heat transfer squeezed flow of carreau fluid over a sensor surface with variable thermal conductivity a numerical study

Results in Physics (2016) 940–945 Contents lists available at ScienceDirect Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics Heat transfer squeezed flow of Carreau fluid over a sensor surface with variable thermal conductivity: A numerical study Mair Khan ⇑, M.Y Malik, T Salahuddin, Imad Khan Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan a r t i c l e i n f o Article history: Received 25 September 2016 Received in revised form 22 October 2016 Accepted 27 October 2016 Available online November 2016 Keywords: Heat transfer Squeezed flow Carreau fluid Sensor surface Variable thermal conductivity Shooting method a b s t r a c t Present phenomenon is dedicated to investigate the heat transfer and squeezed flow of Carreau fluid over a sensor surface The thermal conductivity of the fluid is assumed to be temperature dependent After assimilating these assumptions, the appropriate transformations are used to formulate the partial differential equations into non-dimensional system of ordinary differential equations Solutions for the boundary layer momentum and heat equations are accomplished by a well-known numerical technique namely shooting method The related essential physical parameters are visualized through graphs and tables It is found that the velocity profile increases by increasing squeezed flow parameter b, permeable velocity parameter f , power law index n and Weissenberg number We Similarly, temperature profile increases by increasing small parameter  Ó 2016 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/) Introduction In recent years substantial attention has been paid on heat transfer due to its rapid progress in fluid mechanics and its significance in engineering and industrial processes such as nuclear reactor cooling, space cooling, heat conduction in tissues, energy production, biomedical applications such as magnetic drug targeting, etc Gupta et al [1] presented the numerical solution of flow between two parallel plates approaching each other symmetrically Uddin et al [2] discussed the numerical and experimental analysis of squeezed flow of solid sphere by using Carreau-fluid Rohni et al [3] investigated the two dimensional unsteady flow heat transfer flow of nanofluid towards a shrinking sheet with wall mass suction Mahgoub [2] studied the heat transfer in a porous media with forced convection over a horizontal flat plate Nandy [4] studied the two dimensional heat transfer flow of a nonNewtonian fluid over a stretching sheet with thermal slip They found that temperature field enhances with increasing values of Eckert number and magnetic parameter Bhattacharyya [5] examined the heat transfer and MHD stagnation-point flow of Casson fluid towards a stretching sheet Akbar et al [6] discussed the approximate solution of MHD flow of Eyering-Powell fluid over a stretching sheet Raju et al [7] studied the two dimensional heat transfer properties on flow of steep plats with dissipative in the ⇑ Corresponding author induced area Davarnejad et al [8] presented the concept of convective turbulent flow and heat transfer in the magnesium Oxide–water nanofluid Khan et al [9] analyzed the approximate solution of heat transfer boundary layer flow induced by nonlinear stretching sheet Yasin et al [10] studied the numerical solution of MHD heat transfer flow of viscous fluid with stagnation-point, partial slip and joule heating They found that unique solution exists for the stretching sheet Akbar et al [11] presented the computational study of double-diffusive natural convective boundarylayer flow of a nanofluid over a stretching sheet Sandeep et al [12] studied the two dimensional MHD flow of a dusty nanofluid and heat transfer over an exponentially stretching surface Misra et al [13] presented the concept of chemical reaction as well as mass and heat transfer on the MHD oscillatory flow of blood They found that the mass diffusivity decreases with the increase