inspected the heat transport from flat sheet whose movement is in parallel free-stream. Hussaini et al. [15] considered the boundary-layer in a flat plate, which has a consistent speed opposite in making a beeline for that of the uniform standard. Some basic examinations in this topic are made by Refs. [16 20]. An intensive review of the writing uncovers that when a liquid flows over surface, the boundary-layer showing up from bend of stream-lines, which differ from free-stream because of the weight slope that opposed the course of free-stream flow. The weight within the boundary layer is http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn

778 Communications in Theoretical Physics Vol. 70 not affected because of surface separation, yet some different highlights assume a part, similar to bring down speed, surface of condition, more-keen bent when contrasted with flow of free-stream velocity. Cross flow happens when the boundary-layer velocity part is typical to the bearing of free-stream. The transverse movement is to be completely created in such type flows. The applications of such flows can be visualized in building circumstances like wind flow marvels, mechanical and aviation. Jones [21] analyzed the outcomes of cross flow in which he discussed the impact of sweep back on boundary-layer. Mager [22] contemplated the three-dimensional boundary layer-flow behavior over bended and flat surfaces. Tsung and Hansen [23] have contemplated three-dimensional consistent state fluid flow conditions of power law fluid in rectangular coordinates. Fang and Lee [24] displayed the three-dimensional wall-bounded laminar flow with moving boundary. Bhattacharyya and Pop [25] considered flow and heat transport in cross flow behavior with viscous heating. Haq et al. [26] reported the heat, mass exchange impacts on the cross flow direction. Energized by above analysis, here we describe the melting outcomes for the cross flow direction. The selfpractically identical conditions are comprehended numerically. The physical qualities of flow and heat transport are analyzed. The presentation of this work is arranged as follows. The flow model is presented in Sec. 2. The numerical procedure is discussed in Sec. 3. Section 4 has the results and discussion material. The end remarks are presented in Sec. 5. 2 Flow Modeling We consider a moving plate melt at steady-state temperature. We further assume the three-dimensional (stream wise and transverse) flow into consistent property, hot liquid of similar material. The surface is moving with a consistent speed U w and free-stream speed is represented by U. It is likewise expected that melt surface temperature is T m while T is temperature of free-stream condition in which T > T m. The cross-flow is created at an infinite degree in the span-wise heading. Consequently, the momentum and temperature are free of z-coordinate, hence expressions of boundary layer under viscous heating condition are: [25 26] u x + v y = 0, (1) u u x + v u y = ν 2 u y 2, (2) u w x + v w y = ν 2 w y 2, (3) u T x + v T y = T α 2 y 2 + ν [( u c p y ) 2 + ( w y ) 2 ]. (4) Here u, v, w depict the flow velocities, y denotes the normal coordinate in plane-flow, and z the span-wise coordinate in transverse flow direction, ν(= µ/ρ) for kinematic viscosity, µ for the dynamic viscosity, ρ for the density of liquid, c p for the specific-heat at constant pressure, T for the liquid temperature, α for thermal diffusivity of the liquid. Following by Ishak et al. [4] and Afzal et al., [13] the conditions of under consideration problem are: u = u w, v = 0, w = 0, T = T m, at y = 0, u U, w w 0, T T, as y, (5) ( T ) k = ρ[λ + c s (T m T 0 )] at y = 0, (6) y where k for the thermal conductivity, c s for the heatcapacity of solid body, and T 0 for the surface temperature. To get the similarity solutions of the problem, the transformations are introduced as ψ = U 2νxU f(ζ), ζ = 2νx y, w = w 0 h(ζ), θ(ζ) = T T m T T m. (7) Here ζ be the similarity variable, ψ be the stream function (u = ψ/ y and v = ψ/ x), and θ for dimensionless temperature. Automatically Eq. (1) is fulfilled, and Eqs. (2) to (4) are reduced as d 3 f dζ 3 + d2 f f(ζ) = 0, (8) dζ2 d 2 h dζ 2 + dh f(ζ) = 0, (9) dζ 1 d 2 θ P r dζ 2 + dθ ( d 2 dζ f(ζ) + Ec f ) 2 ( dh ) 2 1 + Ec2 = 0, (10) dζ 2 dζ with df dζ = ϵ, dθ P rf(ζ) + M dζ = 0, h(ζ) = 0, θ(ζ) = 0 at ζ = 0, df 1, θ(ζ) 1, h(ζ) 1 as ζ. (11) dζ From Eqs. (8) to (11), the following parameters are obtained P r = (ν/α) for Prandtl number, Ec 1 = U 2 /c p (T T m ) and Ec 1 = w0/c 2 p (T T m ) are Eckert numbers, M = c p (T T m )/(c s (T m T 0 ) + λ) for melting parameter and ϵ = U w /U for moving parameter, also it is noted that when ϵ > 0 the problem reduces to stretching problem and ϵ < 0 it reduces to shrinking problem. The coefficients of designing interest, which have coordinate effect on mechanical properties inside boundary layer are skin-friction coefficients and Nusselt number, which are individually introduced as (2Re x ) 1/2 Cf x = f (0), (2Re x ) 1/2( w ) 0 Cf z = h (0), U ( Rex ) Nu x = θ (0), (12) 2 where Re x = U x/ν be the Reynolds number. 3 Numerical Approach The boundary value problem of Eqs. (8) to (10) along with the conditions (11) are treated by reducing them into

