Results in Physics 6 (2016) 1126–1135

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Mixed convection dissipative viscous fluid flow over a rotating cone byway of variable viscosity and thermal conductivity

http://dx.doi.org/10.1016/j.rinp.2016.11.0272211-3797/� 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/).

⇑ Corresponding author.E-mail address: raja_minhas88@yahoo.com (H. Jamil).

M.Y. Malik, Hamayun Jamil ⇑, T. Salahuddin, S. Bilal, K.U. Rehman, Zubair MustafaDepartment of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 October 2016Received in revised form 3 November 2016Accepted 14 November 2016Available online 23 November 2016

Keywords:Variable viscosityVariable thermal conductivityViscous dissipationRotating coneMixed convectionHomotopy analysis method

The effects of temperature-dependent viscosity and thermal conductivity on the flow and heat transfercharacteristics of a viscous fluid over a rotating vertical cone are premeditated. The properties of the fluidare assumed to be constant except for the density difference with the temperature. Also, the effect of vis-cous dissipation is considered in the energy equation. The highly nonlinear unsteady equations are con-verted into a system of nonlinear ordinary differential equations which is solved by using Homotopyanalysis method. The interesting findings for different pertinent parameters on momentum, energy, skinfriction coefficient and local Nusselt number are demonstrated in the form of graphs and tables. A com-parison has been made with literature as a limiting case of the well-chosen unsteady problem.� 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

The effect of temperature dependent viscosity on viscous fluidcauses variations in the properties of the fluid. For liquids, the vis-cosity reduces as the temperature increases while for gases the vis-cosity increases as temperature increases. In lubricating fluids, theincrease in temperature causes internal friction which affects theviscosity of the fluid, it will no longer remains constant. Due to thisinsufficiency, many researchers are keen to study the effects of var-ious variable viscosity models under different physical situations.Massoudi and Christie [1] have explored the effects of temperaturedependent viscosity with viscous dissipation on third grade fluidflow in a regular pipe. Numerical solutions have been acquiredby using finite simple difference method. Seddeek [2] analyzedthe MHD free convection unsteady flow with the effects of variableviscosity and aligned magnetic field past a semi-infinite plate. Thefinite difference method was used to solve the governing partialdifferential equations numerically. Pantokratoras [3] presentedthe numerical investigation of magnetohydrodynamics boundary-layer flow with temperature dependent fluid viscosity over aheated stretching sheet. Graphical representations for variationof viscosity and Prandtl number are also included in his work. Freeconvective boundary-layer flow with heat transfer past a verticalporous stretching surface under the effects of variable fluid viscos-

ity and thermal radiation has been investigated by Mukhopadhyayand Layek [4]. They found that effect of viscosity parameter i.e.increase in viscosity parameter causes increment in velocity andreduction in temperature at particular point. Mukhopadhyay [5]inspected the heat transfer and time dependent boundary layerflow of a viscous fluid past a stretching porous sheet. Both viscosityand thermal conductivity are supposed the functions of tempera-ture in her study. Well known shooting method was used toacquire the numerical solution for the governing ordinary differen-tial equations. Nadeem et al. [6] discussed the effects of variableviscosity and heat transfer characteristics for third order peristalticflow, in which two models of viscosity were considered. Bothnumerical and perturbation solutions were achieved in each caseand comparison has been made for both solutions. Turkyilmazoglu[7] examined the impact of time-dependent viscosity and thermalradiation on MHD time-dependent laminar flow of a viscous fluidwith electrical conduction over a porous rotating disk. Rashad [8]focused to study the effects of variable viscosity and radiation ontime dependent MHD boundary layer rotating fluid flow due tostretching surface in porous medium. Runge–Kutta integrationscheme and the local nonsimilarity method with second ordertruncation error was used to get the numerical solutions of theproblem. Article on the flow of two-dimensional second grade fluidunder the influence of temperature dependent viscosity andthermal conductivity over a horizontal stretching sheet with heatsource/sink was presented by Akinbobola and Okoya [9]. Theysupposed viscosity as an inverse function of temperature while

