Meccanica (2014) 49:673–686 DOI 10.1007/s11012-013-9819-3

Unsteady mixed convection boundary layer flow over a vertical cone with non-uniform slot suction (injection) M. Ganapathirao · G. Revathi · R. Ravindran

Received: 31 October 2012 / Accepted: 8 October 2013 / Published online: 22 October 2013 © Springer Science+Business Media Dordrecht 2013

Abstract The non-similar solution of an unsteady mixed convection laminar boundary layer flow over a vertical cone in the presence of non-uniform surface mass transfer through slot has been obtained while the axis of cone is inline with the flow. The unsteadiness is caused by the time dependent free stream velocity. The governing boundary layer equations are transformed into a non-dimensional form by a group of non-similar transformations. The resulting coupled non-linear partial differential equations have been solved numerically by the combination of quasi-linearization technique and an implicit finite difference scheme. Numerical computations are performed for different values of the parameters to display the velocity and temperature profiles graphically. Both accelerating and decelerating free stream velocities are considered. Numerical results are reported to display the effects of nonuniform single and double slot suction (injection) on skin friction and heat transfer coefficients at the wall. Further, the effects of Prandtl number, buoyancy and mass transfer (suction or injection) parameters at different stream-wise locations for various times on velocity and temperature profiles, and on skin friction and heat transfer coefficients are also presented in this paper.

M. Ganapathirao · G. Revathi · R. Ravindran (B) Department of Mathematics, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India e-mail: [email protected]

Keywords Unsteady flow · Mixed convection · Surface mass transfer · Vertical cone · Non-similar solution · Non-uniform slot suction (injection)

1 Introduction Free convection is caused by the temperature difference of the fluid at different locations and forced convection is the flow of heat due to some external applied forces. The combination of both of these phenomena is called the mixed convection. Mixed convection flows, arise in many transport processes both in natural and engineering applications. Convective heat transfer flows over a stationary cone is important for the thermal design of various types of industrial equipments such as stationary heat exchangers, design of canisters for nuclear waste disposal, nuclear reactor cooling system and geothermal reservoirs, etc. In many practical situations, the unsteady mixed convection boundary layer flows do not necessarily allow similarity solution and for the last two decades, a wide range of problems have been considered, where the non-similarity has to be taken into account. Many researchers to confined their studies either a steady non-similar flows or to unsteady semi-similar or selfsimilar flows [1–8] as a consequence of the mathematical difficulties involved in achieving non-similar solutions. In an early work, Hering and Grosh [1] investigated the practical case of steady mixed convection from a

674

vertical cone in an ambient fluid for the Prandtl number Pr = 0.7. Further, Himasekhar et al. [2] found the similarity solution of the mixed convection flow over a vertical rotating cone in an ambient fluid for a wide range of Prandtl numbers. Wang [3] has obtained the similarity solution of boundary layer flows on a rotating cones, discs and axi-symmetric bodies with concentrated heat sources. Anilkumar and Roy [4], and Roy and Anilkumar [5] have obtained, respectively, the self-similar and semi-similar solutions of unsteady mixed convection flows from a rotating cone in a rotating fluid. Kumari et al. [6] obtained non-similar solution on mixed convection boundary layer flow over a vertical cone. Takhar and Nath [7] found the similarity solution on an unsteady boundary layer flow of an electrically conducting fluid in the stagnation region of two dimensional and axi-symmetric bodies by using shooting method. Yih [8] presented non-similar solution to examine the heat transfer characteristics in the mixed convection flow over a vertical cone with porous media. The effect of magnetic field on the rotating infinite permeable cone in a rotating fluid was investigated by Roy et al. [9]. Mixed thermal boundary conditions on spinning cone with magnetic field was given by Ece and Ozturk [10]. Postelnicu [11] was studied the free convection about a vertical frustum of a cone in a micropolar fluid, using the boundary layer assumptions, in the conditions of a uniform heat flux imposed to the wall. Further, the effect of suction or injection on the boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction was studied by Ali and Al-Salem [12]. Recently, Chamkha et al. [13] studied the radiation effect on mixed convection flow over a cone embedded in a porous medium filled with a nanofluid. Ravindran et al. [14] have obtained the non-similar solution to examine the effects of mass transfer (suction or injection) on the steady mixed convection flow over a vertical permeable cone. Similarity solutions for forced convective MHD boundary layer flow with diffusion of chemically reactive solute over a porous flat plate with uniform suction/blowing was investigated by Bhattacharya et al. [15]. More recently, Merkin et al. [16] obtained a similarity solution for forced convective flow over a moving flat plate in an external free stream with strong injection. Some of the above researchers confined their work to uniform mass transfer (suction or injection).

