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Hindawi Journal of Applied Mathematics Volume 2017, Article ID 1697135, 13 pages https://doi.org/10.1155/2017/1697135 Research Article Viscous Dissipation Effects on the Motion of Casson Fluid over an Upper Horizontal Thermally Stratified Melting Surface of a Paraboloid of Revolution: Boundary Layer Analysis T. M. Ajayi, A. J. Omowaye, and I. L. Animasaun Department of Mathematical Sciences, Federal University of Technology, Akure, Ondo State, Nigeria Correspondence should be addressed to T. M. Ajayi; [email protected] Received 30 June 2016; Revised 20 October 2016; Accepted 6 November 2016; Published 4 January 2017 Academic Editor: Igor Andrianov Copyright © 2017 T. M. Ajayi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. eTh problem of a non-Newtonian u fl id flow past an upper surface of an object that is neither a perfect horizontal/vertical nor inclined/cone in which dissipation of energy is associated with temperature-dependent plastic dynamic viscosity is considered. An attempt has been made to focus on the case of two-dimensional Casson uid fl flow over a horizontal melting surface embedded in a thermally stratified medium. Since the viscosity of the non-Newtonian uid fl tends to take energy from the motion (kinetic energy) and transform it into internal energy, the viscous dissipation term is accommodated in the energy equation. Due to the existence of internal space-dependent heat source; plastic dynamic viscosity and thermal conductivity of the non-Newtonian uid fl are assumed to vary linearly with temperature. Based on the boundary layer assumptions, suitable similarity variables are applied to nondimensionalized, parameterized and reduce the governing partial differential equations into a coupled ordinary differential equations. These equations along with the boundary conditions are solved numerically using the shooting method together with 1/2 the Runge-Kutta technique. eTh eeff cts of pertinent parameters are established. A significant increases in Re 𝐶 is guaranteed 1/2 with 𝑆 when magnitude of 𝛽 is large. Re 𝐶 decreases with 𝐸 and 𝑚 . 𝑡 𝑐 1. Introduction Na [2] investigated the laminar free convection heat transfer from a needle. Ahmad et al. [3] examined the boundary layer Within the last thirty years, the study of non-Newtonian flow over a moving thin needle with variable heat ux fl . eTh fluid flow over a stretching surface has received signicfi ant boundary layer flow over a stretching surface with variable attention due to its industrial applications. Such interest is thickness was analyzed by Fang et al. [4]. Recently, the flow of fueled by its pertinent engineering applications in a number different u fl ids over an upper horizontal surface with variable of efi lds as in spinning of filaments, continuous casting of thickness has been investigated extensively in [5–7]. This metal, extrusion of polymers, crystal growing, glass b fi er attracted Makinde and Animasaun [8, 9] to focus on the production, extrusion of plastic sheets, paper production, and case of quartic autocatalysis kind of chemical reaction in process of condensation of metallic plates. Boundary layer the flow of an electrically conducting nanou fl id containing analysis of u fl id flow passing through a thick needle with vari- gyrotactic-microorganism over an upper horizontal surface able diameter was investigated by Lee [1]. Historically, this can of a paraboloid of revolution in the presence and absence of thermophoresis and Brownian motion. In most cases, plastic be referred to as the rst fi report of flow adjacent to a surface with variable thickness where the eeff ct of viscosity is highly dynamic viscosity of non-Newtonian Casson uid fl tends to significant. er Th eaeft r, extensive studies were conducted on take energy away from the motion and transform it into the boundary layer flows over a thin needle. Cebeci and internal energy. However, this area has been neglected. 𝑓𝑥 𝑓𝑥 2 Journal of Applied Mathematics In u fl id mechanics, destruction of u fl ctuating velocity gradient, suction, and injection. It was observed that an gradients due to viscous stresses is known as viscous dissi- increase in the magnetic eld fi parameter leads to an increase pation. This partial irreversible processes is oen ft referred to in u fl id velocity, skin friction, rate of heat transfer, and a fall as transformation of kinetic energy into internal energy of the in temperature. Rajeswari et al. [20] observed that, due to the u fl id (heating up the uid fl due to viscosity since dissipation is uniform magnetic eld fi and suction at the wall of the surface, high in the regions with large gradients). Pop [10] remarked the concentration of the