No. 6 Communications in Theoretical Physics 779 initial-value problem: df dζ = p, dp dζ = q, dq dr + fq = 0, dζ dh dζ = r, dζ + fr = 0, dθ dζ = s, 1 ds ( dq ) 2 P r dζ + fs + Ec 1 + Ec2 r 2 = 0, dζ h(0) = 0, p(0) = ϵ, θ(0) = 0, P rf(0) + Ms(0) = 0, p( ) = 1, θ( ) = 1, h( ) = 1. Table 1 Convergence analysis of the problem when different values of the parameter M = 3, P r = 0.7, Ec 1 = Ec 2 = 0.5, and ϵ = 0.5. ζ f f f h h θ θ 0 0.982 973 0.5 0.081 016 0 0.162 033 0 0.229 360 1 0.428 215 0.623 957 0.165 767 0.247 915 0.331 535 0.294 968 0.345 366 2 0.286 317 0.807 787 0.180 735 0.615 575 0.361 471 0.636 893 0.308 456 4 2.147 428 0.993 270 0.016 845 0.986 541 0.033 691 0.973 418 0.048 880 6 4.145 026 0.999 999 0.000 031 0.999 985 0.000 062 0.999 808 0.000 596 8 6.145 024 1.000 000 0.000 000 1.000 000 0.000 000 1.000 000 0.000 000 Table 2 Convergence analysis of the problem when different values of the parameter M = 3, P r = 0.7, Ec 1 = Ec 2 = 0.5, and ϵ = 2.0. ζ f f f h h θ θ 0 1.694 156 2 0.258 069 0 0.258 069 0 0.395 303 1 0.107 755 1.551 723 0.549 112 0.448 276 0.549 112 0.513 016 0.507 091 2 1.421 744 1.130 040 0.246 662 0.869 959 0.246 662 0.865 196 0.203 235 3 2.475 454 1.012 423 0.034 813 0.987 576 0.034 813 0.976 904 0.047 263 4 3.479 368 1.000 474 0.001 771 0.999 525 0.001 771 0.997 816 0.005 847 5 4.479 486 1.000 007 0.000 033 0.999 992 0.000 033 0.999 891 0.000 360 6 5.479 488 1 0.000 000 1.000 000 0.000 000 1.000 000 0.000 000 Table 3 Convergence analysis of the problem when different values of the parameter M = 3, P r = 0.7, Ec 1 = Ec 2 = 0.5, and ϵ = 0.1. ζ f f f h h θ θ 0 0.037 792 0.099 999 0.002 010 0 0.001 827 0 0.008 818 1 0.136 765 0.097 916 0.002 193 0.001 894 0.001 994 0.009 040 0.009 372 4 0.418 136 0.088 293 0.005 078 0.010 642 0.004 617 0.045 455 0.016 838 8 0.679 322 0.015 976 0.049 749 0.076 384 0.045 226 0.200 727 0.081 344 12 0.318 859 0.709 771 0.272 096 0.736 155 0.247 360 0.836 203 0.150 370 16 4.103 447 0.999 987 0.000 052 0.999 988 0.000 048 0.999 912 0.000 271 4 18 6.103 444 1 0.000 000 1.000 000 0.000 000 1.000 000 0.000 000 Keeping in mind the end goal to coordinate above initial value problem, one requires an incentive for q(0), r(0), and s(0) i.e. f (0), h (0), and θ (0) however no such esteem is given at the limit. The appropriate value for f (0), h (0), and θ (0) is picked, and afterward, integration is carried out. Contrasting the ascertained esteem for f (0), h (0), and θ (0) at ζ = 6 (say) with the given limit condition and changing the assessed esteem, a superior estimate for the arrangement is given. Taking the arrangement of qualities for f (0), h (0), and θ (0) and applying the Runge-Kutta-Fehlberg method with stepmeasure i = 0.01, the above technique is rehashed until the outcome up to the coveted level of precision 10 5 is gotten. Tables 1 3 are made to analyze the convergence values of f(ζ), f (ζ), f (ζ), h(ζ), h (ζ), θ(ζ), and θ (ζ) by setting M = 0.5, P r = 0.7, Ec 1 = 0.5 = Ec 2, and ϵ = 0.5, 2.0, 0.1. With a particular authentic objective to admit the procedure used as a touch of this examination and to judge the exactness of the present examination, relationship with available happens in Ishak et al. [4] (see in Figs. 7 and 8). 4 Results and Discussion Figures 1 3 characterize the behavior of moving parameter ϵ on velocities f (ζ), h(ζ), and temperature θ(ζ). The velocity f (ζ) at the wall is boost up significantly with the increasing values of ϵ. Higher values of moving parameter ϵ enhance the velocity h(ζ) and temperature θ(ζ) remarkably. It is further analyzed that for smaller values of ϵ, the velocity and temperature are away from the wall but the larger values lead these quantities closer to wall. The impact of melting constraint M on velocities f (ζ), h(ζ) and temperature θ(ζ) are explored in Figs. 4