Meccanica (2014) 49:673–686

Mass transfer from a wall slot into the boundary layer is of interest for various prospective applications including thermal protections, energizing the inner portion of the boundary layer in adverse pressure gradient and skin friction reduction on control surfaces. Moreover, mass transfer through a slot strongly influences the development of a boundary layer along a surface and can prevent or at least delay the separation of the viscous region. In case of uniform mass transfer, finite discontinuities arise at the leading and trailing edges of the slot and those can be avoided by choosing a non-uniform mass transfer distribution along a stream-wise slot and it has been discussed in Minkowycz et al. [17]. Different studies have reported the influence of non-uniform mass transfer through single slot on a steady water boundary layer flows over (i) a cylinder and (ii) a sphere [18], rotating sphere [19], yawed cylinder [20] and slender cylinder [21], respectively. Later, Roy and Saikrishnan [22] presented multiple slot on an exponentially decreasing free stream steady flow. Further, several investigators [23–25] have studied the effect of non-uniform double slot into a steady boundary layer flows over a slender cylinder, sphere and cylinder, respectively. All these studies deal with pertain to steady flows. The extensive literature on non-similar solution and mass transfer technique shows that the studies confined only for steady boundary layer flows with nonuniform mass transfer. However, in many practical situations, the flow can be unsteady due to the time dependent free stream velocity or the curvature of the body or the surface mass transfer or even possibly due to all these effects. There are several transport processes with non-uniform mass transfer (suction or injection) in industry where the buoyancy force arises from the thermal diffusion caused by the temperature gradient. Cone shaped bodies are often encountered in many engineering applications and many heat transfer problems of mixed convection flows. Moreover, Heat transfer problems of mixed convection boundary layer flow over a stationary cones are extensively used by auto-mobile and chemical industries. Therefore, as a step towards the eventual development on an unsteady mixed convection flows, it is very useful to study the effect of non-uniform mass transfer through slot suction (injection) over a vertical cone. The aim of the present investigation is to obtain the non-similar solution of an unsteady mixed convection boundary layer flow over a vertical cone through

Meccanica (2014) 49:673–686

675

cone along its generator and the transverse coordinate y is measured normal to it into the fluid, respectively. The physical model and coordinate system are shown in Fig. 1. The thermo-physical properties of the fluid in the flow model are assumed to be constant except the density variations causing a body force term in the momentum equation. The Boussinesq approximation is invoked for the fluid properties to relate the density changes to temperature changes and to couple in this way the temperature field to the flow field. Under these assumptions and imposing Mangler’s transformation, axi-symmetric problem reduced into a twodimensional problem [27], the equations of continuity, momentum and energy governing the unsteady mixed convection boundary layer flow along a vertical cone can be expressed as [26, 28, 29] ∂u ∂v + = 0, ∂x ∂y

(1)

∂u ∂u ∂u +u +v ∂t ∂x ∂y = Fig. 1 The physical model and coordinate system

non-uniform slot suction (injection). Non-similar solutions are obtained numerically by solving a set of coupled non-linear partial differential equations governing the unsteady boundary layer flow using an implicit finite difference scheme in combination with the quasi-linearization technique [30, 31]. Numerical solutions are obtained for different values of the Prandtl number, suction (injection) parameter and buoyancy parameter. Some particular cases of the present results have been compared with Kumari et al. [6], and found in excellent agreement.

2 Mathematical formulation Consider an unsteady mixed convection laminar boundary layer flow of a viscous and incompressible fluid over a vertical circular cone with a half angle γ . It is assumed that the forced flow moves parallel to the axis of the cone in the upward direction with the free stream velocity u∞ and uniform temperature of the ambient fluid T∞ . The surface of the cone is at a uniform higher temperature Tw i.e., Tw > T∞ . The stream wise coordinate x is measured from the apex of the

∂ 2u ∂ue ∂ue + ue +ν 2 ∂t ∂x ∂y + gβ(T − T∞ ) cos γ ,

(2)

∂T ∂T ν ∂ 2T ∂T +u +v = , ∂t ∂x ∂y Pr ∂y 2

(3)

where u and v are the velocity components along the x and y directions, respectively; g is the gravitational acceleration ; β is the coefficient of thermal expansion; T is the temperature; ν is the kinematic viscosity; Pr is the Prandtl number. The initial conditions are given by u(x, y, 0) = ui (x, y),

v(x, y, 0) = vi (x, y),

T (x, y, 0) = Ti (x, y).

(4)

The boundary conditions are given by u(x, 0, t) = 0,

v(x, 0, t) = vw (x, t),

T (x, 0, t) = Tw = Constant,

  u(x, ∞, t) = ue (x, t) = U (x)φ t ∗   = u∞ x m/3 φ t ∗ ,

(5)

T (x, ∞, t) = T∞ = Constant. Here the subscripts i, e, ∞ and w denote the conditions at initial, at the edge of the boundary layer, at

676

Meccanica (2014) 49:673–686

the free stream boundary layer and at the wall, respectively; t and t ∗ are the dimensional and dimensionless times, respectively. vw denotes the surface mass transfer distribution with vw < 0 corresponds to suction, vw > 0 corresponds to injection or blowing and vw = 0 corresponds to an impermeable surface of the cone. For simplicity, we have considered the unsteady free stream distribution of the form φ(t ∗ ) = 1 + ε(t ∗ )2 and ε is the unsteady parameter, where ε > 0 is for an accelerating flow and ε < 0 is for a decelerating flow, respectively. U is the free stream velocity component is given by m U (x) = u∞ x 3 and m is the exponent in the power law variation of the free stream velocity, and the exponent m is related to cone half angle γ by γ=

πm m+1

or

m=

γ . π −γ

Recently, Singh et al. [33] have given the external solution, i.e., the velocity at the edge of the boundary ¯ n R(t ∗ ) which is relayer ue (x, t) = u∗e R(t ∗ ) = u∞ (x) lated to the exponent n by n=

x¯ du∗e , u∗e d x¯

where the exponent n is also related to the included angle of the wedge πβ by n=

β . 2−β

3x due ue dx

or

3x dU , U dx

where the exponent m is related to the cone half-angle γ is given by m=

1 1  6 νx 2 m+3 U 2 , ξ= , η=y 6 νx m+3 U   3γ 3x due m + 3 dU ∗ t, m= = , t = 6 dx π −γ ue dx 1  2     6 νxU φ t ∗ f ξ, η, t ∗ , ψ(x, y, t) = m+3 