u fl id decreases with the increase that understanding the concept of energy dissipation and in chemical reaction parameter. Ghosh et al. [21] found transport in nanoscale structures is of great importance for that an increase in inclination of the applied magnetic eld fi the design of energy-efficient circuits and energy-conversion opposesprimaryflowandalsoreducesGrashofnumbers.Das systems. However, energy dissipation and transport of non- [22] concluded that increasing magnetic eld fi and thermal Newtonianfluidare also of importance to engineersand radiation leads to deceleration of velocity but reverse is the scientists. Motsumi and Makinde [11] examined the effects of effect for the melting parameter when the solid surface and viscous dissipation parameter (i.e., Eckert number), thermal the free stream move in the same direction. Motsa and Ani- diffusion, and thermal radiation on boundary layer flow of masaun [23] presented the behavior of unsteady non-Darcian Cu-water and Al O -water nanou fl ids over a moving flat magnetohydrodynamic uid fl flow past an impulsively using 2 3 plate. In another theoretical study on combined eeff cts of bivariate spectral local linearization analysis. Koriko et al. Newtonian heating and viscous dissipation parameter on [24] illustrated the dynamics of two-dimensional magneto- boundary layer o fl w of copper and titania in water over hydrodynamics (MHD) free convective flow of micropolar stretchable wall, Makinde [12] reported an increase in the u fl id along a vertical porous surface embedded in a thermally moving plate surface temperature and thermal boundary stratified medium. It was concluded that velocity profiles layer thickness. Depending on the admissible grouping of and microrotation profiles are strongly influenced by the variables (parameterization), Eckert number and Brinkmann magnetic field in the boundary layer, which decreases with an number (𝑃𝑟×𝐸 ) may be used to quantify viscous dissipation. increase in the magnitude of magnetic parameter. Recently, In addition, unsteady mixed convection in the flow of air theoretical investigation of MHD natural convection flow in over a semi-infinite stretching sheet taking into account the vertical microchannel formed by two electrically noncon- eect ff of viscous dissipation was carried out by Abd El-Aziz ducting infinite vertical parallel plates and eeff cts of MHD [13]. Both at steady stage (𝐴=0 ) and unsteady stage (𝐴= mixed convection on the flow through vertical pipe with time 1.5), velocity and temperature of the flow increase with an periodic boundary condition was presented explicitly by Jha increase in the magnitude of Eckert number. Recently, effects and Aina [25, 26]. of viscous dissipation, Joule heating, and partial velocity slip Several processes involving melting heating transfer in on two-dimensional stagnation point flow were reported by non-Newtonian u fl ids have promising applications in ther- Yasin et al. [14]. In another study conducted by Animasaun mal engineering, such as melting permafrost, oil extraction, andAluko [15],itisreportedthatwhendynamic viscosity magma solidification, and thermal insulation. As such, a lot of airisassumedtovarylinearlywithtemperature,normal of experimental and theoretical work has been conducted in negligible effect of Eckert number on velocity profiles will the kinetics of heat transfer accompanied with melting or be noticed. Raju and Sandeep [16] focused on the motion of solidification eeff ct. eTh process of melting of ice placed in a Casson fluid over a moving wedge with slip and observed hot stream of air at a steady state was rst fi reported by Roberts a decrement in the temperature field with rising values of [27]. Historically, this report can be referred to as the pio- Eckert number. In the flow of non-Newtonian Casson fluid neering analysis of the melting phenomenon. Another novel over an upper horizontal thermally stratiefi d melting surface report on melting phenomenon during forced convection of a paraboloid of revolution, dissipation of smaller eddies heat transfer when an iceberg drifts in warm sea water was due to molecular viscosity near the wall is significant. In the presented by Tien and Yen [28]. From their investigation, they presence of constant magnetic field, electrically conducting observed that melting at the interface results in a decrease Casson ufl id flowoverobjectthatisneither aperfect hori- in the Nusselt number. Epstein and Cho [29] discussed the zontal/vertical nor inclined/cone is also an important issue. laminar film condensation on a vertical surface. Much later, eTh study of electrically conducting u fl id flow is of melting heat transfer in a nanou fl id boundary layer on a considerable