780 Communications in Theoretical Physics Vol. 70 6. In these figures, the red lines represent the behavior at ϵ = 0.5 while the orange line shows the pattern at ϵ = 2.0. Figure 4 executes that the velocity f (ζ) is enhance for both case ϵ = 2.0 and ϵ = 0.5 corresponding to larger values of melting constraint. In case of ϵ = 2.0 the velocity is close to the wall in absence of melting constraint i.e. M = 0.0 and it starts to away from the wall with increasing M. The velocity component h(ζ) is varied significantly with an increase in melting constraint (see Fig. 5). The velocity curves in case of larger values of moving parameter are closer to wall while the curves for larger ϵ for away from the wall. Figure 6 clearly exploits that the increasing temperature behavior is appeared with an increment in the values of melting constraint. The curves of temperature θ(ζ) are going away from the wall with the use of larger values of melting constraint. Fig. 1 (Color online) Variation of ϵ on f when M = 1, P r = 0.7, Fig. 2 (Color online) Variation of ϵ on h when M = 1, P r = 0.7, Fig. 3 (Color online) Variation of ϵ on θ when M = 1, P r = 0.7, Fig. 4 (Color online) Variation of M on f when P r = 3, Fig. 5 (Color online) Variation of M on h when P r = 3, Fig. 6 (Color online) Variation of M on θ when P r = 3,

No. 6 Communications in Theoretical Physics 781 Figures 7 and 8 explore the impact of f (ζ) on θ(ζ) and further, in these figures, we compare our results with the computations of Ishak et al. [4] The green lines in these figures elucidate the results of Ref. [4]. We notice that our results have very good match with the results of Ref. [4] by setting Ec 1 = 0 = Ec 2. These figures also interpret that the velocity f (ζ) and temperature θ(ζ) are retarded with an enhancement in P r for both cases of f (ζ) and θ(ζ). Thermal diffusivity factor is more effective in comparative to momentum diffusivity. Here thermal diffusivity is weaker for larger P r due to which both f (ζ) and θ(ζ) are decay. Variations in velocities f (ζ), h(ζ) and temperature θ(ζ) in presence of viscous heating effects by considering various values of P r are discussed in Figs. 9 11. In these figures, we use the negative values of moving parameter i.e. ϵ = 0.1. Here we examine that f (ζ), h(ζ), and θ(ζ) are moving away from the wall as we increase the values of P r. Fig. 7 (Color online) Variation of P r on f when M = 1, Fig. 8 (Color online) Variation of P r on θ when M = 1, Fig. 9 (Color online) Variation of P r on f when M = 1, Ec 1 = Ec 2 = 0.5, and ϵ = 0.1. Fig. 10 (Color online) Variation of P r on h when M = 1, Ec 1 = Ec 2 = 0.5, and ϵ = 0.1. Fig. 11 (Color online) Variation of P r on θ when M = 1, Ec 1 = Ec 2 = 0.5, and ϵ = 0.1. Fig. 12 (Color online) Variation of Ec 1 on θ when P r = 0.7, M = 3.0.

