NUMERICAL STUDY OF THERMAL RADIATIONS AND THERMAL STRATIFICATION MECHANISMS IN MHD CASSON FLUID FLOW Khalil Ur REHMAN b c * Noor Ul SABA b Iffat ZEHRA c Muhamma Yousaf MALIK ab an Sarar Muhamma BILAL c a Department of Mathematics Faculty of Science King Abulaziz University PO Box-8 Jeah 589 Saui Arabia b Department of Mathematics Quai-i-Azam University Islamaba 44 Pakistan c Department of Mathematics Air University PAF Complex E-9 Islamaba 44 Pakistan * Corresponing author; E-mail: krehman@math.qau.eu.pk The features of Casson liqui when flow fiel is thermally stratifie are offere in this article. The flow is taken through an incline cyliner with both MHD an thermal raiations assumptions. A mathematical moel is constructe in terms of ifferential equations via funamental laws. Since the resultant system is non-linear so a self-coe computational algorithm is implemente to report numerical solution. The obtain observations in this regar are presente by way of both tabular an graphical trens. It is notice that the Casson flui temperature is ecreasing function of thermal stratification parameter while opposite tren is observe via thermal raiation parameter. Moreover the cylinrical geometry amits enlarge variations towars involve physical parameters as compare to flat surface. Key wors: Casson flui thermal raiations temperature stratification MHD heat generation heat absorption an incline surface. Introuction An analysis on non-newtonian fluis has always become a topic of great interest among the researchers because of its wie range of uses in engineering an inustry. The flui moel use in this analysis is Casson flui an it has huge applications in the fiel of foo processing metallurgy an bioengineering operations to mention just a few. The Casson flui moel has got popularity because of the success of the experimental an theoretical investigations see [-]. In 959 the Casson flui was first ientifie by Casson. The Casson flui is well-efine as a shear thinning flui having an infinite viscosity at zero shear rate []. If a shear stress applie to the flui is less than the yiel stress Casson turns into a soli but it starts moving for the case greater shear stress as compare to yiel stress. By keeping in view all these aspects the Casson flui flow via various geometries is iscusse by the researchers like Kameswaran et al. [4] aresse the ual solution of Casson flui flow yiele by flat surface. The time epenent flow of Casson flui was reporte by Khali et al. [5]. Animasaun [6] offere non-darcy flow of Casson liqui manifeste with n th orer chemical reaction. By consiering the Casson flui moel the effects of ferrous nanoparticles was stuie by Raju an Saneep [7].
Recently Nawaz et al. [8] worke on variable thermal conuctivity aspects subject to Casson flui moel. The recent evelopments regaring the Casson flui flow can be assesse in [9-]. The flui having interaction with magnetic fiel is terme as magnetohyroynamic flui (MHD) flow. Such combination claims significant number of applications like woun healing magnetic reason imaging (MRI) an canter action causing hypothermia etc. Even the metals fusion an cooling process involve the use of externally applie magnetic fiel. Owning the importance of MHD flows various efforts are share by investigators like Chakrabarti an Gupta [] stuie MHD flow of Newtonian flui moel. The non-newtonian flui flow along with MHD effect was iscusse by Akbar et al. []. Since than one can assesse the literature regaring MHD analysis in [-5]. The effects of thermal raiation claims a vital role in the inustrial an engineering processes. These processes inclue performance at an extreme temperature uner ifferent non-isothermal conitions an instance where convective heat transfer coefficient are lower. Thermal raiations are type of electromagnetic raiations that are emitte in the form of energy. These raiations travel with a spee equal to the spee of light an o not require any type of meium for their propagation. The effect of thermal raiation parameter on MHD flow towars heate surface was given by Rahman an Salahuin [6]. Many researchers [7-9] acknowlege the importance of contribution of thermal raiations mixe convection an suggeste efinitive