M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135 1127

thermal conductivity as a direct function of temperature. Umavathiand Ojjela [10] studied the effects of variable fluid viscosity, whichis a function of temperature and the heat transfer of the Newtonianfluid filled in a rectangular duct placed vertically. Finite differencemethod was used to find the numerical results. Manjunatha andGireesha [11] examined the problem of the heat transfer of a dustyviscous, magnetohydrodynamic and electrically conducting fluidflow under the impact of variable viscosity and thermal conductiv-ity on unsteady stretching sheet. Khan et al. [12] proposed theproblem of non-aligned hydromagnetic stagnation point with vis-cous nanofluid and variable viscosity in presence of thermal radia-tion flow over convectively heated stretching surface. Fourth–fifthorder Runge–Kutta–Fehlberg method was used to integrate thegoverning equations. The study of the influence of temperature-dependent viscosity on mass and heat transfer in MHD boundarylayer flow past over a vertical porous plate with thermal radiation,thermophoresis, magnetic field and an nth-order homogeneouschemical reaction field was discussed by Makinde et al. [13].Governing equations were solved with the help of Nachtsheimand Swigert-shooting iteration technique jointly with the sixth-order RK integration scheme. Umavathi et al. [14] analyzed the col-lective effects of variable thermal conductivity and viscosity onmixed convective incompressible flow of viscous fluid in verticalchannel, while chemical reaction of first order is also being studied.

The aptitude of a material to transfer heat is called thermal con-ductivity and it plays imperative role in cooling. It has numerouspractical applications in heat-treated materials traveling betweena wind-up roll and feed roll or on conveyor belt preserves, in thefield of metallurgy and chemical engineering. Thus, the impact oftemperature dependent thermal conductivity in rotating cone isof great interest to researchers. Khan et al. [15] presented the anal-ysis of the steady-state heat transfer of laminar film flow overstretching/shrinking sheet with influence of temperature-basedconductivity and viscosity. Method of Homotopy perturbation isused to tackle the governing equations. Sherief and Latief [16] con-sidered to study the fractional order theory of thermoelasticitywith thermal conductivity vary with temperature on half-space.They used Laplace transform to solve the problem. Inversion isdone by applying numerical technique. Miao and Massoudi [17]studied the heat transfer and slag-type non-Newtonian fluid flowwith variable thermal conductivity and viscosity effects betweentwo flat horizontal plates. Various parametric studies on velocity,temperature and volume fraction were also presented by them.Lin et al. [18] explored the heat transfer and flow of pseudo-plastic nanofluid with viscous dissipation and temperature-dependent thermal conductivity effects in finite thin film over atime-dependent stretching surface. Cu;TiO2;CuO and Al2O3 wereutilized as nano-particles with fluid sodium carboxymethyl cellu-lose (CMC)-water as a base fluid. Devi and Prakash [19] usednumerical approach to investigate the hydromagnetic flow in thepresence of temperature-dependent thermal conductivity and vis-cosity on slandering stretching plate. Animasaun [20] focused tostudy the variable thermal conductivity, variable viscosity, Dufourand thermophoresis effects on the flow of electrically conductingCasson fluid over a porous vertical plate with viscous dissipation,suction and chemical reaction of nth order. Solutions for governingequations were achieved by applying shooting method with Quad-ratic interpolation and Runge–Kutta Gill technique. Umavathi andSheremet [21] utilize numerical approach to study the flow andheat transfer of nanofluid particles with variable conductivity ina vertically placed rectangular duct. Method of finite difference isoperated to develop the solution of equations. Ezzat and El-Bary[22] studied on the Stroke’s flow of unsteady thermoelectric fluidwith temperature-dependent thermal conductivity and fractionalorder of heat transfer. Laplace transform was applied to solve the

stroke’s problem and further inverse process is done by usingnumerical approach. Khan and Malik [23] scrutinized the forcedconvective transfer of Sisko nanofluid subject to thermal conduc-tivity depending on temperature past over a cylinder stretchedhorizontally. With the help of shooting technique they solved thegoverning equations. Ezzat and El-Bary [24] deliberated theimpacts of fractional order heat transfer and temperature-dependent thermal conductivity on a hollow cylinder of infinitelength with perfect conduction and homogeneous axial magneticfield. Laplace transform is used to solve the equations in thedomain of Laplace transform, further Fourier series based numeri-cal method was used for inversion process. Numerical computa-tions for displacement, temperature, stress distribution inducedelectric and magnetic fields were found out and shown graphically.