∂ψ , ∂y

∂ψ , ∂x     fη ξ, η, t ∗ = F ξ, η, t ∗ ,     u = U φ t ∗ F ξ, η, t ∗ , u=

v=−

(6)

  T − T∞ G ξ, η, t ∗ = , T w − T∞  1 6 νU 2  ∗  v = −2−1 φ t m+3 x     3−m m+3 f+ (ξfξ − ηF ) × 3 3     dφ m−3 ∗ −1 t ft ∗ + φ f , +2 3 dt ∗ to Eqs. (1)–(3), it is found that Eq. (1) is identically satisfied, and Eqs. (2) and (3) reduce to nondimensional form, as are given by: 

In our present study, the external solution is given by ue (x, t) = U (x)φ(t ∗ ) = u∞ x m/3 φ(t ∗ ) which is related to the exponent m by m=

Applying the following non-similarity transformations

3γ . π −γ

 3−m ξ φ(F Fξ − Fη fξ ), (7) m+3 m Pr−1 Gηη + φf Gη − Gt ∗ 3    m − 3 ∗ dφ ∗ ∗ t +2 f G + φ(f G − F G ) η t η t m+3 dt ∗ 

=

Hence, the external solution ue (x, t) is related to the cone half-angle γ by x due γ = π −γ ue dx

   2m Fηη + φf Fη + φ 1 − F2 m+3   6 λφ −1 G + m+3  m dφ −1 φ (1 − F ) − Ft ∗ + 3 dt ∗     m − 3 ∗ dφ  +2 t 1 − F 2 + f Fη m+3 dt ∗ + φ(ft ∗ Fη − F Ft ∗ )

or

x dU . U dx

Meccanica (2014) 49:673–686

 =

677

 3−m ξ φ(F Gξ − Gη fξ ), m+3

(8)

where Pr =

μCp , k

u∞ x Rex = ν

Grx = m+3 3

gβx 3 (Tw − T∞ ) cos γ , ν2

and λ =

Grx . Re2x

It may be noted that for m = 32 , λ becomes a constant and numerical solutions are computed for different values of λ as discussed in results and discussion section. Here ξ, η are the transformed co-ordinates; η∞ is the edge of the boundary layer; ψ and f are dimensional and dimensionless stream functions, respectively; F and G are dimensionless velocity and temperature, respectively; μ is the dynamic viscosity; Cp is the specific heat at constant pressure; k is the thermal conductivity; Grx is the local Groshof number; Rex is the Reynolds number; λ is the buoyancy parameter. The transformed boundary conditions are     F ξ, 0, t ∗ = 0, G ξ, 0, t ∗ = 1, at η = 0, (9)     F ξ, ∞, t ∗ = 1, G ξ, ∞, t ∗ = 0, as η → ∞,

η where f = 0 F dη + fw and fw can be calculated from the following equation   3−m φξ(fξ )w φfw + m+3      m−3 ∗ vw dφ +2 t φ(ft ∗ )w + ∗ fw = − ξ m+3 dt ν Hence fw =

   3+m −1 3 + m −1 − 3+m ξ φ ξ 3−m vw (ξ )ξ 3−m dξ ν 3−m 0

Eqs. (7) and (8) under the boundary conditions (9) can be solved numerically. In particular, the effect of nonuniform slot suction (injection) into an unsteady mixed convection boundary layer flow over a vertical cone with single and double slot have been studied. We have assumed here that the flow is steady at time t ∗ = 0 and becomes unsteady for t ∗ > 0 due to the time dependent free stream velocity m where ue (x, t) = U (x)φ(t ∗ ) = u∞ x 3 φ(t ∗ )

φ(t ∗ ) = 1 + ε(t ∗ )2 ; ε > 0 or ε < 0. Hence the initial conditions (i.e., conditions at t ∗ = 0) are given by the steady state equations obtained from the Eqs. (7) dφ and (8) by substituting φ(t ∗ ) = 1, dt ∗ = Ft ∗ = Gt ∗ = 0 ∗ when t = 0. It may be noted that the steady state equations with ξ = 0, λ = 0, m = 0 and A = 0 in the present problem are same as those of Kumari et al. [6]. 2.1 Single slot In the vertical cone, the velocity at the edge of the boundary layer and surface mass transfer are being functions of ξ and t ∗ gives rise to non-similarity. In the case of single slot, the value fw is given by ⎧ 0, ξ ≤ ξ0 ⎪ ⎪ ⎨ − 3+m −1 fw = Aφ ξ 3−m C(ξ, ξ0 ), ξ0 ≤ ξ ≤ ξ0∗ (10) ⎪ ⎪ ⎩ −1 − 3+m Aφ ξ 3−m C(ξ0∗ , ξ0 ), ξ ≥ ξ0∗ where the function   C(ξ, ξ0 ) = 1 − cos ω∗ (ξ − ξ0 ) . Here vw is taken as ⎧ 0, ξ ≤ ξ0 ⎪ ⎪ ⎪ ⎪ ⎨ −ν( 3−m )Aξ − 3+m 3−m ω ∗ sin{ω ∗ (ξ − ξ )}, 0 3+m vw (ξ ) = ∗ ⎪ ⎪ ξ0 ≤ ξ ≤ ξ 0 ⎪ ⎪ ⎩ 0, ξ ≥ ξ0∗ where ω∗ , ξ0 are the two free parameters which determine the slot length and slot location, respectively. The function vw (ξ ) is continuous for all the values of ξ and it has a non-zero value only in the interval [ξ0 , ξ0∗ ]. The reason for taking such a function is that it allows the mass transfer to change slowly in the neighbourhood of the leading and trailing edges of the slot. The discontinuities at the leading and trailing edges of the slot in the case of uniform mass transfer are removed by choosing a non-uniform mass transfer distribution along a stream-wise slot. The surface mass transfer parameter A > 0 or A < 0 indicates the suction or injection, respectively. 2.2 Double slot The use of double slot helps to maintain the maximum values of the velocity and temperature gradients