interest in modern metallurgical. This can be stretching circular cylinder was examined by Gorla et al. traced to the fact that most uids fl in this sector are electrically [30]. In the study of the effect of radiation on MHD mixed conducting u fl id. Historically, the rst fi report on the motion convection flow from a vertical plate embedded in a saturated of electrically conducting u fl id in the presence of magnetic porous media with melting, Adegbie et al. [31] reported field was presented by Rossow [17]. er Th eaeft r, Alfv en ´ [18] that the Nusselt number decreases with increase in melting reported that if a conducting liquid is placed in a constant parameter. Adegbie et al. [31] stated that the temperature of magnetic field, every motion of the liquid generates a force UCMfluidflow over amelting surfaceisanincreasingfunc- called electromotive force (e.m.f.) which produces electric tion of variable thermal conductivity parameter. Omowaye currents. One of the most significant importance of these and Animasaun [32] investigated boundary layer analysis in contributions are its applications in engineering problems the flow of upper convected Maxwell u fl id flow. Due to the such as MHD generators, plasma studies, nuclear reactors, fact that temperature at the wall is zero, classical temperature- and geothermal energy extractions. Soundalgekar and Murty dependent viscosity and thermal conductivity linear models [19] investigated heat transfer in MHD flow with pressure were modified to suit the case of both melting heat transfer Journal of Applied Mathematics 3 and thermal straticfi ation. In another study of micropolar negligible but gravity is sufficiently strong to make the specific u fl id flow in the presence of temperature-dependent and weight appreciably different between any two layers of Casson spaceheatsource,theanalysisofthecasewherevortexviscos- u fl id on the surface; hence the rate flow on this kind of surface ity is a constant function of temperature was reported in [33]. is referred to as free convection. eTh body force term suitable Within the past two decades, the effects of temperature- to induce the flow over a surface which is neither a perfect dependent viscosity on the uid fl flow have become more horizontal/vertical nor inclined/cone is important to engineers dealings with geothermal systems, 𝑚+1 crude oil extraction, and machinery lubrication. Due to (1) − +𝜌𝑔 =𝑔𝛾 (𝑇 −𝑇)+0. 𝑥 ∞ friction and internal heat generated between two layers of u fl id, viscosity and thermal conductivity of u fl id substance Following the theory stated in Casson [45] and boundary may be aeff cted by temperature; for more details see Batchelor layer assumptions, the rheological equation for an isotropic [34], Lai and Kulacki [35], and Abd El-Aziz [36]. Proper Casson u fl id flow together with heat transfer is of the form consideration of this fact in the study on inherent irre- 𝜕𝑢 𝜕 V versibility in a variable viscosity Couette flow by Makinde + =0, (2) and Maserumule [37], numerical investigation of micropolar u fl id flow over a nonlinear stretching sheet taking into 𝜕𝑢 𝜕𝑢 1 1 𝜕 𝜕𝑢 account the effects of a temperature-dependent viscosity by 𝑢 + V = (1 + ) (𝜇 (𝑇 ) )+𝑔𝛾 𝜌 𝛽 Rahman et al. [38], effects of MHD on Casson uid fl flow (3) in thepresenceofCattaneo-Christov heat ufl xbyMalik 𝑚+1 et al. [39], fluid flow through a pipe with variation in 𝑜 ⋅ (𝑇 −𝑇)− 𝑢, viscositybyMakinde[40],Cassonfluidflowwithinboundary 2 𝜌 layer over an exponentially stretching surface embedded in 𝜕𝑇 𝜕𝑇 1 𝜕 𝜕𝑇 a thermally stratified medium by Animasaun [41], steady 𝑢 + V = (𝜅 (𝑇 ) ) 𝑝𝜌𝐶 fully developed natural convection flow in a vertical annular microchannel having temperature-dependent viscosity in the 𝜇 (𝑇 ) 1 𝜕𝑢 𝜕𝑢 𝑄 [𝑇 (𝑥 )−𝑇 ] 𝑜 𝑤 ∞ presence of velocity slip and temperature jump at the annular + (1 + ) + (4) 𝑝𝜌𝐶 𝛽 𝜌𝐶 microchannel surfaces by Jha et al. [42] have enhanced the 𝑝 body of knowledgeonfluidflow,boundarylayer analysis, 𝑚+1 (𝑚−1)/2 and heat/mass transfer. In a recent experiment, Alam et al. 𝑜 ⋅ exp [−𝑛𝑦 (𝑥+𝑏 ) ]. [43] concluded that thermal boundary layer decreases with 2 𝜗 an increasing temperature-dependent viscosity. Hayat et al. Suitable boundary conditions governing the flow along upper [44] discussed the eect ff of variable thermal conductivity on horizontal surface of a paraboloid of revolution are the mixed convective flow over a porous medium stretching surface. In thearticle,the kinematics viscosityofCassonfluid 𝑚 𝑢=𝑈 (𝑥+𝑏 ) , was considered as a function depending on plastic dynamic viscosity, density, and Casson parameter and hence reported 𝜕𝑇 ∗ ∗ 𝜅 (𝑇 ) =𝜌 [𝜆 +𝑐 (𝑇 −𝑇 )]V (𝑥, 𝑦) , that increase in the magnitude of temperature-dependent 𝑏 𝑠 𝑚 𝑜 (5) viscosity parameter leads to an increase in u fl id’s velocity. eTh above literature review shows that there exists no published 𝑇=𝑇 (𝑥 ) article on the eeff cts of viscous dissipation in the flow of non- (1−𝑚)/2 Newtonian Casson u fl id over a upper horizontal thermally at 𝑦=𝐴 (𝑥+𝑏 ) , stratified melting surface of a paraboloid of revolution. 