Viscous dissipation behaves like an energy source and changesthe temperature distributions, which affects the heat transfer rates.The importance of viscous dissipation depends weather the cone isbeing heated or cold. Ragueb and Mansouri [25] described effect ofviscous dissipation on non-Newtonian fluid in elliptical duct withunvarying wall temperature and solved numerically. Barik andDash [26] devoted to explore the time-dependent magnetohydro-dynamic flow of electrically conducted, incompressible viscousfluid influenced by radiation, viscous dissipation and chemicalreaction. Desale and Pradhan [27] examined the boundary layerand hydrodynamic flow past a flat plate with variable plate tem-perature and viscous dissipation. Graphical demonstration for var-ious parameters were also presented by them. Numericalinvestigation of heat transfer rate and MHD flow induced bypower-law shrinking/stretching permeable sheet of nanofluid withviscous dissipation effects were presented by Dhanai et al. [28].Mabood and Mastroberardino [29] conducted a study on MHDwater based nanofluid flow over a stretching sheet subject to melt-ing heat transfer, viscous dissipation and electrical conduction.They solved governing boundary layer equations numerically byutilizing Runge–Kutta–Fehlberg method. Study on heat transferand magnetohydrodynamic flow effected by heat source and vis-cous dissipation on a stretching sheet was explored by Reddyet al. [30]. Solution of equations were carried out with the helpof Keller box method. Effects of some important parameters werealso discussed. Mabood et al. [31] studied mass transfer, heattransfer and MHD stagnation point flow of nano-particles Cu andAl2O3 with water as base fluid in porous medium under the effectsof radiation, chemical reaction and viscous dissipation.

Rehman et al. [32] discussed double stratified medium effectson mixed convective boundary layer Eyring-Powell fluid flow gen-erated by an inclined stretched cylinder in presence of heat absorp-tion/generation. Fifth order RK algorithm with shooting techniqueis utilized to solve the problem numerically. Article on axi-symmetric flow of electrically conducting viscous fluid with mag-netic field past a non-linear stretching sheet presented by Aliet al. [33]. They used Homotopy analysis method for analyticalsolution and shooting method to obtain numerical solution andmade comparison between both results. Shazad and Ali [34] pre-sented methematical analysis for heat transfer and magneto–hy-drodynamic flow of power law fluid over vertically stretchedsurface. Homotopy analysis method is applied on transformed gov-erning ordinary equation to obtain the solution. Khan et al. [35]deliberated the analytical solution of MHD mixed convectedFalkner-Skan flow past a porous mediumwith convected boundaryconditions by utilizing Homotopy analysis method. Shahzad andAli [36] examined the power law model of viscous incompressiblesteady state MHD flow on vertically stretched sheet. Ahmed et al.[37] focused on exact solution for MHD flow with convective heattransfer of Jeffrey fluid over a stretching sheet. Thermal radiation,internal heat source/sink and viscous dissipation with transverse

1128 M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135

magnetic field are taken in account. Ahmed et al. [38] investigatedheat transfer and axi-symmetric. flow of power law fluid modelpast a radially stretching sheet. Both numerical and analytical solu-tions are obtained by using shooting method and Homotopy anal-ysis method respectively. Also they demonstrate the effects ofvarious parameters on velocity and temperature profiles throughgraphs and tables.

In view of all the above mentioned effects and their applica-tions, the purpose of the present analysis is to examine the variableviscosity and thermal conductivity effects on mixed convectiveheat transfer in viscous fluid over a vertical rotating cone in pres-ence of viscous dissipation. Homotopy analysis method is used tofind out the solution and the behavior of different physical flowparameters such as the mixed convection, variable viscosityparameter, variable thermal conductivity, Prandtl number, Eckertnumber and unsteady parameter are studied. Moreover, the graphsare plotted and deliberated for the heat transfer rate and skin fric-tion coefficient.

