Variable Viscosity on Unsteady Dissipative Carreau Fluid over a Truncated Cone Filled with Titanium Alloy Nanoparticles

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1 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 Variable Viscosity on Unsteady Dissipative Carreau Fluid over a Truncated Cone Filled with Titanium Alloy Nanoparticles CSK Raju 1, KR Sekhar, SM Ibrahim 3, G Lorenzini 4, G Viswanadha Reddy, E Lorenzini 5 1 Department o Mathematics, VIT University, Vellore, India Department o Mathematics, Sri Venkateswara University, Andhra Pradesh, India 3 Department o Mathematics, GITAM University, Visakhapatnam, India 4 Department o Engineering and Architecture, University o Parma, Parma, Italy 5 Department o Industrial Engineering, University o Bologna, Bologna, Italy Abstract: In this study, we proposed a theoretical investigation on the temperature dependent viscosity eect on magnetohydrodynamic dissipative nanoluid over a truncated cone with heat source/sink. The involving set o nonlinear partial dierential equations is transorming to set o nonlinear ordinary dierential equations by using selsimilarity solutions. The transormed governing equations are solved numerically using Runge-Kutta based Newton s technique. The eects o various dimensionless parameters on the skin riction coeicient and the local Nusselt number proiles are discussed and presented with the support o graphs. We also obtained the validation o the current solutions with existing solution under some special cases. The water based titanium alloy have a lesser riction actor coeicient as compared with kerosene based titanium alloy, whereas the rate o heat transer is higher in water based titanium alloy compared with kerosene based titanium alloy. From this we can highlight that depending on the industrial needs cooling/heating choose the water or kerosene based titanium alloys. Keywords: Viscous dissipation, Nanoluid, Temperature dependent viscosity, Truncated cone, Heat source/sink, Magneticield parameter I. INTRODUCTION Now days, it is well recognized that numerous luids have their own implication in transpiration structures, drug delivery system, electronics, uel cells, nuclear reactor heating up and down classiications and biological sensor systems etc. For physical as well as the industrial applications, the geometry o the problem demands vital role. In particular, the low over a cone has various engineering as well as biomedical applications such as aerosols engines, solar collectors, aeronautics, rotating heat exchangers, geophysics, heartbeat controlling systems and auto mobile industries etc. In view o this the revolution o laminar low over a cone was started Tien [1] in 1960 s. Later on, Lin and Rubin [] analyzed the three dimensional low due to cone illed with viscous nature at one incidence. The low o micropolar luid over a rotating cone with mixed convective conditions was illustrated by Subba and Gorla [3] and highlighted that rotation parameter improves the heat transer rate. Chamkha [4] studied the heat and mass transer characteristics o magnetohydrodynamic low over a cone with radiation. The variable viscosity and thermal radiation on magnetohydrodynamic low over a truncated cone in the presence o double-diusive convection was examined by Mahdy et al. [5]. The water based unsteady nanoluid low over a rotating vertical cone was investigated by Saleem et al. [6]. The researchers studied the heat and mass transer characteristics o Newtonian and non-newtonian luids with various low geometries and eects such as cross-diusion, convective conditions, with rotation or without rotation [7-1]. With that they concluded the rotation have a tendered to encourages the heat transer rate and cross diusion is controlled the mass transer rate.