in mass transfer rate Makanda et al [14] investigated the Casson fluid over a two dimensional MHD free convection flow with partial slip in non-darcy porous medium Raju et al [15] studied the heat and mass transfer boundary layer flow of Casson fluid past an permeable exponentially stretching surface Balla et al [16] investigated the heat transfer of a viscous fluid with unsteady Newtonian, incompressible fluid past a precipitately underway semi-infinite vertical plate Akbar et al [17] analyzed the MHD nanofluid flow over a stretching surface containing gyrostatic microorganisms The analysis of squeezing flow finds several industrial and practical applications in different areas such as physical, chemical engineering, biophysical, food engineering, polymer processing, E-mail address: mair.khan@math.qau.edu.pk (M Khan) http://dx.doi.org/10.1016/j.rinp.2016.10.024 2211-3797/Ó 2016 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 941 M Khan et al / Results in Physics (2016) 940–945 comparison and injection molding and many others Hayat et al [18] investigated the squeezed flow of second grade fluid over a sensor surface Akbar et al [19] studied the incompressible flow of two dimensional viscous nanofluid over a vertical stretching sheet Akbar et al.[20] investigated the physical analysis for variable thermal conductivity over a stretching sheet with convective slip boundary conditions Ahmed et al [21] studied the effects of MHD Casson fluid over a squeezed flow between two parallel plats Haq et al [22] presented the concept of MHD squeezed flow of non-Newtonian fluid namely Casson fluid over a sensor surface Rana et al [23] deliberated the computational study of Casson fluid over a stretching sheet in the presence parietal slip and internal heating Current analysis aims to deliberate the variable thermal conductivity effect on squeezing flow of Carreau fluid model over a sensor surface The numerical solutions are obtained by using shooting method The influence of embedded physical parameters temperature and velocity profiles are shown graphically Mathematical formulation Consider incompressible, two-dimensional unsteady boundary layer flow of Carreau fluid towards a continuously permeable sensor surface Fig shows the closed squeezed channel with height hðtÞ assumed to be much larger than the boundary layer thickness Further it is assumed that the squeezing free stream is supposed to be happened from the tip to the surface Moreover, it is assumed that the lower plate is while immovable upper plate is squeezed The continuity, momentum and energy equations after applying the boundary layer approximation will be [22,24] @u @ v ỵ ẳ 0; @x @y ð1Þ "    2 !# @u @u @u @p @2u ðn À 1Þ @u @ u ; ỵu ỵv ẳ ỵm ỵ3 C @t @x @y @y2 @y @y2 q @x kinematic viscosity, C is the time constant, p is the fluid pressure, aðTÞ is the variable thermal conductivity, T is temperature, t is time, a is the thermal diffusivity and q is the density After eliminating the pressure gradient the momentum Eqs (2) and (3) takes the form "  2 !# @u @u @u @U @U @2u ðn À 1Þ @u @ u ỵu ỵv ẳ ỵU ỵm ỵ3 C ; @t @x @y @t @x @y2 @y @y2 5ị with boundary conditions ux;0; tị ẳ 0; v x; 0; tị ẳ m0 tị; ux; 1;tị ẳ Ux;tị; k @Tx;0;tị ẳ qxị; @y Tx;1; tị ẳ T : ð6Þ Here T ; Uðx; tÞ and qðxÞ are the free stream temperature, wall heat flux and free stream velocity, respectively Here thermal conductivity aðTÞ is of the form a ẳ a1 ỵ hị, where  is small quantity Suppose that wall is a function of a heat flux qðxÞ The velocity at the sensor surface is represented by reference velocity v ðtÞ and is considered when the surface is permeable Eqs (4) and (5) can be transformed into differential form by using following similarity transformations: v pa p w ẳ x amf gị; a ẳ sỵbtị ; u ẳ axf gị; p ẳ f gị am; U ẳ ax; hgị ẳ TTp1m ; gẳy m; q0 x p where qxị ẳ q0 x and v tị ẳ v a Where b is a squeezed flow index, s is a arbitrary constant, a is strength squeezing flow parameter, q0 is heat flux and k is thermal conductivity Note that Eq (1) is identically satisfied while Eqs (4) and (5) takes the form   bg 000 À Á2 ðn À 1Þ 000 00 000 We2 f ị f ị ỵ ẳ 0; f f ỵ bf 1ị ỵ f ỵ fỵ 2 8ị 2ị   @U @U @p ; ỵU ẳ @t @x q @x 3ị   @T @T @T @ @T ỵu ỵv ẳ ; aTị @t @x @x @y @y 4ị     bg b h ẳ 0; h ỵ Prh0 ị Pr f ỵ ỵ hịh00 ỵ Pr f ỵ 2 9ị and the corresponding transformed boundary conditions are where u and v are the velocity components along the x and y directions respectively, U is the free stream velocity, n power law index, m f 0ị ẳ f ; f 0ị ẳ 0; h0 0ị ẳ 1; 10ị f 1ị ! 