678

Meccanica (2014) 49:673–686

or equivalently skin friction and heat transfer coefficients as lower order of magnitudes compared to those quantities in single slot case [24]. In the case of double slot, where ω∗ is the slot length, ξ0 and ξ1 are the two free parameters which determine the starting point of the first and second slot locations, respectively. Hence the continuous function vw (ξ ) has a non-zero value only in the intervals [ξ0 , ξ0∗ ] and [ξ1 , ξ1∗ ]. The effect of non-uniform single and double slots on skin friction and heat transfer coefficients are compared in Fig. 11. The value fw is given by ⎧ 0, ξ ≤ ξ0 ⎪ ⎪ ⎪ ⎪ 3+m ⎪ ⎪ ⎪ Aφ −1 ξ − 3−m C(ξ, ξ0 ), ξ0 ≤ ξ ≤ ξ0∗ ⎪ ⎪ ⎪ 3+m ⎪ ⎪ ⎪ Aφ −1 ξ − 3−m C(ξ0∗ , ξ0 ), ξ0∗ ≤ ξ ≤ ξ1 ⎪ ⎪ ⎨ 3+m fw = Aφ −1 ξ − 3−m C(ξ0∗ , ξ0 ) (11) ⎪ ⎪ 3+m ⎪ − ⎪ + Aφ −1 ξ 3−m C(ξ, ξ ), ξ ≤ ξ ≤ ξ ∗ ⎪ 1 1 ⎪ 1 ⎪ ⎪ ⎪ 3+m ⎪ − ∗ −1 ⎪ Aφ ξ 3−m C(ξ , ξ0 ) ⎪ 0 ⎪ ⎪ ⎪ ⎩ − 3+m −1 + Aφ ξ 3−m C(ξ1∗ , ξ1 ), ξ ≥ ξ1∗ Here vw is taken as ⎧ 0, ξ ≤ ξ0 ⎪ ⎪ ⎪ ⎪ 3+m ⎪ ⎪ −ν( 3−m )Aξ − 3−m ω∗ sin{ω∗ (ξ − ξ0 )}, ⎪ ⎪ 3+m ⎪ ⎪ ⎪ ∗ ⎪ ⎪ ξ0 ≤ ξ ≤ ξ 0 ⎨ vw (ξ ) = 0, ξ0∗ ≤ ξ ≤ ξ1 ⎪ ⎪ ⎪ − 3+m ⎪ 3−m ω ∗ sin{ω ∗ (ξ − ξ )}, ⎪ −ν( 3−m 1 ⎪ 3+m )Aξ ⎪ ⎪ ⎪ ∗ ⎪ ξ1 ≤ ξ ≤ ξ 1 ⎪ ⎪ ⎪ ⎩ 0, ξ ≥ ξ1∗ The main physical quantities of interest are the local skin friction coefficient Cf and the local Nusselt number Nu, which represent the wall shear stress and the heat transfer rate, respectively [29]. These quantities are defined as: Cf =

2[μ( ∂u ∂y )]w ρU 2

= 2(Rex )

−1 2



m+3 6

1 2

  φ t ∗ (Fη )w .

Thus, 

m+3 (Rex ) Cf = 2 6 1 2

1 2

  φ t ∗ (Fη )w .

(12)

Here ρ is the density. Nu = −

[x( ∂T ∂y )]w Tw − T∞

= −(Rex )

1 2



m+3 6

1 2

(Gη )w ,

Thus, (Rex )

−1 2

 Nu = −

m+3 6

1 2

(Gη )w .

(13)

3 Numerical method and validation The set of non-linear coupled partial differential equations (7) and (8) along with the boundary conditions (9), represent a non-linear two point boundary value problem for partial differential equations which is solved by an implicit finite difference scheme in combination with the quasi-linearization technique [30, 31]. Applying quasi-linearization technique, the nonlinear coupled partial differential equations (7) and (8), are replaced by the following system of linear partial differential equations: i+1 + X1i Fηi+1 + X2i F i+1 + X3i Fξi+1 Fηη

+ X4i Fti+1 + X5i Gi+1 = X6i ∗

(14)

i i+1 i i+1 + Y3i Gi+1 Gi+1 ηη + Y1 Gη + Y2 Gξ t∗

+ Y4i F i+1 = Y5i

(15)

The coefficient functions with iterative index i are known and the functions with iterative index (i + 1) are to be determined. The corresponding boundary conditions of equations (14) and (15) are F i+1 = 0,

Gi+1 = 1

at η = 0,

(16)

F i+1 = 1,

Gi+1 = 0

at η = η∞ ,

(17)

where η∞ is the edge of the boundary layer. The coefficients in Eqs. (14) and (15) are given by     3−m m − 3 ∗ dφ ξ φfξ + 2 t f X1i = φf + m+3 m+3 dt ∗