2 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 A metallic solid or liquid that can be composed rom a homogeneous or non-homogeneous combination o two or more non-metals or metals or metalloid nanometer sized particles is called an alloy. Recent days, these can be extensively used or conveying the speciic physical characteristics o mixtures. The steel, phosphor bronze, gold, brass and solder are the examples o alloys. Alloys have variety o wide range applications such as aerospace science technology, hip joint replacement processes, advanced powder technology, surgical implantation and various biological treatments. The brie summary o alloys and its application are given [13-15]. Frequently these are used in abrication handling systems, construction o cold and hot rolling sheets etc. Finally, the combination o and alloys has various industrial as well science and engineering applications. These alloys supports the weight savings in place o aluminum and lesser strengthen aerospace type steel model. Later on, the researchers presented the experimental as well as theoretical investigations on titanium alloys or various low geometries and eects [16-19]. In heat transer equipment, luids are oten utilized as heat carriers. For instance, a heat transer luid plays crucial role in several continuous industrial applications like electronics industry, petroleum, polymer, automotive industry, etc. So the thermal conductivity o heat transer luids is important in the heat transer equipment. But conventional heat transer luids like water, ethylene glycol and oil are usually inadequate or heat transer. We know that solid metals like copper have more thermal conductivity than those conventional luids like water. Hence metallic liquids have more thermal conductivity equated to nonmetallic liquids. So researchers attempted to dispense solid metal/metallic-liquid particles in conventional luids to enhance thermal conductivity and heat transer rate. In 1881, Maxwell [0] discovered that the thermal conductivity o suspensions consist o spherical particles enhanced with the volume raction o the solid particles. Later several researchers continued this work. But in a small passage, the luid lows might have clogging problems when mini and micro sized particles mixed with the base luids. Thereore, very small quantity o particle is needed to dispense in the luids to get rid o clogging problems. Choi [1] introduced the term nanoluid which reers to the luid in which nanometer-sized (less than 50 nm) particles disseminated. But current technology is acceptable to manuacture nanoparticle o 10 nm. Later some authors measured the thermal conductivity o nanoluids using dierent methods [,3]. Recently, several researchers investigated the nanoluid low through dierent channels by considering water as a base luid and metallic particles like Cu, Cuo, Al O 3, TiO as nanoparticles by viewing various parameters [4-30]. Some o their discoveries are mixture o water and Al O 3 exhibited more enhancement compare to the mixture o water and Cuo, the Nusselt number raises with the rise in nanoparticle volume raction and enhancing thermal radiation parameter lessen the temperature o the nanoluid. The magnetohydrodynamic is the study o conducting liquids. It has various applications such as controller in MHD generators, nuclear reactor system, activating the blood cells by applying external magnetic ield etc. Keeping view into this the authors studied the magnetic ield with various eects o low controlling parameters like (magnetic Nano properties, cross-diusion, thermal radiation and non-darcy porous layer) and dierent geometries (cone, plate, wavy surace, sheet etc.) [31-35]. With this they highlighted that the magnetic ield act as a controller in the low ield. Similarly the heat source/sink is also is have signiicance in various heat treatment processes such as heat exchanger systems, sugar actories, airconditioning equipment s, crystallization process etc. Because o this signiicance the authors investigated the heat source or sink on low over various geometries (sheet, plate, wedge cone etc.) with various low controlling parameters (chemical reaction, dissipation, non-newtonian properties etc.) [36-41]. In that papers they concluded that the heat source/sink is useul in heat treatment controlling processes. In most o the real time applications combination o luids are very helpul to enhancing or decreasing the temperature as well as heat transer rate. It has various applications such as cooling organisms, heating developments and also biomedical as well as aerospace science and technologies. With the inspirations o the above stated applications and studies, this paper address the temperature dependent viscosity eect on dissipative unsteady magnetohydrodynamic nanoluid low over a truncated cone in the presence o heat source/sink. The set o nonlinear partial dierential equations are transorming to ordinary dierential equations and then ound the solution by using Runge-Kutta based shooting technique.

3 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 II. FORMULATION OF THE PROBLEM In this study, we consider the heat source/sink eect on unsteady dissipative nanoluid low due to a truncated cone in rotating rame with temperature dependent viscosity. We consider the rectangular curvilinear ixed coordinate system. The geometry o the problem is shown in Fig. 1. Let, x and y be the velocity components along the (tangential), (circumerential or azimuthal) and z (normal) directions, respectively. The hal angle o is. Both the luid and the cone are in a rigid body rotation about the axis o cone with timed dependent angular velocity. Their rotation is either in same or in opposite direction, which causes the unsteadiness in the luid low. The temperature variations are responsible or the buoyancy orces in the luid low. The surace o cone is an electrically-insulate. The low is considered to be axisymmetric. The magnetic Reynolds number is taken to be small such that the eect o induced magnetic ield is negligible. The wall temperature is a unction o x. According to above assumptions the boundary layer equations o momentum and energy or an incompressible, unsteady low Ching Yang Cheng [9], Patralescu et al. [11] and Anil kumar and Roy [8] are given by: Fig. 1. Physical coniguration. ( ru) ( rw) 0, x z u u u v u ve 3( n 1) u u n u w n g( ) n ( T T )sin B u, t x z x z z x z z v v v vu u dve n u w n n B v, t x z x z z dt T T T T u c p n u w k n n Q0 T T ( ) ( ), t x z z z (1) () (3) (4) With the boundary conditions: u v x t w z 1 0, 1sin (1 sin( ) ), 0, T=T w at 0, u v v x t (1 sin( ) ) 1 0, e sin( ), T T as z, Where u, v are the velocity components along the x, y directions respectively. n is the density o the nanoluid, g is the acceleration due to gravity, v is ree stream velocity, () n is the thermal expansion coeicient due to temperature (5)