Mathematical formulation

Consider unsteady incompressible flow of viscous fluid over arotating cone with time-dependent angular velocity X. Let thevelocity components u;v and w be in direction of x; y and z respec-tively, where x-axis is in tangential direction, y-axis is along withazimuthal direction and z- axis is normal to the cone. Further flowis supposed to be axi-symmetric. The buoyancy forces appearbecause of temperature variation, gravity acts in downward direc-tion along cone’s axis. Moreover, variations in wall temperature(Tw) depends on x (tangential coordinate), temperature far awayfrom the surface of cone (T1) is assumed to be constant. Physicalmodel of problem is shown below in Fig. 1.

Variable viscosity and variable thermal conductivity areassumed as follow:

l ¼ l0e�nðT�T1Þ;

l ¼ l0ð1� AhÞ; ð1Þ

where A ¼ nðTw � T1Þ,

Fig. 1. Physical model of the problem.

k ¼ k0e�cðT�T1Þ;

k ¼ k0ð1þ �hÞ; ð2Þwhere � ¼ �cðTw � T1Þ,

where l0 is viscosity and k0 conductivity of fluid,Under above suppositions and by applying Boussinesq approx-

imation theory of boundary layer, the governing system of partialdifferential equations for momentum and partial diffenential equa-tion for energy are as follow:

@ xuð Þ@x

þ @ xvð Þ@z

¼ 0; ð3Þ

@u@t

þ u@u@x

þw@u@z

� v2

x¼ 1q

@

@zl @u@z

� �þ gbðT � T1Þ cosa�; ð4Þ

@v@t

þ u@v@x

þw@v@z

þ uvx

¼ 1q

@

@zl @v@z

� �; ð5Þ

@T@t

þ u@T@x

þw@T@z

¼ 1qcp

@

@zk@T@z

� �þ lqcp

@u@z

� �2

þ @v@z

� �2" #

; ð6Þ

where q and g are density and gravity respectively, a� is semi ver-tical angle of cone, b is the coefficient for temperature expansion,thermal conductivity is denoted by a and specific heat of fluid isdenoted by cp

The appropriate boundary conditions are

uðx; o; tÞ ¼ 0; uðx;1; tÞ ¼ 0;

vðx;0; tÞ ¼ 1ð1� st�ÞXx sina�; vðx;1; tÞ ¼ 0;

Tðx;0; tÞ ¼ Tw; Tðx;1; tÞ ¼ T1; ð7Þhere X represents dimensionless angular velocity,temperature offluid away from the surface of cone is represented by T1, while t�

denotes dimensionless time.It is appropriate to convert partial differential equations into

system of non linear ordinary differential equations by using fol-lowing transformations:

u ¼ � 12ð1� st�ÞXx sina�f 0 gð Þ; v ¼ 1

ð1� st�ÞXx sina�g gð Þ;

w ¼ 1ð1� st�ÞXm0 sina

�� �1

2

f gð Þ; T ¼ T1 þ ðTw � T1ÞhðgÞ;

Tw � T1 ¼ ðT0 � T1Þð1� st�Þ2

xL

� �; t� ¼ ðX sina�Þt; g ¼ X sina�

m0ð1� st�Þ� �1

2

z:

ð8ÞEqs. (4)–(7) takes the following form

1� Ahð Þf 000 � Ah0f 00 þ 12f 02 � ff 00 � 2g2 � 2kh� s f 0 þ g

2f 00

� �¼ 0; ð9Þ

ð1� AhÞg00 � Ag0h0 � fg0 þ gf 0 � s g þ g2g0

� �¼ 0; ð10Þ

1Pr

1þ �hð Þh00 þ � h0ð Þ2h i

þ 12f 0h� fh0 � s 2hþ g

2h

� �þ ð1� AhÞEc 1

4f 00� �2 þ g0ð Þ2

� ¼ 0; ð11Þ

where the parameters appearing in Eqs. (9)–(11) are

Pr ¼ m0a; ReL ¼ X sina� L

2

m0; Gr ¼ gðT0 � T1Þ L

3

m20b cosa�; ð12Þ

k ¼ GrReL

; Ec ¼ xLðX sina�Þ2cpðT0 � TwÞ :

Fig. 2. h-curve for tangential velocity f.

Fig. 3. h-curve for azimuthal velocity g.

Fig. 4. h-curve for temperature field.