𝑢→0, 2. Formulation of the Problem 𝑇→𝑇 𝑥 ( ) (6) Consider a steady, incompressible, laminar flow of an electri- as 𝑦→∞. cally conducting non-Newtonian (Casson) u fl id over a melt- ing surface on upper horizontal paraboloid of revolution in The formulation of the second term in boundary (5) states the presence of viscous dissipation and thermal stratification. that the heat conducted to the melting surface on paraboloid The 𝑥 -axis is taken in the direction of motion and 𝑦 -axis of revolution is equaltothe combinationofheatofmelting is normal to the flow as shown in Figures 1(a) and 1(b). and the sensible heat required to raise the solid temperature Auniform magnetic efi ld of strength 𝐵 is applied normal 𝑇 to its stratified melting temperature 𝑇 (𝑥) (for details, 𝑜 𝑚 to theflow.Theinduced magnetic efi ld duetothe motion see [29]). For lubricating uids, fl heat generated by the inter- of an electrically conducting Casson u fl id is assumed to be nal friction and the corresponding increase in temperature so small; hence it is neglected. eTh stretching velocity is aeff cts the viscosity of the u fl id; hence, it may not be 𝑈 =𝑈 (𝑥+𝑏) andthe wall is assumedimpermeable.It realistic to constant function of plastic dynamic viscosity 𝑤 𝑜 is further assumed that the immediate uid fl layer adjacent to knowing fully well that space-dependent internal heat source (1−𝑚)/2 the surface is specified as 𝑦 = (𝐴 𝑥+𝑏) ,where 𝑚<1 . is significant. In order to account for this variation, it is valid Using Boussinesq approximation, the difference in inertia is to consider the modied fi mathematical models of both the 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝜎𝐵 𝜕𝑦 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑥 𝜕𝑝 4 Journal of Applied Mathematics As y→ ∞ m (1−m)/2 u→ U [x + b] T→ T (x) = T +m (x + b) o ∞ o 2 ⃗ ⃗ ⃗ T (x) > T (x) F= j× B ∞ m (1−m)/2 At the surface y= A[x + b] 𝜕T Slot 𝜅 =𝜌 [𝜆 +c (T −T )](x, y) u(x, y) = 0 b s m o 𝜕y (1−m)/2 T(x, y) = T (x) = [T +m (x + b) ] m o 1 Viscous dissipation is highly significant Nonporous and melting upper horizontal object of paraboloid of revolution 2 (m−1)/2 ⃗ ⃗ ⃗ Lorentz force F= j× B = 𝜎[B(x)] =𝜎 (B (x + b) ) u where is the induce current and B is the interaction of magnetic field (a) 𝜂= 𝜍 + 𝜒 𝜍= 𝜂 − 𝜒 For the case m< 1 fl Upper horizontal thermally stratified melting object of a paraboloid of revolution (b) Figure 1: (a) eTh coordinate system of Casson uid fl flow over upper horizontal thermally stratified melting surface of a paraboloid of revolution. (b) Graphical illustration of uid fl domain and conversion of domain from [𝜒, ∞) to [0, ∞). temperature-dependent viscosity and thermal conductivity stratification ( 𝑇 )atthe meltingwalland at thefreestreamis models proposed in [46] and adopted in [31, 32] as defined as (1−𝑚)/2 𝜇 (𝑇 )=𝜇 [1 + 𝑏 (𝑇 −𝑇)], 𝑏 1 ∞ 𝑇 (𝑥 )=𝑇 +𝑚 (𝑥+𝑏 ) , 𝑚 𝑜 1 (8) (1−𝑚)/2 𝜅 𝑇 =𝜅 [1 + 𝑏 (𝑇 − 𝑇 )], ( ) 𝑏 𝑏 2 𝑚 𝑇 (𝑥 )=𝑇 +𝑚 (𝑥+𝑏 ) . (7) ∞ 𝑜 2 𝑇−𝑇 Using similarity variable for temperature 𝜃(𝜂) in (7) to 𝜃(𝜂) = . 𝑇 −𝑇 ∞ 𝑜 simplify [1 − (𝜂)] 𝜃 we obtain suitable temperature difference in flow past thermally stratified horizontal melting surface of Meanwhile, the models are still in good agreement with paraboloid of revolution as experimental data of Batchelor [34]. It is worth mentioning that the rfi st and second terms in (7) are valid and reliable (1−𝑚)/2 (9) 𝑇 −𝑇 = (1−𝜃 )[𝑇 −𝑇 ]−𝑚 (𝑥+𝑏 ) . ∞ ∞ 𝑜 1 since 𝑇 (𝑥) > 𝑇 (𝑥) in this study, whereas thermal ∞ 𝑚 Transformation of kinetic energy into internal energy (heating up the uid) due to plastic dynamic viscosity occurs within this domain Casson fluid flow Journal of Applied Mathematics 5 2 2 𝑑 𝑓 𝑑 𝑓 From thermal stratica fi tion models in (9), the following 𝑚+1 1 +𝑃 𝐸 [1 + 𝜉 − 𝜃𝜉 − 𝑆𝜉 ](1 + ) 𝑟 𝑐 𝑡 2 2 relations can be easily obtained 2 𝛽 𝑑𝜂 𝑑𝜂 2𝑃 Γ [−𝑛𝜂] (1−𝑚)/2 𝑟 + 𝑒 =0. 