Meccanica (2014) 49:673–686

679

 m−3 ∗ t φft ∗ , m+3     3−m 4m φF + ξ φFξ X2i = − m+3 m+3 

+2

m −1 dφ φ 3 dt ∗     m − 3 ∗ dφ t 2 ∗ F + φFt ∗ , +2 m+3 dt   3−m X3i = − ξ φF, m+3    m−3 ∗ m X4i = − +2 t φF , 3 m+3   6 λφ −1 , X5i = m+3       3−m 2m i 2 φ 1+F − ξ φF Fξ X6 = − m+3 m+3 +

m −1 dφ φ 3 dt ∗     m − 3 ∗ dφ  2 ∗ t 1 + F + φF F −2 t , m+3 dt ∗   3−m ξ φfξ = Pr φf + Pr m+3     m − 3 ∗ dφ t + 2 Pr f + φft ∗ , m+3 dt ∗   3−m ξ φF, = − Pr m+3    m−3 ∗ m +2 t φF , = − Pr 3 m+3     3−m m−3 ∗ ξ φGξ + 2 t φGt ∗ , = − Pr m+3 m+3   3−m ξ φF Gξ = − Pr m+3   m−3 ∗ t φF Gt ∗ . +2 m+3 −

Y1i

Y2i Y3i Y4i Y5i

At each iteration step, the system of linear partial differential equations (14) and (15) were expressed in difference form using the forward difference scheme in the η-direction and backward difference scheme in ξ and t ∗ -directions. Thus, in each step, the resulting equations were then reduced to a system of linear alge-

Fig. 2 Comparison of velocity and temperature profiles with those of Kumari et al. [6] when ξ = 0, λ = 0, m = 0, t ∗ = 0 and A=0

braic equations with a block tri-diagonal matrix, which is solved by using Varga’s algorithm [32]. To ensure the convergence of the numerical solution to the exact solution, the step sizes η, ξ and t ∗ are optimized and taken as 0.01, 0.005 and 0.1, respectively. The results presented here are independent of the step sizes at least up to the fourth decimal place. A convergence criteria based on the relative difference between the current and previous iteration values of the velocity and temperature gradients at the wall are employed. The solution is assumed to have converged and the iterative process is terminated when the difference reaches less than 10−4 . In order to validate the accuracy of our numerical method, solutions have been obtained for the steady state case (t ∗ = 0) with ξ = 0, λ = 0, A = 0 and m = 0 to compare the velocity and temperature profiles (F and G) with those Kumari et al. [6] for different values of Prandtl number, Pr = 0.733 and 6.7. The results are found to be in good agreement and the comparison is shown in Fig. 2.

4 Results and discussion In this section, representative numerical results are displayed with the help of graphical illustrations. Computations have been carried out for various values of physical parameters such as Pr(0.7 ≤ Pr ≤ 7.0),

680

Meccanica (2014) 49:673–686

Fig. 3 Effect of A and t ∗ on velocity profiles at ξ = 0.75 for φ(t ∗ ) = 1 + ε(t ∗ )2 when ε = 0.5, Pr = 0.7, m = 1.5 and λ = 1. Slot position [0.5, 1]

Fig. 4 Effect of A and t ∗ on temperature profiles at ξ = 0.75 for φ(t ∗ ) = 1 + ε(t ∗ )2 when ε = 0.5, Pr = 0.7, m = 1.5 and λ = 1. Slot position [0.5, 1]

λ(0 ≤ λ ≤ 1), A(−0.5 ≤ A ≤ 1), ε(−0.1 ≤ ε ≤ 0.5), t ∗ (0 ≤ t ∗ ≤ 1) and ω∗ = 2π . In all numerical computations m is taken as 1.5 and the edge of the boundary layer η∞ is taken between 4 and 6 depending on the values of the parameters. The results have been obtained for both accelerating (φ(t ∗ ) = 1 + ε(t ∗ )2 ; ε > 0, 0 ≤ t ∗ ≤ 1) and decelerating (φ(t ∗ ) = 1 + ε(t ∗ )2 ; ε < 0, 0 ≤ t ∗ ≤ 1) free stream velocity distributions of the fluid. The effect of surface mass transfer parameter A (A > 0 for suction and A < 0 for injection) on velocity and temperature profiles (F and G) for accelerating flow case φ(t ∗ ) = 1 + ε(t ∗ )2 with ε = 0.5, when λ = 1.0 and Pr = 0.7 at ξ = 0.75 (middle of the slot) are shown in Figs. 3 and 4. The velocity overshoot is observed for injection (A < 0) away from the wall within the boundary layer and the overshoot is decreased for suction (A > 0). Also, the overshoot in the velocity profiles reduced by the time dependent upstream velocity (see Fig. 3). Injection causes a decrease in the steepness of the velocity and temperature profiles. Further, the velocity and thermal boundary layer thicknesses increase with injection but suction reduces both velocity and thermal boundary layer thicknesses. Moreover, the velocity boundary layer thickness increase with time t ∗ but the thermal boundary layer thickness decrease with the increasing of time t ∗ . Figure 5 shows, the effect of buoyancy parameter (λ) and time t ∗ for accelerating and decelerating free

stream flows (φ(t ∗ ) = 1 + ε(t ∗ )2 , ε = 0.1 and ε = 1 −0.1) on skin friction coefficient (Cf (Rex ) 2 ) when Pr = 0.7, A = 0.5 and ξ = 0.75. In both accelerating and decelerating flow, the skin friction coefficient enhances with the buoyancy parameter. The physical reason is that the positive buoyancy force (λ > 0) implies favorable pressure gradient, and the fluid gets accelerated, which result in thinner momentum boundary layer and hence, a higher skin friction rate. Further, the skin friction coefficient increase with time t ∗ for accelerating flow, but it decrease for decelerating flow, for any λ. In particular, when λ = 1 and ξ = 0.75 the skin friction coefficient increase by approximately 5.5 %, for accelerating flow (ε = 0.1) but it decrease by approximately 6.1 % for decelerating flow (ε = −0.1), as time t ∗ increases from 0 to 0.8. Also, positive buoyancy implies favorable pressure gradient, and the fluid gets accelerated, which result in thinner thermal boundary layer and hence, a higher heat transfer rate at the wall. However, the figures are not displayed for the sake of brevity. Figure 6 displays the effect of Prandtl number for accelerating and decelerating free stream flows (φ(t ∗ ) = 1 + ε(t ∗ )2 , ε = 0.1 and ε = −0.1) on skin 1 friction coefficient (Cf (Rex ) 2 ) when λ = 1, ξ = 0.5 and A = 1.0. It is found from Fig. 6 that the skin friction coefficient decreases with increase of Prandtl number because the higher Prandtl number fluid means more viscous fluid which increases the boundary layer thickness and consequently reduce the shear