Table 1Convergence of Homotopy analysis method.

Approximation order �f 00ðgÞ �g0ð0Þ �h0ð0Þ1 0.5666 1.2994 1.50364 0.6616 1.3526 1.38838 0.6633 1.3738 1.390912 0.6642 1.3758 1.390316 0.6645 1.3761 1.389920 0.6645 1.3761 1.3899

Fig. 5. Variation in �f 0ðgÞ for different values of A.

Fig. 6. Variation in �f 0ðgÞ for different values of k.

M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135 1129

where k is used for mixed convection, s is the parameter ofunsteadiness, Pr is the Prandtl number and Ec is the Eckert number.

The boundary conditions for the concerned problem in non-dimensional form are

f ð0Þ ¼ 0; f 0ð0Þ ¼ 0; g 0ð Þ ¼ 1; h 0ð Þ ¼ 1;

f 0ð1Þ ! 0; g 1ð Þ ! 0; h 1ð Þ ! 0: ð13ÞThe skin friction coefficient and coefficient of heat transfer

(Nusselt number) are defined as

Cfx ¼ 2sxzjz¼0

q X sina�1¼st�ð Þ

h i2 ; Cfy ¼ �2syzjz¼0

q X sina�1¼st�ð Þ

h i2 ; ð14Þ

Nux ¼x @T

@z

� �jz¼0

ðTw � T1Þ ; ð15Þ

dimensionless form of skin friction coefficient and Nusselt numberare:

CfxRe1=2x ¼ �ð1� AhÞf 00 gð Þ �

g¼0;

CfyRe1=2x ¼ �ð1� AhÞg0 gð Þ½ �g¼0; ð16Þ

Fig. 7. Variation in �f 0ðgÞ for different values of s.

Fig. 8. Variation in gðgÞ for different values of A.

Fig. 9. Variation in gðgÞ for different values of k.

Fig. 10. Variation in gðgÞ for different values of s.

1130 M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135

NuxRe�1=2x ¼ �h0ðgÞg¼0: ð17Þ

where Rex ¼ 1m0ð1�st�ÞXx2 sina�.

Solution methodology

Eqs. (9)–(11) subject to boundary conditions defined in Eq. (13)are solved analytically with the help of Homotopy analysismethod. Initial guesses and linear operators are defined in Eqs.(19)–(24) i.e. f 0; g0 are initial guesses for velocities componentsand h0 is initial guess for temperature field. Homotopy analysismethod is valid for large as well as small values of parameters.For the sake of convergent series solution, �h- curves for velocitiesand temperature fields are sketched. Because convergent seriessolutions strictly depends on non-zero �h (auxiliary parameter).Figs. 2–4 illustrate the �h- curves for various values of parameters.The admissible values of �hf ; �hg and �hh are �1:7 6 �hf 6 �0:2,

�1:6 6 �hg 6 �0:4 and �1:5 6 �hh 6 �0:1. For �hf ¼ �1:0, �hg ¼ �1:1and �hh ¼ �0:9, convergent solutions in all over the region of garedetermined. Convergence of Homotopy analysis method is givenin Table 1. It is clear from table that the 16 iterations are sufficientfor convergence.

f 0ðgÞ ¼ 0; ð19Þ

g0ðgÞ ¼ expð�gÞ; ð20Þ

h0ðgÞ ¼ expð�gÞ; ð21Þ

Lf ¼ f 000 � f 0; ð22Þ

Lg ¼ g00 � g; ð23Þ

Lh ¼ h00 � h: ð24Þ

Fig. 11. Variation in hðgÞ for different values of A.

Fig. 12. Variation in hðgÞ for different values of Ec.

Fig. 13. Variation in hðgÞ for different values of �.

Fig. 14. Variation in hðgÞ for different values of Pr.

Fig. 15. Variation in hðgÞ for different values of s.

Fig. 16. Variation in Nussetle number with respect to s for different values of k.

M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135 1131

Fig. 17. Variation in Nussetle number with respect to s for different values of A.

Fig. 18. Variation in Nussetle number with respect to s for different values of �.

Fig. 19. Variation in Nussetle number with respect to s for different values of Pr.

Fig. 20. Variation in tangential skin friction with respect to s for different values ofk.