𝑏 (𝑇 −𝑇 )=𝑏 𝑚 (𝑥+𝑏 ) , 1 𝑚 𝑜 1 1 𝑚+1 (10) (1−𝑚)/2 (14) 𝑏 (𝑇 −𝑇 )=𝑏 𝑚 (𝑥+𝑏 ) , 1 ∞ 𝑜 1 2 In (13) and (14), melting parameter 𝛿 ,Prandtl number 𝑃 , magnetic parameter 𝐻 ,Eckertnumber 𝐸 ,temperature- where 𝑇 is known as reference temperature. A significant 𝑟 𝑎 𝑐 dependent thermal conductivity parameter 𝜀 , space-depend- difference between (𝑇 −𝑇 ) and (𝑇 −𝑇 ) can be easily 𝑚 𝑜 ∞ 𝑜 obtained from (10). This can be traced to the fact that the ent internal heat source parameter Γ, skin friction coefficient 𝐶 ,Nusselt number Nu , and buoyancy parameter depend- linear stratification occurs at all points of “ 𝑥 ”onthe wall (1−𝑚)/2 ing on volumetric-expansion coefficient due to temperature (𝑦 = (𝑥 𝐴 + ) 𝑏 )and at all points of “𝑥 ”atthe free stream 𝐺 are defined as as (𝑦 → ∞) . In view of this, it is valid to define temperature- 𝑡 dependent viscous parameter 𝜉 by considering the second (𝑚+1)/2 𝑈 (𝑥+𝑏 ) term in (10) since 𝑇 (𝑥) > 𝑇 (𝑥) . Mathematically, the ratio 𝐸 = , ∞ 𝑚 𝐶 (𝑇 −𝑇 ) 𝑝 𝑤 ∞ of the rst fi two terms in (10) can thus produce dimensionless thermal stratica fi tion parameter ( 𝑆 )as 𝑡 (1−𝑚)/2 𝐶 𝑚 (𝑥+𝑏 ) 𝑝 2 𝛿= , (1−𝑚)/2 [𝜆 +𝑐 𝑚 (𝑥+𝑏 ) ] 𝜉=𝑏 (𝑇 −𝑇 ), 1 ∞ 𝑜 𝑠 1 𝑏 (𝑇 −𝑇 )=𝜉𝑆 , 1 𝑚 𝑜 𝑡 (11) 𝐻 = , 𝜌𝑈 𝑆 = . 𝜀=𝑏 (𝑇 −𝑇 ), 2 ∞ 𝑜 (15) The stream function 𝜓(𝑥, 𝑦) and similarity variable 𝜂 are of Γ= , 𝑚−1 the form 𝜌𝐶 𝑈 (𝑥+𝑏 ) 𝑝 𝑜 𝑔𝛾 𝜕𝜓 𝑢= , 𝐺 = , 2𝑚−1 𝑏 𝑈 (𝑥+𝑏 ) 𝜕𝜓 𝐶 = , V =− , 𝜌 √(𝑚+1 )/2 (𝑈 ) (12) 1/2 𝑚+1 𝑈 (𝑥+𝑏 )𝑞 (𝑚−1)/2 𝜂=𝑦 ( ) (𝑥+𝑏 ) , Nu = , 2 𝜗 𝜅 [𝑇 −𝑇 ] (𝑚+1 )/2 𝑏 ∞ 𝑜 1/2 1/2 where 𝜏 is the shear stress (skin friction) between Casson (𝑚+1)/2 𝜓=𝑓 ( ) (𝜗𝑈 ) 𝑥+𝑏 . ( ) fluid and upper surface of horizontal paraboloid of revolution 𝑚+1 and 𝑞 is theheatflux at allpointsonthe surface It is important to note that the stream function 𝜓(𝑥, 𝑦) 1 𝜕𝑢 𝜏 =𝜇 (1 + ) , automatically satisefi s continuity (2). eTh nonlinear partial 𝑤 𝑏 𝛽 (1−𝑚)/2 𝑦=𝐴(𝑥+𝑏) differential equations (3) and (4) are reduced to the following (16) nonlinear coupled ordinary dieff rential equations 𝜕𝑇 𝑞 =− (𝜅 . 𝑤 𝑏 (1−𝑚)/2 𝑦=𝐴(𝑥+𝑏) 𝑑 𝑓 [1 + 𝜉 − 𝜃𝜉 − 𝑆𝜉 ](1 + ) 𝑡 In order to nondimensionalize the boundary conditions (5) 𝛽 𝑑𝜂 and (6), it is pertinent to note that the minimum value of 2 2 𝑦 is not the starting point of the slot. This implies that all 1 𝑑 𝑓 𝑑 𝑓 2𝑚 (13) −𝜉 (1 + ) +𝑓 − the conditions in (5) are not imposed at 𝑦=0 .Asshown 2 2 𝛽 𝑑𝜂 𝑑𝜂 𝑑𝜂 𝑚+1 𝑑𝜂 𝑑𝜂 in Figures 1(a) and 1(b), it is obvious that it may not be 2𝐻 realistic to say that 𝑦=0 at all points on upper horizontal + +𝐺 [(1−𝜃 )𝜉−𝜉𝑆 ]=0, 𝑡 𝑡 melting surface of a paraboloid of revolution. Hence, it is 𝑚+1 𝑑𝜂 not valid to set 𝑦=0 in similarity variable 𝜂 .Uponusing (1−𝑚)/2 𝑑 𝜃 1−𝑚 𝑦=𝐴(𝑥+)𝑏 the minimum value of 𝑦 which accurately [1+𝜀𝜃 ] +𝜀 −𝑃 𝑆 +𝑃 𝑓 𝑟 𝑡 𝑟 corresponds to minimum value of similarity variable 𝑑𝜂 𝑑𝜂 𝑑𝜂 𝑚+1 𝑑𝜂 𝑑𝜂 1/2 𝑚+1 𝑈 1−𝑚 𝑜 (17) 𝜂=𝐴 ( ) =𝜒. −𝑃 𝜃 2 𝜗 𝑚+1 𝑑𝜂 𝑑𝑓 𝑑𝜃 𝑑𝑓 𝑑𝜃 𝑑𝜃 𝑑𝑓 𝑑𝑓 𝑑𝑓 𝑑𝜃 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝑓𝑥 𝜕𝑦 𝜎𝐵 𝑓𝑥 6 Journal of Applied Mathematics 2 2 (1−𝑚)/2 1 𝑑 𝐹 𝑑 𝐹 This implies that, at the surface (𝑦 = (𝑥 𝐴 + ) 𝑏 ),the 2 −𝜉𝑆 ] (1 + ) +𝑃 [1 +𝜉−Θ𝜉 𝑡 𝑟∞ 2 2 boundary condition suitable to scale the boundary layer flow is 𝜂=𝜒 . eTh boundary condition becomes 2Γ [−𝑛𝜍] −𝜉𝑆 ] 𝑒 =0. 𝑚+1 =0, (22) Due to the fact that Θ(𝜍) = 0, the influence of temperature- 𝛿 1+𝜃𝜀 𝑚−1 [ ] 𝑓 (𝜒) + +𝜒 =0, dependent thermal conductivity on heat conduction during (18) 𝑃 𝑚+1 melting process diminishes. Dimensionless boundary condi- tion reduces to 𝜃 (𝜒) = 0 at 𝜒=𝜂. =0, → 1, 𝛿 𝑑Θ 𝑚−1 𝐹 (𝜍 )+ +𝜒 =0, (23) 𝑃 [1 +𝜉−𝜉𝑆 ] 𝑚+1 𝑟∞ 𝑡 (19) 𝜃 (𝜒)→ 1 − 𝑆 Θ 𝜍 =0 ( ) as 𝜒→∞. at 𝜍=0. Moreover, dimensionless governing equations (13) and (14) are depending on 𝜂 while the boundary conditions (18) and → 1, (19) are functions and/or derivatives depending on 𝜒 . In order (24) to transform the domain from [𝜒, ∞) to [0, ∞)it is valid to Θ (𝜍 ) → 1 − 𝑆 adopt 𝐹(𝜍) = 𝐹(𝜂−𝜒) = 𝑓(𝜂) and Θ(𝜍) = Θ(𝜂 − ) 𝜒 = 𝜃(𝜂) ; for more details, see Figure 1(b). Considering the fact that as 𝜍→∞. Prandtl number is strongly dependent on plastic dynamic Upon substituting the similarity variables (12) and models of viscosity and thermal conductivity, and it is assumed that physical quantities (i.e., 𝐶 and Nu )atthe wall into (16) we both properties vary linearly with temperature; hence for obtain more accurate analysis of boundary layer as suggested in [47, 48], the 𝑃 in (14) is 1/2 Re 𝐶 =(1 + )𝐹 (0), ∗ 𝛽 (25) 𝜇 𝐶 𝜇 𝐶 𝑏 𝑝 𝑏 𝑝 𝑃 = =[1 + 𝜉 − 𝜃𝜉 − 𝑆𝜉 ] 𝑟 𝑡 ∗ ∗ −1/2 𝜅 𝜅 Nu Re =−Θ (0). 𝑏 𝑏 (20) 𝑥 𝑥 =𝑃 [1 + 𝜉 − 𝜃𝜉 − 𝑆𝜉 ] 𝑟∞ 𝑡 3. Numerical Solution Equation (18) reveals that Prandlt number at free stream is Numerical solutions of the boundary valued problem (21)– denoted as 𝑃 . eTh na fi l dimensionless governing equation 𝑟∞ (24) are obtained using classical Runge-Kutta method with (coupled system of nonlinear ordinary differential equation) shooting techniques and MATLAB package (bvp5c). The is boundary value problem cannot be solved on an inn fi ite interval and it would be impractical to solve it for even a 1 𝑑 𝐹 very large n fi ite interval; hence 𝜍 at inn fi ity is 10.Using the [1 +𝜉−Θ𝜉−𝜉𝑆 ](1 + ) −𝜉 (1 method of superposition by Na [49], the boundary value problem of ODE has been reduced to a system of vfi e 2 2 1 𝑑Θ 𝑑 𝐹 𝑑 𝐹 2𝑚 simultaneous equations of rfi st order (IVP) for vfi e unknowns (21) + ) +𝐹 − 2 2 𝛽 𝑚+1 following the method of superposition. In order