1; h1ị ! 0; where Pr ẳ rt is Prandtl number, f ¼ pffiffiffi t is the permeable velocity a3 x2 C2 and We ¼ m is Weissenberg number Using the boundary approximations, the wall shear stress at the sensor surface is given by Squeezing sw ẳ Flow 7ị a  3 @u n 1ị @u @T ỵ C2 ; qw ẳ k ; @y @y @y at y ẳ 0: 11ị the coefficient of skin friction and Nusselt number are defined as y,v V V x,u Flow h(t) L Fig Physical model of sensor surface Cf ¼ sw pffiffi ; ax am Nu ẳ qw k : qxK 12ị In the dimensionless form, the skin friction and Nusselt number are defined as h i pffiffiffiffiffiffiffi 00 00 Rex cf ẳ f gị ỵ n1ịWe f gịị gẳ0; p Nu Rex ẳ ẵh0 gịgẳ0 ; p where Rex ẳ x am ð13Þ 942 M Khan et al / Results in Physics (2016) 940–945 0 Fig Variation of f ðgÞ with n Fig Variation of f ðgÞ with We Numerical solutions The non-linear ordinary differential equations for momentum and heat i.e Eqs (8) and (9) subject to the boundary conditions (10) are solved numerically using fifth order Runge–Kutta method along with shooting technique for unlike values of parameters Let 00 u1 ¼ f ; u2 ¼ f ; u3 ¼ f ; u4 ¼ h and u5 ¼ h00 Hence the leading equations become u01 ẳ u2 ; 11ị u02 ẳ u3 ; 12ị u03 ẳ h !   bg i u1 ỵ u3 ỵ u22 bu2 1ị ; 2 ỵ 3n 1ịu2 We2 13ị u04 ẳ u5 ; u05 ẳ 14ị   !   bg b u4 ¼ 0; u5 Pru25 ỵ Pr u2 ỵ Pr u1 ỵ ỵ u4 ị 2 15ị with the analogous initial conditions: u1 0ị ẳ f 0; u2 0ị ẳ 0; u4 0ị ẳ 1; u2 gị ! 1; u4 ðgÞ ! when g ! 1; ð16Þ In order to solve this modified system of five first order ordinary differential Eqs (11)–(15) by Runge–Kutta method five initial conditions are compulsory whereas only three of them are given Two more initial conditions are required to replace u2 ðgÞ ! 1; u4 ðgÞ ! when g ! Thus it is assumed that u2 0ị ẳ s1 and u4 0ị ¼ s2 These unknown initial conditions u2 ð0Þ and u4 ð0Þ are first predicted and consequently resolute using Newton–Raphson’s method for individual parameters with respect to the given free stream conditions These slopes are modified such that the boundary conditions are satisfied for g ! The resulting initial value problem is attempted numerically using a fifth order Runge– Kutta-Fehlberg integration scheme The convergence criteria and step size are taken to be 10À6 and 0:1 for all numerical solutions Fig Variation of f ðgÞ with f Results and discussion Numerical computations are carried out for various sets of pertinent parameters on skin friction coefficient, local Nusselt number, velocity and temperature profiles The behavior of power law index n on velocity profile is illustrated in Fig It is examined that the increase in power law index n causes reduction in both velocity of the fluid and boundary layer thickness This proves that the increment in power law index n gives the field reduction and prevents the motion of the fluid The dimensionless velocity profile for different values of Weissenberg number Weis sketched in Fig It is noticed that large values of Weissenberg number We dimensionlies velocity profile for n > Fig shows the velocity distribution for different values of permeable velocity f It is noticed that, velocity profile reduces for suction case (i.e f > 0) Because suction causes the fluid to be more attached to the sensor surface, so velocity decreases The velocity and temperature profiles against g for different values of squeezed flow index b are shown in Figs and It is clear that increment in squeezed flow index 943 M Khan et al / Results in Physics (2016) 940–945 Fig Variation of f ðgÞ with b Fig Variation of hðgÞ with Pr Fig Variation of hðgÞ with  Fig Variation of h(g) with b b causes reduction