Fig. 21. Variation in tangential skin friction with respect to s for different values ofA.

1132 M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135

Results and discussion

This portion is devoted to study the graphical effects of differentparameters like mixed convection k, coefficient of variable viscos-ity A, coefficient of variable thermal conductivity �, Prandtl numberPr, Eckert number Ec and unsteady parameter s on velocities, sur-face stresses (tangential and azimuthal), temperature and coeffi-cient of heat transfer rate.

The effects of variable viscosity parameter A on primary velocity�f 0ðgÞ is shown in Fig. 5. In this figure it is noticed that primaryvelocity is zero on the surface of the cone (g ¼ 0Þ, velocity graphmoves to its peak value then moves downward and finally it tendsto zero when g! 1. It is also noticed that variation in variable vis-cosity gives increase in �f 0ðgÞ. In Fig. 6, almost similar behavior of�f 0ðgÞ is observed under the influence of parameter k, but hereboundary layer thickness increases sharply with growing k. Otherdimensionless parameters are taken as constant. Fig. 7 shows the

Fig. 22. Variation in azimuthal skin friction with respect to s for different values ofk.

Fig. 23. Variation in azimuthal skin friction with respect to s for different valuesof A.

Table 2Comparison of skin friction coefficient and local Nusselt number with previous work for A

Pr k CfxRe1=2x 0:5CfyRe

0 1.0253 0.6150.7 1 2.2007 0.849

10 8.5041 1.3990 1.0255 0.615

10 1 1.5626 0.68310 5.0820 0.984

0 1.0255 0.6150.7 1 2.2010 0.849

10 8.5042 1.3990 1.0255 0.615

10 1 1.5630 0.68310 5.0820 0.984

0 1.0255 0.6150.7 1 2.2012 0.849

10 8.5041 1.3990 1.0256 0.615

10 1 1.5636 0.68310 5.0821 0.984

M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135 1133

behavior of primary velocity �f 0ðgÞ for unsteady parameter s. Forlarge values of s velocity profile �f 0ðgÞ goes downward, highestvalue of s slow down the tangential velocity of fluid and velocitygoes to zero far away from the cone surface.

Figs. 8–10 are drafted to investigate the behavior of secondaryvelocity or circumferential velocity gðgÞ under the impact of somephysical parameters A; k and s. In Fig. 8, velocity profile gðgÞ issketched for various values of A (=0.5, 1.0, 1.5, 2.0) with� ¼ 0:1;Pr ¼ 0:7; k ¼ 0:3; s ¼ 0:4 and Ec ¼ 0:3. At the wall of conecircumferential velocity gðgÞ is 1 and graph decreases graduallyas g increases and finally approaches to zero. Magnitude of sec-ondary velocity reduces for increasing values of A. In Fig. 9, varia-tion has given to mixed convection k (=2, 4, 6, 8) withA ¼ 0:7; � ¼ 0:1; Pr ¼ 0:7; s ¼ 0:4 and Ec ¼ 0:3, and result of thisvariation has been obtained in the form of descending graph. It isalso observed that velocity attains its peak value along the wall.Fig. 10 demonstrate that circumferential velocity gðgÞ also hasdecreasing effect for increasing values of unsteady parameter s(=0.5, 1.0, 1.5, 2.0).

Behavior of temperature field hðgÞ under different influencingparameters like unsteady parameter s, Eckert number Ec, PrandtlnumberPr, coefficients of variablefluid viscosityA andvariable ther-mal conductivity � is drafted in Figs. 11–15. In Fig. 11, Temperatureprofile hðgÞ is one at g ¼ 0 and goes downward with increasing g.hðgÞ has decreasing amplitude for increasing A (=0.5, 1.5, 2.5, 3.5),whilemounting values of Eckert number Ec (0.1, 0.4, 0.7, 1.0) boost-ing up the graph of hðgÞ, which is displayed in Fig. 12. Different val-ues of coefficient of variable thermal conductivity � (0.3, 0.6, 0.9, 1.2)enhances the graph of temperature (observe Fig. 13). Effect ofPrandtl number Pr is revealed in Fig. 14. It is clear that thickness ofthermal boundary layer and hðgÞ decreases large Prandtl number Pr.