Meccanica (2014) 49:673–686

681

Fig. 5 Effect of λ and t ∗ on skin friction coefficient for φ(t ∗ ) = 1 + ε(t ∗ )2 when ξ = 0.75, Pr = 0.7, A = 0.5, m = 1.5 and λ = 1. Slot position [0.5, 1]

Fig. 6 Effect of Pr and t ∗ on skin friction coefficient for φ(t ∗ ) = 1 + ε(t ∗ )2 when ξ = 0.5, A = 1.0, λ = 1 and m = 1.5. Slot position [0.5, 1]

stress. To be more specific, for λ = 1 at t ∗ = 0.4, as Pr 1 increases from 0.7 to 7.0, Cf (Rex ) 2 decrease by approximately 16 %, for accelerating flow (ε = 0.1) and approximately 17 %, for decelerating flow (ε = −0.1). In case of accelerating flow, Fig. 6 shows that the skin friction coefficient increase with time t ∗ . In par1 ticular, for Pr = 0.7 the value of Cf (Rex ) 2 increase by approximately 4.4 %, as time t ∗ increases from 0

to 0.8. However, the skin friction coefficient decrease with time t ∗ for decelerating flow. For example, for 1 Pr = 0.7 the Cf (Rex ) 2 decrease by approximately 4.5 %, as time t ∗ increases from 0 to 0.8. Also, the higher Prandtl number fluid has lower thermal conductivity which results in thinner thermal boundary layer and hence a higher heat transfer rate at the wall. However, the effect of Pr on velocity and tempera-

682

Meccanica (2014) 49:673–686

Fig. 7 Effect of non-uniform slot suction on skin friction and heat transfer coefficients for φ(t ∗ ) = 1 + ε(t ∗ )2 when ε = 0.4, Pr = 0.7, λ = 1 and m = 1.5. Slot position [ξ0 = 0.5, ξ0∗ = 1]

Fig. 8 Effect of non-uniform slot injection on skin friction and heat transfer coefficients for φ(t ∗ ) = 1 + ε(t ∗ )2 when ε = 0.4, Pr = 0.7, λ = 1 and m = 1.5. Slot position [ξ0 = 0.5, ξ0∗ = 1]

ture profiles (F and G), and on heat transfer coeffi−1 cient (Nu(Rex ) 2 ) is not displayed here for the sake of brevity. Figures 7 and 8 shows, the effects of non-uniform slot suction (A > 0) and injection (A < 0) on skin 1 friction and heat transfer coefficients (Cf (Rex ) 2 and

−1

Nu(Rex ) 2 ) for different time levels t ∗ = 0 and t ∗ = 1. It is observed from Fig. 7, the upstream velocity gradient effect carries over beyond the location ξ = 1, displaying slight oscillation in skin friction profile at time t ∗ = 0 but there is no oscillation at time t ∗ = 1 and the temperature gradient effect carries over

Meccanica (2014) 49:673–686

683

Fig. 9 The downstream movement of single slot suction on skin friction and heat transfer coefficients for φ(t ∗ ) = 1 + ε(t ∗ )2 when ε = 0.4, Pr = 0.7, λ = 1 and m = 1.5. Slot positions are [ξ0 = 0.4, ξ0∗ = 0.9] and [ξ0 = 0.8, ξ0∗ = 1.3]

beyond the location ξ = 1, displaying slight oscillation in the heat transfer profile at time t ∗ = 1 but there is no oscillation at time t ∗ = 0. Both the skin friction and heat transfer coefficients are increase with time t ∗ . To be more specific, when time t ∗ increases from 0 to 1, for λ = 1, A = 0.5, Pr = 0.7 and ε = 0.4, the 1 −1 increase of Cf (Rex ) 2 and Nu(Rex ) 2 are approximately 20.5 % and 23 %, respectively, at the middle of the slot (ξ = 0.75). On the other hand, when time t ∗ increases from 0 to 1, for λ = 1, A = −0.5, Pr = 0.7 and 1 −1 ε = 0.4, the increase of Cf (Rex ) 2 and Nu(Rex ) 2 are approximately 12 % and 66 %, respectively, at the middle of the slot (ξ = 0.75). When t ∗ ≥ 0, in case of non-uniform slot suction (A > 0), the skin friction and heat transfer coefficients increase as the slot starts and attain their maximum values before the trailing 1 −1 edge of the slot. Finally Cf (Rex ) 2 and Nu(Rex ) 2 decrease from their maximum values and reaches a finite value. Non-uniform slot injection (A < 0) has the reverse effect. Suction thins the boundary layer and greatly increases the wall slope (skin friction and heat transfer). Blowing thickens the boundary layer and makes it reduces the skin friction and heat transfer coefficients. Figure 9 displays the effect of the downstream movement of the single slot locations at different time levels t ∗ = 0 and t ∗ = 1. It has a significant effect on 1 skin friction and heat transfer coefficients (Cf (Rex ) 2 and Nu(Rex )