to integrate the corresponding IVP, the values of 𝐹 (𝜍 = 0) and Θ (𝜍 = 2𝐻 0)are required. However, such values do not exist aer ft the + +𝐺 [(1−Θ)𝜉−𝜉𝑆 ]=0, 𝑡 𝑡 𝑚+1 nondimensionalization of the boundary conditions (5) and (6). The suitable guess values for 𝐹 (𝜍 = 0) and Θ (𝜍 = 0) 𝑑 Θ 𝑑Θ 𝑑Θ [1+𝜀Θ ] +𝜀 +[1+𝜉−Θ𝜉 −𝜉𝑆 ] are chosen and then integration is carried out. The calculated values of 𝐹 (𝜍) and Θ(𝜍) at inn fi ity ( 𝜍=10 )are compared with the given boundary conditions in (24) and the estimated 1−𝑚 𝑑Θ ⋅(−𝑃 𝑆 +𝑃 𝐹 𝑟∞ 𝑡 𝑟∞ values 𝐹 (𝜍 = 0) and Θ (𝜍 = 0) areadjustedtogiveabetter 𝑚+1 approximation for the solution. Series of values for 𝐹 (𝜍 = 0) 1−𝑚 𝑚+1 and Θ (𝜍 = 0) are considered and applied with fourth-order −𝑃 Θ )+𝑃 𝐸 [1 + 𝜉 − Θ𝜉 𝑟∞ 𝑟∞ 𝑐 classical Runge-Kutta method using step size Δ𝜍 = 0.01 . 𝑚+1 2 𝑑𝜍 𝑑𝐹 𝑑𝜍 𝑑𝜍 𝑑𝐹 𝑑𝜍 𝑑𝜍 𝑑𝜍 𝑑𝜍 𝑑𝐹 𝑑𝜍 𝑑𝜍 𝑑𝜍 𝑑𝜍 𝑑𝜍 𝑑𝐹 𝑑𝐹 𝑑𝜍 𝑓𝑥 𝑓𝑥 𝑑𝜍 𝑑𝐹 𝑑𝜍 𝑑𝜍 𝑑𝜒 𝑑𝐹 𝑑𝑓 𝑑𝜍 𝑑𝐹 𝑑𝜒 𝑑𝜒 𝑑𝑓 𝑑𝜃 𝑑𝜒 𝑑𝑓 𝑑𝜍 𝑑𝜍 Journal of Applied Mathematics 7 Table 1: Validation of numerical technique: comparison between the solutions of classical Runge-Kutta together with shooting (RK4SM) and MATLAB solver bvp5c for the limiting case. 𝑆 𝐹 (𝜍 = 0) (RK4SM) 𝐹 (𝜍 = 0) (bvp5c) Θ (𝜍 = 0) (RK4SM) Θ (𝜍 = 0) (bvp5c) 0 0.140452622100684 0.140452622451879 2.244340556008000 2.244340556781432 0.5 0.140381952095092 0.140381952471126 2.090185332523411 2.090185332413216 1 0.140582481552633 0.140582481451697 1.932324537463769 1.932324537412539 eTh above procedure is repeated until asymptotically con- 8 −6 verged results are obtained within a tolerance level of 10 .It At metalimnion level of thermal is very important to remark that setting 𝜍 =10, all profiles stratification S = 0.5 ∞ t are compatible with the boundary layer theory and asympto- tically satisfies the conditions at free stream as suggested by fi Pantokratoras [50]. It is worth mentioning that there exist no relatedpublished articles that canbeusedtovalidatethe 4 accuracy of the numerical results. In view of this, (21)–(24) can easily be solved using ODE solvers such as MATLAB’s bvp5c as explained in Kierzenka and Shampine [51] and Gok ¨ han [52]. 3.1. Verification of the Results. In order to verify the accuracy of the present analysis, the results of classical Runge-Kutta 0 2 4 6 8 10 together with shooting have been compared with that of Dimensionless distance (𝜍) bvp5c solution for the limiting case when 𝜉 = 0.07 , 𝜀 = 0.1 , 𝛽 = 0.1 , 𝐸 = 0.3, 𝐻 = 0.25, Γ=1, 𝑛 = 0.09 , 𝑃 =1, 𝑚= m = 0.05 m = 0.15 𝑐 𝑎 𝑟∞ m = 0.1 m = 0.2 0.17, 𝛿 = 0.2 , 𝜒 = 0.3 ,and 𝐺 =1atvariousvaluesof 𝑆 within 𝑡 𝑡 the range 0≤𝑆 ≤1. As shown in Table 1, the comparison in Figure 2: Effect of 𝑚 on 𝐹(𝜍) . the above case is found to be in good agreement. This good agreement is an encouragement for further study of the effects of other parameters. 4. Results and Discussion 0.8 eTh numerical computations have been carried out for vari- fi ousvaluesofmajor parameters usingthe numericalscheme 0.6 discussed in the previous section. This section presents the effect of different embedded physical parameters on the flow. Following Mustafa et al. [53], the ratio of momentum 0.4 diffusivity to thermal diffusivity is considered to be unity (i.e., 𝑃 =1) due to the fact that Casson uid fl flow under 𝑟 At metalimnion level of thermal 0.2 consideration possesses substantial yield stress. stratification S = 0.5 4.1. Inu fl ence of Velocity Index Parameter 𝑚 and Eckert 0 0 2 4 6 8 10 Number 𝐸 . At metalimnion level of thermal stratica fi tion Dimensionless distance (𝜍) (𝑆 = 0.5), when 𝜉 = 0.07 , 𝜀 = 0.1 , 𝛽 = 0.1 , 𝐸 = 0.3, 𝐻 = 𝑡 𝑐 𝑎 0.25, Γ=1, 𝑛 = 0.09 , 𝑃 =1, 𝛿 = 0.2 , 𝜒 = 0.3 ,and 𝐺 = m = 0.05 m = 0.15 𝑟∞ 𝑡 1, it is worth noting that both vertical and horizontal velo- m = 0.1 m = 0.2 cities decrease with 𝑚 ;see Figures2and3.Combination Figure 3: Effect of 𝑚 on 𝐹 (𝜍). of these practical meanings of velocity index parameter 𝑚 justifies the decrease in both vertical and horizontal velocities profiles we obtained. Meanwhile, temperature distribution Θ(0) = 0and as 𝜍→10 , Θ(𝜍) → 1 − 𝑆 . In addition, within the flow increases with 𝑚 from a few distance after 𝑡 graphical illustration of the function which describes the wall till free stream; see Figure 4. With an increase in the immediate layer of Casson fluid next to an upper horizontal magnitude of 𝑚 , Figure 5 reveals that temperature gradient (1−𝑚)/2 profile Θ (𝜍) increases near the wall 0 ≤ 𝜍 ≤ 3.1 and decreases surface of a paraboloid of revolution 𝑦=𝐴(𝑥 + )𝑏 thereaeft r as 𝜍→10 . It is very important to notice that velo- against 𝑥 (i.e., 𝐴=1 and 𝑏 = 0.8 )atvarious values of 𝑚 city profile when 𝑚 = 0.15 perfectly satisfies free stream less than 1 and stretching velocity at the wall 𝑈 =𝑈 (𝑥 + 𝑤 𝑜 condition asymptotically. This can be traced to the fact that 𝑏) when stretching rate (i.e., 𝑈 =3) are presented in Horizontal velocity proles F (𝜍) Vertical velocity proles F() 8 Journal of Applied Mathematics −1/2 Table 2: Variations in Nusselt number Nu Re with Eckert number at various values of velocity index parameter when 𝜉 = 0.07 , 𝜀 = 0.1 , 𝑥 𝑥 𝛽 = 0.1 , 𝐸 = 0.3, 𝐻 = 0.25, Γ=1, 𝑛 = 0.09 , 𝑃 =1, 𝑆 = 0.5, 𝛿 = 0.2 , 𝜒 = 0.3 ,and 𝐺 =1. 