4 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 dierence, is the electric conductivity o the luid, s is the electric conductivity o the solid, is the semi-vertical angle o the cone, v is the kinematic viscosity. n is the dynamic viscosity coeicient. Where (c p ) n is the heat capacity o the nanoluid, T is the luid temperature, T w, T are the luid temperatures near and the ar away rom the boundary, K n is the thermal conductivity o the nanoluid, c p is the speciic heat capacitance at constant pressure, Q 0 is the heat source/sink parameter. Here we assume that the viscosity changes with temperature in the orm as given by Ching Yang Cheng [9]. 1 ( T T ) (6) Where is viscosity o the ambient luid, is the constant, E=( T w-t ) is the viscosity variation parameter. To convert the nonlinear partial dierential equations into ordinary nonlinear dierential equations we introduce the selsimilarity transormations are given by: v x St v St z e 1 1/ 1/ 1/ sin (1 sin ), ( sin ) (1 sin ), u x St v x St g sin (1 sin ) ', sin (1 sin ), 1/ 1/ T T w sin (1 Stsin ),, T T w (7) Here s is the unsteady parameter, 1 and are the angular velocities o the cone and ree stream luid respectively, = 1 + is the composite angular velocity, Here in equation (5), u and v are automatically satisying the continuity equation. The nanoluid constants Buongiorno [30] are given by: n (1 ) s,( cp) n (1 )( cp) ( cp) s, n,.5 (1 ) kn ( ks k ) ( k ks ) 3( 1) s, n 1, k ( ks k ) ( k ks ) ( ) ( 1) (8) By using equation (5) and (6), the equations (1) to (4) are transormed as ollows: 1 1 ' 1 s ( n1) ''' '' g S ' S '' (1 ) (1.5 1) 3 We ''' '' 1 E (1 ) 1 ( ) s M ' ' '' (1 ) sin( ) 0, (1 E ) ( ) 1 1 s 1 '' ( ' ' ' ) (1 ).5 g S g g g g S S1 M ' ' g ' 0 1 E (1 ) (1 E ), (9) (10)

5 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 k ( c ) ' 1 1 k c E n p s '' QH Pr Pr (1 ) S( ' ') Ec '' 0,.5 ( p) (1 ) (1 ) The transormed boundary conditions are: (0) 0 '(0), g(0), (0) 1, '( ) 0, g( ) 1 1, ( ) 0, (1) 3 w sin 1 1 gt cos T T L L Gr,Re, L 1,Pr k /( cp), QH Q0(1 st sin ) / sin, 3 x sin Gr B0,, (1 sin ) / sin (1 st sin ) ReL We M st Where 1 is ratio o the angular velocity to composite velocity, Gr is the grasho number, Re L is the Reynolds number, is the buoyancy orce parameter,, s are the densities o the luid and solid particles respectively, T w -T >0 is or heated cone and T w -T <0 is or cooled cone. Where Pr = k /(c p ) is the Prandtl number, S is the unsteadiness parameter and Q H = Q 0 (1-stsin)/sin is the heat source/sink parameters. For physical quantities o interest, the riction actor coeicient and the rate o heat transer are given by: 1/ 4 1 C xgr (1 E)(1 ) 1/ 4 1 C ygr (1 E)(1 ) k 1/ 4 n Gr Nux ' 0. k.5.5 '' 0, g ' 0, (11) (13) (14) (15) III. METHOD OF SOLUTION The nonlinear dierential equations (9), (10) and (11) with the boundary conditions (1) are solved numerically using Runge-Kutta and Newton s methods. Initially, the set o nonlinear ordinary dierential equations converted to irst order dierential equations, by using the ollowing procedure: G ' y, G '' y3, H ' y5, G y1, H y4, y6, ' y7, s ( ) s A 1, B 1, ( ) 3( 1) ( c ) p s C 1, D 1, ( ) ( 1) ( cp) (16)