−1 2

). It is observed from Fig. 9 that, when

the slot moves downstream at different time levels t ∗ = 0 and t ∗ = 1, the skin friction and heat transfer coefficients are decrease by suction. Non-uniform slot injection has the reverse effect. To be more specific, when λ = 1, Pr = 0.7, A = 0.5, ε = 0.4 and t ∗ = 1 1 −1 the decrease of Cf (Rex ) 2 and Nu(Rex ) 2 are approximately 38.9 % and 38.8 %, respectively, at the middle of the slots. Figure 10 shows the effect of non-uniform double slot suction on skin friction and heat transfer coeffi1 −1 cients (Cf (Rex ) 2 and Nu(Rex ) 2 ) at different time levels t ∗ = 0 and t ∗ = 1. In the case of double slot suction (A > 0), the skin friction and heat transfer coefficients increase as the slot begins and attain their maximum values before the trailing edge of the slot. 1 −1 Next, Cf (Rex ) 2 and Nu(Rex ) 2 decrease their maximum values at the trailing edge of the slot. Similar variations of skin friction and heat transfer coefficients are observed in the second slot, but the effect of suc1 −1 tion on Cf (Rex ) 2 and Nu(Rex ) 2 is less in the second slot when compared to the first slot, and finally beyond 1 the trailing edge of the second slot Cf (Rex ) 2 and −1

Nu(Rex ) 2 remain finite in magnitude. Non-uniform double slot injection (A < 0) has the reverse effect. Figure 11 displays, the effect of mass flow rate on skin friction and heat transfer coefficients, when the single slot and double slot are compared. The mass flow rates A = 0.25 and A = 0.25 are applied into the double slot locations at ξ0 = 0.5 and ξ1 = 1.5, re-

684

Meccanica (2014) 49:673–686

Fig. 10 Effect of non-uniform double slot suction on skin friction and heat transfer coefficients for φ(t ∗ ) = 1 + ε(t ∗ )2 when ε = 0.4, Pr = 0.7, λ = 1 and m = 1.5. Slot positions are [ξ0 = 0.5, ξ0∗ = 1] and [ξ1 = 1.5, ξ1∗ = 2]

Fig. 11 Effect of single and double slot suction on skin friction and heat transfer coefficients for φ(t ∗ ) = 1 + ε(t ∗ )2 when ε = 0.4, t ∗ = 1, Pr = 0.7, λ = 1 and m = 1.5. (—) double slot; (− − −) single slot; (− · − · −·) no slot. Slot positions are [ξ0 = 0.5, ξ0∗ = 1] and [ξ1 = 1.5, ξ1∗ = 2]

spectively. Further, the total mass flow rate A = 0.5 (0.25 + 0.25) in double slot located at ξ0 = 0.5 and ξ1 = 1.5 is applied into the single slot location at ξ0 = 0.5. It is noticed in Fig. 11 that the total mass flow rate of double slot is less effective on skin friction and heat transfer coefficients when compared to the single

slot having the same total mass flow rate. Thus, the result presented in Fig. 11 indicates that the mass flow rate and position of the slot having significant effect on skin friction and heat transfer coefficients. But it may be remarked that the use of double slot helps to maintain the maximum values of the skin friction and

Meccanica (2014) 49:673–686

heat transfer coefficients as lower order of magnitudes compared to those quantities in single slot case (see Fig. 11). For example, in Fig. 11 the maximum values of skin friction and heat transfer coefficients are approximately 4.8 and 1.2, respectively, for double slots whereas the maximum values of those corresponding quantities are approximately 6.5 and 1.8, respectively, for single slot.

5 Conclusion Non-similar solution of an unsteady mixed convection boundary layer flow over vertical cone through nonuniform slot suction (injection) has been investigated numerically. We conclude as follows: • The over shoot in the velocity profiles reduced by the time dependent upstream velocity as well as suction. • Skin friction coefficient decrease with the increase of Prandtl number, whereas the Nusselt number increase with Prandtl number. • The skin friction and heat transfer coefficients increase with the increase of mass transfer rates, i.e., the value of A (A > 0). • Non-uniform slot injection helps to reduce skin friction and heat transfer coefficients at a particular stream-wise location on the cone surface. • When the slot moves downstream direction, the skin friction and heat transfer coefficients reduced by suction. Non-uniform slot injection has the reverse effect. 1 • Double slot is less effective on Cf (Rex ) 2 and −1

Nu(Rex ) 2 when compared to single slot having the same total mass flow rate. • The velocity and thermal boundary layer thicknesses increase with non-uniform slot injection but the non-uniform slot suction reduces both velocity and thermal boundary layer thicknesses. • Skin friction and heat transfer coefficients are strongly affected by the time dependent free stream velocity which confirms the importance of present investigation of unsteady mixed convection flow. It is worth observing that, non-similar solutions have been obtained through non-uniform slot injection (suction) over (i) a cylinder and (ii) a sphere [18], rotating sphere [19] and yawed cylinder [20], respectively. Later, non-uniform double slot injection (suction) over a diverging channel [22], sphere [24] and

685

cylinder [25], respectively. In all these studies, flow separation is identified. Separation can be delayed by non-uniform single slot suction and also moving the slot downstream but the non-uniform slot injection is just the opposite. Double slot suction is found more effective compared to single slot suction in delaying the separation. But the separation phenomena is not observed in the present study. Acknowledgements Authors express sincere thanks to the anonymous reviewers for their detailed and very useful comments in improving the quality of the manuscript. One of the authors (M. Ganapathirao) is thankful to the Ministry of Human Resource Development, the Government of India for the grant of a fellowship to pursue this work.