significances. From the above mentione limite literature stuy it is etermine that few attempts are available to encountere the Casson flui flow towars a cylinrical surface. To be more precise the combine aspects of thermal raiations an stratification phenomena in a magnetize flow fiel is not iscusse yet. An attempt is attractive in this sense it contains simultaneous analysis for both an incline cyliner an flat geometry. The flow is accomplishe by consiering no slip conitions. The physical effects involve in this paper inclues stagnation point mixe convection magnetic fiel an heat generation/ absorption. A numerical solution is presente by means of shooting metho an ultimate finings are reporte with the ai of both graphs an tables.. Mathematical formulation In the present work magnetohyroynamic incompressible bounary layer flow of a Casson flui uner the region of stagnation point is consiere. The no slip conition is applie on the flui flow that is the stretching velocity of geometry resemble with the velocity of the flui particles. In aition heat generation thermal raiation an stratification phenomenon mixe convection effects are also taken into account. It is important to note that the estruction of fluctuation velocity graients by the action of viscous stresses in a laminar bounary layer flow of Casson flui is assume to be small so that the viscous issipation is ignore [9]. Near the cylinrical surface the strength of temperature is higher than that of ambient flui. For the geometrical representation we have taken the cyliner axial line along X -axis an the raial irection is ajuste normal to the flui flow ( R -axis). The most acknowlege ifferential equations in the fiel of flui science are the energy an the momentum equations. They are enough to emonstrate the flow fiel properties. So that the complete moel of these equations [9-] base on the bounary layer approximations are given as: RU RV X R ()
U U U U Ue B U V U e U U X R R R R X T g T T cos T T T T Q U V Rq T T X R R R R cpr R c 4 4 T here q R k R reorganize Eq. () as: ( ) ( ) R p represents the Rosselan raiative heat flux. Accoringly we can 4 4 Q ( ) ( ) p p T T T T T U V R T T X R R R R c R k R R c with: U cx U U X X V T X R T X T R b L L U U Ue X T ( X R) T ( X ) T X as R L L ( ) ( ) w( ) at here V ( X R)an U( X R ) are velocity components in R an X irection respectively. Moreover c L Tw ( X ) Q T T cp T B g Ue an represents imensionless constants reference length arbitrary surface temperature heat generation coefficient ambient temperature flui temperature specific heat capacity at constant pressure thermal iffusivity an inclination thermal expansion coefficient acceleration ue to gravity uniform magnetic fiel electrical conuctivity free stream velocity Casson flui parameter flui ensity an kinematics viscosity respectively. For the solution of Eq. ()-(4) along with the bounary conitions given by Eq. (5) we have assume the transformation [9- -] UX b U R b U U F V F L R L b L U X T T bf T( ) L T T where w T F F b an U represents stream function reference temperature flui velocity imensionless variable raius of cyliner an reference velocity respectively. The Eq. () ientically fulfills an acknowlege by the stream function are: U V R R R X 7) use of Eq. (6) into Eqs. ()- (5) results: ( ) ( ) F( ) ( A) A T cos F( ) F( ) F( ) F( ) K K F e () () (4) (5) 6) 8)
with: T T K 4R 6K 4R T F Pr F QT F F F T at A T when here Q Pr R m A an K heat generation/absorption thermal stratification Prantl number thermal raiation mixe convection velocities ratio magnetic fiel an curvature parameters respectively an they are escribe as: L B L Gr 4 T U K R A R U U k k U ReX gt Tw T X LQ Pr Gr an Q. c U c At the cylinrical surface the skin friction coefficient (SFC) is written as: w U CF w U ) R Rb where an w expresses the flui viscosity an the shear stress corresponingly. The non imensional expression we have: UX here Re X vl number (LNN) is given as: F C F X Re at escribes the local Reynols number. The expression for the local Nusselt Xq T NU q k q w X w R Rb k( Tw T ) R Rb the imensionless form is pre-arrange as: NU X 4 T R Re at. X p 9) ) ) ) 4) 5). Computational scheme For the implementation of the computational scheme firstly we have converte the partial ifferential equations into system of ODE s. To be more specific Eqs. (8)-(9) are couple nonlinear ODE s with Eq. () an are solve by making use of the shooting scheme [-]. For this purpose we have reuce the given equations into a system of five first orer ODE s by captivating: 4