Figs. 16–19 are drafted to express behavior of Nusselt numberwith respect to s against some important physical parameters, itis illustrated that variation in k enhances the local Nusselt number�h0ð0Þ (Fig. 16). It is discovered from Figs. 17 and 18 that rise in Aand � gives decrease in the rate of heat transfer. Moreover, Fig. 19demonstrates that rate of heat transfer rises for higher values ofPrandtl number Pr. Figs. 20 and 21 are devoted to manifest theimpact of k (mixed convection) and A (variable viscosity parame-ter) on coefficient of tangential skin friction CfxRe

1=2 with respect

to s. It is clear from these figures that CfxRe1=2 growing up as k

increases while it behaves decreasingly with increasing A. Influ-ence of k and A on azimuthal skin friction 0:5CfyRe

1=2 is plotted

in Figs. 22 and 23 respectively. It is depicted that 0:5CfyRe1=2 has

¼ 0; s ¼ 0; � ¼ 0 and Ec ¼ 0.

1=2x NuRe�1=2

x

3 0.4295 Present values2 0.61210 1.00977 1.41105 1.56601 2.3581

4 0.4299 Saleem and Nadeem [39]3 0.61212 1.39928 1.41115 1.56615 2.3581

8 0.4299 Chamka et al. [40]6 0.61205 1.00978 1.41107 1.56620 2.3580

Table 3Skin friction coefficient for various values of k and A.

k A CfxRe1=2x 0:5CfyRe

1=2x

1 1.0783 0.58675 2.2133 0.873110 8.2103 1.6571

0.2 0.6789 0.74190.4 0.5374 0.65100.6 0.4137 0.5706

Table 4Local Nusselt number for unlike values of Ec, Pr, A and �.

Ec Pr A � NuRe�1=2x

0 0.45720.5 0.40831.0 0.3440

3 1.58925 1.84107 1.9206

1 0.88512 0.85733 0.8177

0.1 0.78530.3 0.67610.7 0.5372

1134 M.Y. Malik et al. / Results in Physics 6 (2016) 1126–1135

rising behavior for growing k (observed in Fig. 22 and decreasingbehavior for growing A can seen in Fig. 23.

Comparison of the present values of skin frictions and Nusseltnumber for Ec ¼ A ¼ � ¼ s ¼ 0with previous studies [32,33] is givenin Table 2. In Table 3, numerical values of tangential skin frictionCfxRe

1=2x and azimuthal skin friction 0:5CfyRe

1=2x is given for different

values of variable viscosity parameter A and mixed convectionparameter k. Numerical values of both CfxRe

1=2x and 0:5CfyRe

1=2x

increases for increasing k, while both has opposite behavior for vari-able viscosity parameter A.Table 3 explored the behavior of Nusseltnumber for influencing parameters Pr;A; � and Ec. It is found thatincreasing values of Pr enhances the Nusselt numberwhile oppositebehavior is noticed for physical parameters A; � and Ec (Table 4).

Conclusion

Rotating flow of a viscous fluid over a vertical cone is addressed.A comparative analysis for variable thermal conductivity and vis-cosity is also presented. The main features of this study are listedbelow:

� The magnitude of primary velocity �f 0ðgÞ boosts up for largevalues of variable viscosity parameter A,on the other hand sec-ondary velocity gðgÞ behaves oppositely for growing A.

� Mixed convection parameter k has significant impact on veloc-ity profiles. Primary velocity profile �f 0ðgÞ enhances while sec-ondary velocity profile gðgÞ reduces for large values of mixedconvection parameter k.

� Both velocity profiles have decreasing behavior for large valuesof unsteady parameter s. Temperature field hðgÞ also decreaseswith increasing s.

� Skin-friction coefficients CfxRe1=2 and 0:5CfyRe

1=2x is raised with

increasing k, while A (viscosity parameter) causes reduction inboth CfxRe

1=2 and 0:5CfyRe1=2x .

� Heat transfer rate NuRe�1=2x decreases for increasing values of

Eckert number Ec, variable-viscosity parameter A and variablethermal conductivity parameter �, whereas opposite behavioris noticed in the case of Prandtl number.

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