References 1. Hering RG, Grosh RJ (1963) Laminar combined convection from a rotating cone. J Heat Transf 85:29–34 2. Himasekhar K, Sarma PK, Janardhan K (1989) Laminar mixed convection from a vertical rotating cone. Int Commun Heat Mass Transf 16:99–106 3. Wang CY (1990) Boundary layers on rotating cones, discs and axi-symmetric surfaces with a concentrated heat source. Acta Mech 81:245–251 4. Anilkumar D, Roy S (2004) Unsteady mixed convection flow on a rotating cone in a rotating fluid. Appl Math Comput 155:545–561 5. Roy S, Anilkumar D (2004) Unsteady mixed convection from a rotating cone in a rotating fluid due to combined effects of thermal and mass diffusion. Int J Heat Mass Transf 47:1673–1684 6. Kumari M, Pop I, Nath G (1989) Mixed convection along a vertical cone. Int Commun Heat Mass Transf 16:247–255 7. Takhar HS, Nath G (1997) Similarity solution of unsteady boundary layer equations with a magnetic field. Meccanica 32:157–163 8. Yih KA (1999) Mixed convection about a cone in a porous medium: the enire regime. Int Commun Heat Mass Transf 26:1041–1050 9. Roy S, Takhar HS, Nath G (2004) Unsteady MHD flow on a rotating cone in a rotating fluid. Meccanica 39:271–283 10. Ece MC, Ozturk A (2009) Boundary-layer flow about a vertical spinning cone under mixed thermal boundary conditions and a magnetic field. Meccanica 47:177–187 11. Postelnicu A (2012) Free convection from a truncated cone subject to constant wall heat flux in a micro polar fluid. Meccanica 47:1349–1357 12. Ali M, Al-Salem K (2013) The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction. Meccanica. doi:10. 1007/s11012-012-9687-2 13. Chamkha AJ, Abbasbandy S, Rashad AM, Vajravelu K (2013) Radiation effects on mixed convection about a cone embedded in a porous medium filled with a nanofluid. Meccanica 48:275–285

686 14. Ravindran R, Roy S, Momoniat E (2009) Effects of injection (suction) on a steady mixed convection boundary layer flow over a vertical cone. Int J Numer Methods Heat Fluid Flow 19:432–444 15. Bhattacharya K, Layek GC (2012) Similarity solution of MHD boundary layer flow with diffusion and chemical reaction over a porous flate plate with suction/blowing. Meccanica 47:1043–1048 16. Merkin JH, Bachok N, Jaradat MA, Pop I (2013) A similarity solution for the flow and heat transfer over a moving permeable flat plate in an external free stream: case of strong injection. Meccanica 48:297–306 17. Minkowycz WJ, Sparrow EM, Schneider GE, Pletcher RH (1988) Handbook of numerical heat transfer Wiley, New York, pp 140–144 18. Saikrishnan P, Roy S (2003) Non-uniform slot injection (suction) into water boundary layers over (i) a cylinder and (ii) a sphere. Int J Eng Sci 41:1351–1365 19. Roy S, Saikrishnan P (2003) Non-uniform slot injection (suction) into steady laminar water boundary layer flow over a rotating sphere. Int J Heat Mass Transf 46:3389–3396 20. Roy S, Saikrishnan P (2004) Non-uniform slot injection (suction) into water boundary layer flow past yawed cylinder. Int J Eng Sci 42:2147–2157 21. Datta P, Anilkumar D, Roy S, Mahanti NC (2006) Effect of non-uniform slot injection (suction) on a forced flow over a slender cylinder. Int J Heat Mass Transf 49:2366–2371 22. Roy S, Saikrishnan P (2008) Multiple slot suction/injection into an exponentially decreasing free stream flow. Int Commun Heat Mass Transf 35:163–168

Meccanica (2014) 49:673–686 23. Roy S, Datta P, Ravindran R, Momoniat E (2007) Non-uniform double slot injection (suction) on a forced flow over a slender cylinder. Int J Heat Mass Transf 50:3190–3194 24. Roy S, Saikrishnan P, Pandey BD (2009) Influence of double slot suction (injection) into water boundary layer flows over sphere. Int Commun Heat Mass Transf 36:646–650 25. Saikrishnan P, Roy S, Rizwan Sadiq M, Pandey BD (2009) Non-uniform double slot suction (injection) into water boundary layer flows over a cylinder. Int J Heat Mass Transf 52:894–898 26. Roy S, Datta P, Mahanti NC (2007) Non-similar solution of an unsteady mixed convection flow over a vertical cone with suction or injection. Int J Heat Mass Transf 50:181–187 27. Schlichting H, Gersten K (2000) Boundary-layer theory Springer, New York, pp 86–89 28. Bejan A (2004) Convection heat transfer. Wiley, New York 29. Pop I, Ingham DB (2001) Convective heat transfer: mathematical and computational modelling of viscous fluids and porous media. Pergamon, Oxford 30. Inouye K, Tate A (1974) Finite difference version of quasilinearization applied to boundary layer equations. AIAA J 12:558–560 31. Bellman RE, Kalaba RE (1965) Quasilinearization and nonlinear boundary-value problems. Elsevier, New York 32. Varga RS (2000) Matrix iterative analysis. Springer, New York 33. Singh PJ, Roy S, Ravindran R (2009) Unsteady mixed convection flow over a vertical wedge. Int J Heat Mass Transf 52:415–421