5 F M F M F M M M 4 T M M 4 T by making the use of above replacements the ientical form of Eqs. (8)-(9) epening on the new variables is given by: M M 7) with: M M 5 M A A cos M 4 ( M ) MM KM ( ) M ( )( K) M 4 M 4 M Pr M M M M M QM 6K 4R M 5 K R 4 5 4 5 M M M M 4 M 5 8) where an are esignate as the initial guesse values. For the integration of Eq. (7) it must be manatory that we have: M 5 F T an M when 9) besie this we have observe that the two initial conitions explicitly M an M when are not known but we o have 5 M ( ) A an M 4( ) when. ) The integration of Eq. (7) are carrie in such a way that the Eq. () satisfies completely. 4. Results an iscussion 4.. Velocity profiles The computational algorithm is implemente with following values of involve parameters that is. A. K.. 45 or. Pr.7. Q. an R.. The numerical values of the SFC are provie with the help of Tables - for the positive values of ifferent parameters namely K A Q. In etail it is foun the SFC (in absolute sense) is foun to be an increasing function of K an while it shows an opposite attitue for A an. However the Table clearly epicts that the skin friction coefficient is inepenent of ifferent values of Q. In other wors the SFC shows constant values for ifferent positive values of Q. Negative sign involve in Table an physically represents the amount of rag force offere by 6) 5
Table. Variations in SFC via K an A. K A F.5CF Re F X... -.495 -.89... -.568 -.56... -.654 -.8... -.46 -.85..4. -.449 -.8498..6. -.4448 -.8896... -.495 -.89... -.4 -.6866...5 -.66 -.5 Table. Variations in SFC via an Q. Q F.5C F Re X F.4.. -.687 -.74.6.. -.69 -.84.8.. -.745 -.487... -.444 -.888..5. -.99 -.7986..7. -.94 -.7886...4 -.495 -.89...5 -.495 -.89...6 -.495 -.89 cylinrical surface to the Casson flui particles. Further the negative sign subject to Table an 4 inicate the rate of heat transfer normal to the cylinrical surface. The effects of various physical parameters namely K an A on the Casson flui velocity (CFV) are represente with the ai of Figs. -4. To be more specific in Fig. the impact of Casson flui parameter on the velocity profile is given. It has been observe through the figure that the velocity profile shows a ecrease in the behavior for both the surfaces by increasing the value of the. Fig. is plotte to observe the attitue of the K on CFV. It shows that the increasing values of the K represents an increase in the CFV for both surfaces (flat an cyliner). For the raius of curvature the curvature parameter shows a reverse attitue. An increment in curvature parameters brings ecreasing values of raius of cyliner. This reuces the contact surface area of surface with the Casson flui particles which yiels less resistance an as a results CFV increases. Fig. represents the impact of the on the velocity profile for the Casson flui for both surfaces. It has been observe that the higher values of the brings an increase in the CFV which is in fact ue to the appealing behavior of the thermal buoyancy forces. Fig. 4 represents the influence of the magnetic fiel parameter on the CFV. The increasing values of the magnetic fiel parameter brings a ecrease in the velocity profile for cylinrical an the flat surfaces. In real exercise by increasing the magnetic fiel parameter a resistive force calle the Lorentz force subsiize effectively which offers resistance to the flui particles an as a result of this the horizontal velocity of the flui ecreases. For Fig. 5 it is clear that the CFV is increasing function of velocity ratio parameter. 6