6 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 1/ y y1 y3 y4 S( y 1/ y3) A 1 3( n 1) ( y4 (1 1) ) G ''' 1/ We ''.5 (1 ) (1 Ey 6) yy 3 7 sin( ) By.5 6My (1 ) (1 ye 6 ).5 yy 5 7 H '' (1 ) (1 Ey6) Ay4 y5 y1 y5 yy4 S S1 MCy.5 4 (1 ) (1 ye 6 ) k yy '' Pr DS y6 y7 y1y7 QH Pr y6 Ecy.5 3 k n (1 ) (1 E) With boundary conditions as: y1 y 0, y, y 1, at y 0, y 1, y 0 at 4 6 (17) (18) (19) (0) We guess the values o y 3 (0),y 5 (0),y 7 (0) which are not given at the initial conditions. The equations (17)-(19) are integrated by taking the help o Runge-Kutta method with the successive iterative step length is For this we used ODE45 MATLAB solver to solve the irst order nonlinear coupled dierential equations. The correctness o the supposed values is checked by equating the calculated values y,y 4,y 6,y 8 at = max with their given values at = max. I there is any dierence exist the process is continued upto the required good values. Alternatively, we are using the Newton s Raphson method to get the accurately ound the initial values o y 3 (0),y 5 (0),y 7 (0) and then integrate Eq. (17)- (19) by using the Runge-Kutta Newton s Raphson method. This process is repeated until the settlement between the designed value and the condition given at is within the speciied degree o accuracy In order to validates the precision o the present solutions with Raju et al. [7], Anikumar and Roy [8] solutions. We ound worthy agreement with their solutions. IV. RESULTS AND DISCUSSION The nonlinearordinary dierential equations (9)-(11) together with boundary conditions (1) are solved numerically by using Runge-Kutta based shooting technique. The numerical computations have been carried out or dierent values o involving non-dimensional governing parameters. Results describe the eect o various non-dimensional governing parameters on the riction actor coeicient and local Nusselt number or with viscous variation and without viscous variation cases. For numerical solutions we have chosen the non-dimensional parameter values as M=0., =0.1, Q H =0.1, Ec=0., 1 =0., =/4, We=0.5, n=, =5, = these values are kept as ixed in entire study except the variations in the corresponding igures and tables. Skin Friction Factor Coeicients Distribution The variations o the non-dimensional governing parameters on riction actor coeicient or both the E=0 and E= cases were observed in Figs. -5. Fig. displays the buoyancy orce, volume raction o nanoparticle and Eckert number on riction actor coeicient or the E=0 and E= cases. It is noted that the rising values o buoyancy and volume raction o nanoparticles are improves riction actor coeicient (C x Gr 1/4 ) monotonically, whereas there is no change in with improving values o Eckert number. Generally, increasing values o buoyancy orce generates high pressure orces in the low. Due to this we saw improvement in riction actor coeicient. Similarly the rising value o volume raction o nanoparticles encourages the size o the particles, which can lead to improve the riction actor coeicient (C x Gr 1/4 ) or the E=0 and E= cases. Fig. 3 shows the eects o W e, and Q H on riction actor coeicient

7 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 or the E=0and E= cases. It is noted that the Weissenberg number and hal angle parameters are improves the riction actor coeicients (C x Gr 1/4 ), but heat source/sink parameter depreciates the riction actor coeicient. It is ound that the riction actor coeicient (C x Gr 1/4 ) is more in E=0 case when compared with E= case. From these results we can able to say that viscous variation parameter modulating the riction actor coeicients depending on the industrial needs. The eect o, Ec and We on riction actor coeicient (C y Gr 1/4 ) is plotted in Fig. 4. The buoyancy parameter and weissenberg number are increases the riction actor coeicient (C y Gr 1/4 ). But, the Eckert number minimizes the riction actor coeicient or the E=0and E=cases. The variations o, and Q H on (C y Gr 1/4 ) is shown in Fig. 5. It is clear that due to the dominance o rotation we have seen depreciation in azimuthal riction actor coeicient with rising values o. Fig.. The variations o, Ec and. Fig. 3. The variations o We, and Q H. Fig. 4. The Variations o, Ec and We.

8 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 Fig. 5. The variations o, and Q H. Heat Transer Rate Distribution The variation o local Nusselt number on various non-dimensional governing parameters or E=0 and E=cases was shown in Figs. 6 and 7. Fig. 6 displays the eects o, and Ec and We on local Nusselt number or the E=0 and E= cases. With improving values o volume raction o nanoparticle encourage the heat transer rate proiles, whereas depreciates the Nusselt number increasing values o Eckert number. Physically, by mixing o nanoparticles into the base luid improves the interaction between the particles, this can leads to improves the heat transer rate. The variations o W e, and Q H on the heat transer rate proiles is plotted in Fig. 7. It is noted that the heat transer rate is rises with increasing values o heat source/sink parameter, but minimizes with hal angle and weissenberg number. It is interesting to mention that the heat transer rate is higher in E=0 case compared with E= case, this can help to highlight that the viscous variation parameter control the heat transer rate. Generally, higher viscosity variations lead to improve the boundary layer due to this we seen depreciation in the heat transer rate. Fig. 6. The variations o, Ec and. Fig. 7. The variations o We, and Q H.