in both velocity and temperature distributions Fig.7 shows the behavior of Prandtl number Pr on temperature profile It is clear from figure that an increase in the values of Prandtl number Pr, the temperature profile reduces This is due to the fact that by increasing Prandtl number Pr the thermal diffusivity of the fluid reduces, so temperature decreases Fig demonstrates the influence of small parameter  on temperature profile An increase in small parameter  corresponds to increase in kinetic energy of the fluid particles which increases the variation of thermal characteristics, so the fluid temperature and boundary layer thickness increases Fig.9 shows the effect of skin friction coefficient against squeezed flow parameter b for different values of power law index n It is observed that skin friction coefficient increases for increasing values of power law index n Fig.10 displays the skin friction coefficient against squeezed flow parameter b for various values of Weissenberg number We It is noticed that the skin friction increases for large values of Weissenberg number We Fig 11 illustrates the effect of local Nusselt number against squeezed flow parameter for different values of small parameter  It is clear that, Fig Variation of skin friction coefficient against squeezed flow parameter b for different values of power index n 944 M Khan et al / Results in Physics (2016) 940–945 local Nusselt number increases on increasing values of small parameter  Tables and shows the behaviors of local Nusselt number and skin friction coefficient for different values of pertinent parameters Conclusions Two dimensional squeezed flow of Carreau fluid in presence of variable thermal conductivity is considered The existing investigation are mostly useful in applications involving solar collectors heat transfer, oil recover, etc The main points are summarized below: Fig 10 Variation of skin friction coefficient against squeezed flow parameter b for different values of Weissenberg number We  The temperature profile reduces for large values of Prandtl number Pr  The temperature profile enhances by increasing small parameter   The increasing values of squeezed flow parameter b enhances the velocity and temperature profiles  The skin friction coefficient enhances for large values of Weissenberg number We and power law index n  The increasing values the power law index n and Weissenberg number We enlarges the velocity profile  The velocity profile increases for increasing values of permeable velocity parameter f References Fig 11 Variation of local Nusselt number against squeezed flow parameter b for different values of variable thermal conductivity Table 1 Variation of the skin friction coefficient Re2x C f against different values of Weissenberg We and power law index n We n C f Rex 1=2 0.1 0.5 0.9 0.1 0.1 0.1 1.2 1.2 1.2 1.5 1.8 2.1 0.7234 0.7320 0.7394 0.7238 0.7241 0.7244 Table Variation of the Nusselt number Àhð0Þ against various values of Prandtl number Pr and variable thermal conductivity Pr  Àh0 ðOÞ 0.1 0.5 0.9 0.1 0.1 0.1 0.5 1.2 1.2 1.5 1.8 2.1 0.2500 0.2514 0.2563 0.3463 0.3482 0.3514 [1] Gupta PS, Gupta AS Squeezing flow between parallel plates Wear 1977;45:177–85 [2] Uddin J, Marston JO, Thoroddsen ST Squeeze flow of a Carreau fluid during sphere impact AIP Phys Fluids 2012;24:073104 [3] Rohni AM, Ahmad S, Pop L Flow and heat transfer over an unsteady shrinking 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Obaidat S Squeezed flow and heat transfer in a second grade fluid over a sensor surface Therm Sci 2014;18:357–64 [19] Akbar NS, khan ZH Effect of variable thermal conductivity and thermal radiation... Heat Fluid Flow 2016;26(1):108–21 [12] Das M, Mahato R, Nandkeolyar R Newtonian heating effect on unsteady hydro magnetic Casson fluid flow past a flat plate with heat and mass transfer Alexandria... presence parietal slip and internal heating Current analysis aims to deliberate the variable thermal conductivity effect on squeezing flow of Carreau fluid model over a sensor surface The numerical