𝑐 𝑎 𝑟∞ 𝑡 𝑡 −1/2 −1/2 −1/2 −1/2 Nu Re Nu Re Nu Re Nu Re 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 When 𝑚 = 0.05 When 𝑚 = 0.15 When 𝑚 = 0.25 When 𝑚 = 0.35 0 −1.9834 −2.0331 −2.0471 −2.0414 1 −2.2543 −2.2092 −2.1769 −2.1460 2 −2.4838 −2.3594 −2.2869 −2.2333 3 −2.6799 −2.4879 −2.3799 −2.3055 4 −2.8485 −2.5981 −2.4582 −2.3646 5 −2.9940 −2.6924 −2.5237 −2.4120 At metalimnion level of thermal At metalimnion level of thermal stratification S = 0.5 stratification S = 0.5 1.5 1.5 fi fi 0.5 0.5 −0.5 0 2 4 6 8 10 0 2 4 6 8 10 Dimensionless distance (𝜍) Dimensionless distance (𝜍) m = 0.05 m = 0.15 m = 0.05 m = 0.15 m = 0.1 m = 0.2 m = 0.1 m = 0.2 Figure 4: Effect of 𝑚 on Θ(𝜍). Figure 5: Effect of 𝑚 on Θ (𝜍). Figure 6. It is noticed that as 𝑚 increases within 0.05 ≤ 𝑚 ≤ 0.2, the thickness of the paraboloid of revolution decreases wall. Figure 10 reveals that a distinct significant increase in 1/2 but corresponding influence of stretching on the flow is Re 𝐶 is guarantee with an initial increase in 𝛽 from 0.2 to an increasing function. eTh variation in local skin friction 0.25. Physically, increase in the magnitude of non-Newtonian coefficient and local Nusselt number which is proportional to Casson parameter (𝛽→∞ implies sharp transition in the localheattransferrateasstatedin(25)asafunctionofviscous flow behavior from non-Newtonian u fl id flow to Newtonian dissipation term and velocity index parameter is shown in u fl id flow. In view of this, resistance in the u fl id flow is Figure 7 and Table 2. Figure 7 shows that, at a xfi ed value of 𝑚 , produced. It is worth mentioning that an increase in 𝛽 implies 1/2 Re 𝐶 decreases with Eckert number 𝐸 .Atconstantvalue 𝑥 𝑐 adecreaseinyield stress 𝑃 of theCassonfluidand increase 1/2 in the magnitude of plastic dynamic viscosity 𝜇 . of 𝐸 ,unequal decrease in Re 𝐶 with 𝑚 is also observed. 𝑐 𝑏 −1/2 It is observed that the present study complements related In addition, Nu Re decreases with 𝐸 at various values of 𝑥 𝑥 𝑐 studies on Casson uid fl flow with temperature-dependent −1/2 𝑚 .Table 2shows that Nu Re increases highly significant plastic dynamic viscosity on nonmelting surface; see Figures when magnitude of 𝐸 is large. 8 and 9 in [41], Figures 3 and 4 in [54], and Figure 2 reported by Jasmine Benazir et al. [55]. The relationship between non- 4.2. Inu fl ence of Non-Newtonian Casson Parameter 𝛽 ,Thermal Newtonian Casson parameter 𝛽 and Eckert number is sought Straticfi ation Parameter 𝑆 , and Eckert Number 𝐸 . Figures 𝑡 𝑐 for and illustrated graphically in Figures 13 and 14. Within 8–12 illustrate the eeff ct of increasing the magnitude of non- 1/2 0≤𝐸 ≤ 1.5, there exists no significant difference in Re 𝐶 Newtonian Casson parameter 𝛽 on all the vfi e profiles when 𝑥 with 𝛽 . As shown in Figure 13, when the magnitude of 𝛽= 𝜉 = 0.07 , 𝜀 = 0.1 , 𝐸 =2, 𝐻 = 0.25, Γ=1, 𝑛 = 0.09 , 𝑃 =1, 𝑐 𝑎 𝑟∞ 1/2 0.25, a distinct significant increase in Re 𝐶 is observed 𝑚 = 0.35 , 𝑆 = 0.5, 𝛿 = 0.2 , 𝜒 = 0.3 ,and 𝐺 =1.With 𝑡 𝑡 duetoanincreaseinthe magnitudeofviscous dissipation an increase in the magnitude of 𝛽 ,itisobservedthatvertical parameter. At small magnitude of 𝛽 ,local Nusseltnumber velocity decreases, horizontal velocity decreases, temperature −1/2 distribution increases only within the u fl id domain ( 2≤ (Nu Re ) which is proportional to local heat transfer rate 𝜍≤8 ), and the temperature gradient increases only near the is found to be decreasing with 𝐸 .Atlarge valueof 𝛽 , Temperature proles Θ(𝜍) Temperature gradient proles Θ (𝜍) 𝑓𝑥 𝑓𝑥 𝑓𝑥 𝑓𝑥 𝑓𝑥 Journal of Applied Mathematics 9 (ii) Different patterns of stretching velocity at 4.5 the wall when U = 3, b = 0.5 𝜍 4 3.5 fi m 𝛽 2.5 1.5 (i) Different profiles of (1−m)/2 1 0 y| =A(x + b) min 0.5 0 2 4 6 8 10 0 2 4 6 8 10 Dimensionless distance (𝜍) Spatial x-direction 𝛽 = 0.2 𝛽 = 0.3 (1−𝑚)/2 Figure 6: Graphical illustrations of (i) 𝑦 =𝐴(𝑥 + )𝑏 ; (ii) min 𝛽 = 0.25 𝛽 = 0.35 stretching wall velocity 𝑈 =𝑈 (𝑥 + ) 𝑏 at various values of 𝑚 . 𝑤 𝑜 Figure 8: Effect of 𝛽 on 𝐹(𝜍) . 0.22 0.2 0.8 0.18 fi 0.16 0.6 0.14 0.4 0.12 0.1 0.2 0.08 0.06 0 2 4 6 8 10 0 1 2 3 4 5 Viscous dissipation parameter E Dimensionless distance (𝜍) 𝛽 = 0.2 𝛽 = 0.3 m = 0.05 m = 0.25 𝛽 = 0.25 𝛽 = 0.35 m = 0.15 m = 0.35 1/2 Figure 9: Effect of 𝛽 on 𝐹 (𝜍). Figure 7: Variations in Re 𝐶 with 𝐸 at various values of 𝑚 . 