9 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 Special Cases Table 1 depicts the thermo physical properties o nanoluids and base luid. Table displays the comparison o the new solutions with the existed literature under some special restricted cases. We ound better coincidence with the existed and available literature. This authenticates the current results and the numerical procedure we used in the current study. Table 3 demonstrates the variations o riction actor and local Nusselt number with varied values o Eckert number and magnetic ield parameter with S=0, S=0.8 and =0, =0.1 cases or two type o Ti6Al4V+water and Ti6Al4V+kerosene. The titanium alloy has higher heat transer rate in unsteady case compared with steady case; however the riction actor coeicient is higher in steady case as compared with unsteady case. Similarly, the kerosene based titanium alloy has higher riction actor coeicients as compared with water based titanium alloy, whereas water based titanium alloy has greater heat transer rate comparing kerosene based titanium alloy. From this we can say depending on the societal needs we can use steady, unsteady and base, nanoluids. Thermo Physical Properties Water Kerosene Ti6A14V C p (J/KgK) k(w/m K) Table 1. Thermo physical properties o water and Ti6A14V. Pr=0.7 Pr=1 Pr=10 - (0) - (0) - (0) - (0) - (0) - (0) - (0) - (0) - (0) Anilkumar and Roy [8] Raju et al. [7] present Anilkumar and Roy [8] Raju et al. [7] Present Anilkumar and Roy [8] Raju et al. [7] Table. The validation o the current results with already existed literature some limited case M= We=Ec=E= Q H ==0, n=1with dierent values o prandtl number. Velocity and Temperature Distribution Fig. 8 displays the variations o volume raction o titanium alloy nanoparticles on temperature ield. This proves physically behavior o. The increasing values o improve the particle to particle distribution; this can helps to encourages the temperature ield. The similar perormance was seen in Ec. The increasing value o Ec advances the distribution o particles. This is displayed in Fig. 9. The eects o on velocity and temperature ields are plotted in Figs. 10 and 11. It is noted that both the velocity and temperature ields are encouraged with higher values o hal angle parameter. This may happen due to the rising values o hal angle parameter changes the channel position vertical to horizontal, which helps to boost up the velocity and temperature ields. The variation in temperature ields against dierent values o heat source/sink and weissennberg parameter (Q H and W e ) is plotted in Figs. 1 and 13. It is clear that the temperature ield is enhanced with improving values o Q H and W e. Generally, the rising values o Q H and we generate a high pressure orces in the low. Because o this cause we seen enhancement in temperature ield. Figs. 14 and 15 show the inluence o on velocity and temperature ields. Due to the domination o rotation and unsteadiness in the low, the velocity and temperature ields are improved with increasing values o. Present

10 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 Nanoluid type Ec M S=0 S=0.8 Skin riction coeicient Ti6A14V+water Ti6A14V+kerosene Ti6A14V+water 10 Ti6A14V+kerosene Local Nusselt number Table 3. Variations o riction actor and nusselt number with various values o viscous dissipation parameter and magnetic ield parameter or the steady, unsteady cases and base luid, nanoluid cases.

11 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 Fig. 8. The variations o temperature proiles. Fig. 9. The variations o Ec on temperature ield. Fig. 10. The variations o velocity ield.

12 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 Fig. 11. The variations o on temperature ield. Fig. 1. The variations o Q H on temperature ield. Fig. 13. The variations o We on temperature ield.

13 ISSN(Online): ISSN (Print): International Journal o Innovative Research in Science, (An ISO 397: 007 Certiied Organization) Vol. 6, Issue 11, November 017 Fig. 14. The variations o on velocity ield. Fig. 15. The variations o on temperature ield. V. CONCLUSION Quantiying the viscosity is a signiicant eect in state o the low property. It plays an important role in the quality controlling in various industrial applications such as ood, petrochemical, chemical, pharmaceutical, coatings, paints, cosmetics and auto motives. Keeping view into this we analyzed the heat source/sink and temperature dependent viscosity eect on titanium alloy nanoluid due to heated cone in the presence o viscous dissipation. Arising set o governing dierential equations are solved numerically using Runge-Kutta and Newton s method. The eects o various dimensionless parameters on the riction coeicient and the local Nusselt number are deliberated with the help o graphs. The conclusions are as ollows: The viscous dissipation parameter have tendency to minimizing the rate o heat transer and improves the temperature ield. The heat source/sinkparameter and volume raction o nanoparticleare boost up the heat transer rate. The water based titanium alloy has lesser riction actor compared with kerosene based titanium alloy. The water based titanium alloy has higher local Nusselt number compared with kerosene based titanium alloy. The dispersion nanoparticle interaction is very high in unsteady case with base and nanoluid cases. The magneticield parameter have higher heat transer rate compared with Eckert number.

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