782 Communications in Theoretical Physics Vol. 70 Figure 12 depicts the behavior of temperature θ(ζ) for different values of Eckert number Ec 1 by setting ϵ = 0.1, ϵ = 0.5, and ϵ = 2.0. The red lines denote the curves against the case when ϵ = 0.1 and Ec 1 = 0.0, 0.5, 5.0. The temperature θ(ζ) is enhanced in this case. The green and blue lines represent the results for ϵ = 0.5 and ϵ = 2.0 when Ec 1 = 0.0, 5, 10. Here we note that the maximum peak is occur corresponding to ϵ = 2.0 and Ec 1 = 10 at ζ = 2.0. The temperature curve is more closer to wall when the values of ϵ and Ec 1 are larger. The changes in temperature θ(ζ) for multiple values of Ec 2 against ϵ = 0.1, ϵ = 0.5, and ϵ = 2.0 are elaborated in Fig. 13. The increasing values of Ec 2 and ϵ lead the temperature curves near to wall. The maximum peak in case of ϵ = 2.0 is observed at ζ = 1.9 when Ec 2 = 10. On the other side, this peak for ϵ = 0.5 is found at ζ = 2.5 when Ec 2 = 10. Fig. 13 (Color online) Variation of Ec 2 on θ when P r = 0.7, M = 3.0. Fig. 14 (Color online) Variation of P r vs. M on f (0), h (0), θ (0) when Fig. 15 (Color online) Variation of P r vs. M on f (0), h (0), θ (0) when Fig. 16 (Color online) Variation of P r vs. M on f (0), h (0), θ (0) when Fig. 17 (Color online) Variation of ϵ vs. M on f (0), h (0), θ (0) when Ec 1 = Ec 2 = 0.5, P r = 0.7. Fig. 18 (Color online) Streamline pattern of moving parameter when ϵ = 0.5.

No. 6 Communications in Theoretical Physics 783 Figure 14 discusses the values of f (0), h (0), θ (0) when Ec 1 = Ec 2 = 0.5, M = 0, 1, 3 and ϵ = 0.1. We have noted that increase in M has no effect on f (0), h (0), and θ (0) because we obtained solid lines against various values of it. In addition, the values of f (0) are higher as comparative to h (0) and θ (0). In Fig. 15, we consider ϵ = 0.5 while all other values remain same as in Fig. 14. Here we note that the increase in M leads to retardation in the values of f (0), h (0), and θ (0). It is also observed that the values of f (0) are smaller as comparative to h (0) and θ (0) when ϵ = 0.5. Figure 16 is presented for the situation when ϵ = 2.0. This figure clearly reports that the values of f (0) become negative when use much larger moving parameter i.e. ϵ = 2.0. From these figures, we conclude that the negative values of ϵ have no effect on f (0), h (0), and θ (0) while the impact of increasing moving parameter is remarkable as shown in Figs. 15 and 16. Figure 17 characterizes the features of various values of melting constraint M and moving parameter ϵ on the quantities f (0), h (0) and θ (0). These quantities are retard with an enhancement in ϵ and M. Finally the streamlines pattern is displayed in the Figs. 18 20. Fig. 19 (Color online) Streamline pattern of moving parameter when ϵ = 0.1. 5 Final Remarks In this work, consolidated comparability numerical procedure is utilized to explore about the melting heat transport over a cross stream. The diminished conditions are settled numerically by adopting shooting scheme in process to Runge Kutta Fehlberg method. A parametric report is performed to investigate the impacts of different administering parameters on the stream and heat exchange qualities. The accompanying conclusions can be drawn from this examination: Rising benefits of melting parameter diminishes the velocity (f, h) and temperature (θ) at ϵ = 0.5 and 2. Return slant is discovered just for the velocity (f ) when ϵ = 2. The friction factor and Nusselt number decline with Fig. 20 (Color online) Streamline pattern of moving parameter when ϵ = 2. the increments of melting parameter but opposite trend at ϵ = 2. At the point ϵ < 0 both velocity (f, h) and temperature (θ) curves diminish with an expansion of Prandtl number. Presence of Eckert number builds the velocity and temperature curves at ϵ = 0.1, 0.5, 2.0 (see Fig. 8). Acknowledgement The authors are very much thankful to the editor and referee for their encouraging comments and constructive suggestions to improve the presentation of this manuscript. References [1] L. Roberts, J. Fluid Mech. 4 (1958) 505. [2] M. Epstein and D. H. Cho, J. Heat Trans. 98 (1976) 531. [3] W. T. Cheng and C. H. Lin, Int. Commun. Heat and Mass Tran. 35 (2008) 1350. [4] A. Ishak, R. Nazar, N. Bachok, and I. Pop, Heat Mass Transfer 46 (2010) 463. [5] K. Ganesh Kumar, B. J. Gireesha, B. C. Prasannakumara, et al., Diffusion Foundations 11 (2017) 33. [6] T. Hayat, A. Kiran, M. Imtiaz, et al., European Phys. J. Plus 132 (2017) 265. [7] M. Sheikholeslami and H. B. Rokni, Chin. J. Phys. 55 (2017) 1115. [8] M. Sheikholeslami and H.B. Rokni, Int. J. Heat and Mass Tran. 114 (2017) 517.

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