Table. Variations in LNN via K an A. K A T NU Re X 4 R T... -.479.485.4.. -.766.88.6.. -.9898.7... -.8.44... -.5.44..5. -.8.4... -.8.44... -.67.476... -.98.5 4.. Temperature profiles Table an Table 4 expresses the impact of K A Q on the LNN. It is notice that the LNN shows an inciting attitue towars the higher values of K A however it shows a ecline behavior towars increasing values of an Q. Table 4. Variations in LNN via an Q. Q T NU Re X 4 R T.7.. -.99.9.8.. -.9..9.. -.98.7... -.4.446..4. -.4.449..5. -.44.45... -.8.44...4 -.9.87...7 -.758.6 The influences of an involve parameters namely Q Q R K an Pr on the Casson flui temperature (CFT) are represente by means of Figs. 6-. In etail the behavior of the temperature istribution towars is ientifie in Fig. 6. It is notice that the temperature profile shows ecline curves for both the surfaces for increasing values of the. This actually happens ue to the rop in temperature ifference between ambient flui an surface of cyliner an hence the temperature profile shows a ecreasing values. Fig. 7 is plotte to observe the impact of the Q on the CFT. It is notice that the CFT increases while opposite behavior is observe for the heat absorption parameter epicte in Fig. 8. This all happens because heat energy is prouce via heat generation process that brings an improvement in temperature while in the case of heat absorption parameter heat energy is release an as a results ecrease in temperature istribution is witnesse for the Casson flui. Fig. 9 portrays the attitue of the temperature profile towars R. It is observe that for both 7
surfaces the temperature profile shows an inciting attitue towars the R. The large values of thermal raiation parameter correspons significant amount of transfer of heat so that the temperature of flow regime enhance. Fig. epicts the temperature variation towars the K. It is clearly observe that by increasing the values of the curvature parameter the temperature profile also shows an inciting behavior for cylinrical as well as the flat surface. The positive values of K reflects ecrease in raius of cyliner so that lesser resistance is face by Casson flui particles an average kinetic energy enhances so that the flui temperature shows inciting traits because Kelvin temperature is efine as an average kinetic energy. The influence of the temperature profile against the Pr is emonstrate in Fig.. It is clear from the figure that an increase in the Pr causes a strong reuction in the temperature istribution which makes the thermal bounary layer thin. As Pr amits inverse relation with the thermal conuctivity so increasing values of Pr correspons less iffusion of energy because of which a ecrease in the CFT is notice. The obtain results are valiate via comparison with existing literature. We foun an excellent match. Table 5 is constructe in this irection. Table. 5. Comparative values of Pr Biin an Nazar [] ( EK ) NU Re X X Mukhopahyay [] ( St S M ) T ' towars Pr. Present outcomes ( K Q A R )..9547.9547.9547..474.474.474..896.896.896 5. Graphical results.9.9.8.8 F' ( ).7.6.5 Green Curve = Flat Surface F' ( ).7.6.5.4 =....4 K =...4..... 4 5 6 Fig.. Impact of on CFV.. 4 5 6 Fig.. Impact of K on CFV. 8
.9.9.8.8.7.7 F' ( ).6.5 F' ( ).6.5.4 =...5.4 =..4.6..... 4 5 6 Fig.. Impact of on CFV.. 4 5 6 Fig. 4. Impact of on CFV..6.5.4 A =...7.9.8 F ' ( )....9 T ( ).7.6.5.4 =....8.7.6.5 A =..7.....9 4 5 6 4 5 6 Fig. 5. Impact of A on CFV. Fig. 6. Impact of on CFT..9.8.8.7.7.6.6 T ( ).5.4 T ( ).5.4. Q + = +. +. +.. Q - = -. -. -..... 4 5 6 Fig. 7. Impact of Q on CFT. 4 5 6 Fig. 8. Impact of Q on CFT. 9
.9.9.8.8.7.7 T ( ).6.5.4 T ( ).6.5.4. R =..4.5. K =...4.... 4 5 6 Fig. 9. Impact of R on CFT. 4 5 6 Fig.. Impact of K on CFT..9.8.7.6 T ( ).5.4.. Pr =..5.7. 4 5 6 Fig.. Impact of Pr on CFT. 6. Concluing remarks The article is mae to offer a numerical results on Casson flui flow towars both flat an cylinrical geometries. The key outcomes of the presents stuy are assemble as follow: The CFV shows ecline curves via both an. The CFV shows an inciting values for increasing values of both an K. The CFT shows increasing traits for the positive values of both the K an the R. The CFT istribution is foun to be ecreasing function towars Pr Q an but an inverse tren is seen for the Q. The cylinrical surface amits enlarge variations towars an involve physical parameters as compare to flat surface. References [] Batra R. L. Bigyani J. Flow of a Casson flui in a slightly curve tube International Journal of Engineering Science 9 (99) pp. 45-58.
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