5. Conclusion −1/2 Nu Re increases with 𝐸 ;see Figure 14.Thesimulation 𝑥 𝑥 𝑐 eTh boundary layer analysis of non-Newtonian Casson fluid was further extended to unravels the relationship between flow over a horizontal melting surface embedded in a ther- non-Newtonian Casson parameter 𝛽 , thermal stratification mally stratified medium in the presence of viscous dissipation parameter 𝑆 , and local skin friction coefficients when 𝜉= internal space heat source has been investigated numerically. 0.07, 𝜀 = 0.1 , 𝐸 =2, 𝐻 = 0.25, Γ=1, 𝑛 = 0.09 , 𝑃 =1, 𝑐 𝑎 𝑟∞ The effects of the velocity power index, melting parameter, 𝑚 = 0.17 , 𝛿 = 0.2 , 𝜒 = 0.3 ,and 𝐺 =1.Itisrevealedin temperature-dependent viscous parameter, Eckert number, 1/2 Figure 15 that Re 𝐶 increases with 𝛽 at epilimnion stage thermal conductivity, and magnetic interaction parameter which is known as the highest and warmest layer (𝑆 =0). In were examined. Conclusions of the present analysis are as 1/2 addition, a significant decrease in Re 𝐶 is observed with follows: an increase in 𝛽 at hypolimnion stage which can be referred to as the coolest layer. Mathematically, when 𝑆 =0,this (1) Increaseinthe magnitudeofvelocityindex param- implies that Θ(𝜍) = 1 and maximum wall temperature at eter leads to a decrease in velocity and increase in 1/2 the wall explains the increase in Re 𝐶 since increase in temperature due to combine practical influence of the 𝛽 corresponds to a decrease in yield stress 𝑃 . parameter. (1−m)/2 1/2 (i) y =A(x + b) and Re C min x fx (ii) Stretching wall velocity U =U (x + b) w o Horizontal velocity proles F (𝜍) Vertical velocity proles F() 𝑓𝑥 𝑓𝑥 𝑓𝑥 𝑓𝑥 10 Journal of Applied Mathematics 0.18 2.5 0.16 0.14 1.5 fi 0.12 fi 0.1 0.08 0.5 0.06 0.04 −0.5 0.02 −1 0 2 4 6 8 10 0 2 4 6 810 Dimensionless distance (𝜍) Dimensionless distance (𝜍) 𝛽 = 0.2 𝛽 = 0.3 𝛽 = 0.2 𝛽 = 0.3 𝛽 = 0.25 𝛽 = 0.35 𝛽 = 0.25 𝛽 = 0.35 Figure 12: Effect of 𝛽 on 𝐹 (𝜍). Figure 10: Eeff ct of 𝛽 on 𝐹(𝜍) . 0.1 3.5 0.08 2.5 0.06 fi 0.04 1.5 0.02 0.5 −0.02 0 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 810 Viscous dissipation parameter E Dimensionless distance (𝜍) 𝛽 = 0.15 𝛽 = 0.2 𝛽 = 0.3 𝛽 = 0.2 𝛽 = 0.25 𝛽 = 0.35 𝛽 = 0.25 1/2 Figure 11: Effect of 𝛽 on 𝐹 (𝜍). Figure 13: Variations in Re 𝐶 with 𝐸 at various values of 𝛽 . 𝑥 𝑐 −1/2 (5) In the case of Casson u fl id flow over an upper hori- (2) Nu Re decreases with 𝐸 at various values of 𝑚 𝑥 𝑐 1/2 zontal thermally stratified melting surface of a para- within the interval 0.05 ≤ 𝑚 ≤ 0.35 .Re 𝐶 boloid of revolution, decrease in horizontal velocity is decreases with 𝐸 and 𝑚 . guaranteed with an increase 𝑚 and 𝛽 . (3) At various values of viscous dissipation within 0≤ (6) With an increase in the magnitude of 𝑚 ,the inufl ence 𝐸 ≤4 anddue to thenatureofthe flowpastmelting 𝑐 of stretching velocity at the wall 𝑈 =𝑈 (𝑥 + ) 𝑏 on 𝑤 𝑜 surface, a valid non-Newtonian parameter falls within horizontal andverticalvelocitiesisstrongerthanthat (1−𝑚)/2 the interval 0.000001 ≤ 𝛽 < 0.21 .EquivalentNewto- of 𝑦 = ( 𝐴 𝑥+𝑏) which describes the immediate nian u fl id flow is guaranteed for 𝛽 ≥ 0.23 . u fl id’s layer next to upper horizontal surface of a para- boloid of revolution due to melting heat transfer. 1/2 (4) Local skin friction coefficient Re 𝐶 increases negligible with 𝑆 when magnitude of 𝛽 is small. A An extension of the present study to the case of Williamson 1/2 significant increases in Re 𝐶 is guaranteed with 𝑆 and Prandtl u fl id flow over an upper horizontal thermally when magnitude of 𝛽 is large. stratified melting surface of a paraboloid of revolution Shear stress proles F (𝜍) Temperature proles Θ(𝜍) 1/2 Re C Temperature gradient proles Θ (𝜍) x fx 𝑓𝑥 𝑓𝑥 𝑓𝑥 𝑓𝑥 Journal of Applied Mathematics 11 𝑞 :Heattransfer 𝑇 : Dimensional u fl id temperature −2.05 𝑈 :Stretchingvelocityatthewall 𝑏 :Thermalpropertyoftheuflid −2.1 𝐸 :Eckertnumber 𝑓, 𝐹 : Dimensionless vertical velocity 𝜅 : eTh rmal conductivity of the Casson fluid −2.15 𝑏 𝑚 : Velocity power index 𝑇 :Meltingtemperatureatwall 𝛽 = 0.2 −2.2 𝛽 = 0.15 𝑇 :AmbientTemperature Re : Local Reynolds number −2.25 𝛽 : Non-Newtonian Casson parameter 𝜉 : Wall thickness parameter 𝛽 = 0.25 𝜇 : Plastic dynamic viscosity −2.3 𝜆 : Latentheatofthe ufl id 0 1 2 3 4 𝛿:Meltingparameter Viscous dissipation parameter E 𝜃, Θ : Dimensionless temperature −1/2 Figure 14: Variations in Nu Re with 𝐸 at various values of 𝛽 . 𝑥 𝑐 𝜗 : Kinematic viscosity 𝜉 : Temperature-dependent viscosity parameter 0.09 𝜀 : Temperature-dependent thermal conductivity parameter 𝜌 : Density of Casson fluid 0.085 𝜎 : Electrical conductivity of the u fl id 𝜓(𝑥, 𝑦) :Streamfunction. 0.08 Competing Interests 0.075 eTh authors declare that they have no competing interests. 0.07 References [1] L. 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