The study of boundary layer flow over a stretching sheet in the presence of chemical reaction and heat generation or absorption has many practical applications such as extrusion process,drawing of copper wires,polymer industries,and fibre industry. Sakiadis^[1] was the first to study the boundary layer flow on moving solid surfaces. He obtained a numerical solution of the governing boundary layer equation by the similarity transformation. Tsou et al.^[2] investigated analytically and experimentally the flow and heat transfer problem on a continuous moving surface. Consequently,various researchers^{[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]} have studied the aspects of the stretching surface and plate with heat transfer characteristics. All the above mentioned studies dealt with the steady flow.

In many practical situations,the flow could be unsteady due to the time dependent free stream velocity. In general,unsteady mixed convection boundary layer flows do not necessarily allow similarity solutions where non-similarity has to be taken into account. The unsteadiness and non-similarity in such flows may be due to the time dependent free stream velocity,the surface mass transfer,the curvature of the body,or all these effects. Since the mathematical difficulties are involved in obtaining non-similar solutions for such flow problems,many researchers have confined their studies either to steady non-similar flows or to unsteady semisimilar or self-similar flows^{[19, 20, 21, 22, 23, 24, 25, 26]}.

Moreover,the study of mass transfer from a wall slot into the boundary layer is of interest for various prospective applications such as thermal protection,energizing of 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 affects the development of a boundary layer along a surface of the stretching sheet. If we choose uniform mass transfer in a slot,finite discontinuities arise at the leading and trailing edges of the slot,and those can be avoided by choosing non-uniform mass transfer along a stream-wise slot. Some researchers focused their work on non-uniform slot mass transfer into an unsteady boundary layer flow problem. For example,recently,the unsteady mixed convection flow over a vertical cone with non-uniform slot suction/injection was investigated by Ganapathirao et al.^[27]. So far,no attempt has been made to study the effect of non-uniform slot suction/injection into an unsteady mixed convection flow over a vertical stretching sheet.

In certain applications such as those involving heat removal from nuclear fuel debris,underground disposal of radioactive waste materials,storage of food stuffs,and exothermic chemical reactions,the working of fluid heat generation or absorption effects are important. In addition,in many chemical engineering processes,chemical reactions take place between a foreign mass and the working fluid which moves due to the stretching of a surface. These processes take place in numerous industrial applications,namely,polymer production,manufacturing of ceramics or glassware,food procession,etc. The order of chemical reaction depends on many factors. One of the simplest chemical reactions is the first-order reaction in which the rate of reaction is directly proportional to the species concentration. Recently,a first-order chemical reaction and heat generation or absorption effects on the unsteady mixed convection flow over a vertical cone and wedge with non-uniform slot mass transfer were studied by Ravindran et al.^[28] and Ganapathirao et al.^[29],respectively.

The aim of the present work is to study the effect of non-uniform slot suction/injection into an unsteady mixed convection flow over a vertical stretching sheet in the presence of chemical reaction and heat generation/absorption. The governing boundary layer equations are first transformed into a set of non-similar equations and then solved numerically by an implicit finite difference scheme in combination with the quasi-linearization technique. The numerical results are compared with the previously published data,and the results are found to be in good agreement.

Let us consider the unsteady mixed convection laminar boundary layer flow along a vertical stretching sheet,which moves in a vertically upward direction. We also consider the presence of a first-order chemical reaction and the influence of heat generation or absorption. The x-axis is taken along a vertical stretching sheet,and the y-axis is normal to it. The coordinate system and the flow model are shown in Fig. 1.

Fig. 1 Physical model and coordinate system

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It is assumed that both the free stream velocity U_∞ and the sheet velocity U_w are moving in the same direction. The continuous stretching surface is assumed to have the power-law velocity U_w(x) = U_w0x^m as well as the power-law wall temperature T_w(x) = T_∞ + Bxⁿ,where T_∞ is the uniform ambient temperature,and B is a constant. The power-law wall concentration C_w (x) = C_∞ + B*xⁿ,where C_∞ is the uniform ambient concentration,and B* is a constant. The surface of the sheet is maintained at a variable wall temperature T_w and a concentration C_w when B > 0 (T_w > T_∞) corresponding to a heated sheet (assisting flow),B < 0 (T_w < T_∞) corresponding to a cooled sheet (opposing flow),and B* > 0 (C_w > C_∞).

The buoyancy force arises due to the temperature and concentration differences in the fluid. A heat source or sink is placed within the flow to allow possible heat generation or absorption effects. The concentration of the diffusing species is assumed to be very small in comparison with other chemical species far away from the surface of the sheet C∞ and is infinitesimally small.

Under the foregoing assumptions,along with the Boussinesq approximation,the governing equations for continuity,momentum,energy,and concentration are as follows:

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The initial conditions are given by

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The associated boundary conditions are

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where U ^w(x) ≥ 0,U^∞(x) ≥ 0,and vw is the mass transfer velocity. Here,we consider the unsteady free stream distribution of the form φ(τ) = 1+α(τ)2,where α > 0 is for an accelerating flow and α < 0 for a decelerating flow. We note that the velocity of the stretching sheet and the free stream velocity are given by U ^w(x) = U^w0x^m and U_∞(x) = U_∞0x^m,respectively.

The composite reference velocity is defined by

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Introduce the following transformations to Eqs. (1)-(4):

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The continuity equation (1) is automatically satisfied,and Eqs.(2)-(4) reduce to

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where

subject to the boundary conditions

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in which ε corresponds to the ratio of the free stream velocity to the composite reference velocity,and ε is a constant. The similarity transformation is based on the composite reference velocity U(x) (see Abdelhafez^[3]).

(i) When ε = 1,i.e.,U_w = 0,it is the classical Blasius flat-plate flow problem which means that the sheet is at rest,and the fluid is in motion.

(ii) When ε = 0,i.e.,U_∞ = 0,the fluid is at rest,and the motion is created by the sheet (see Soundalgekar and Murty^[4]) and the equations for the stretching sheet (as stipulated by Afzal and Varshney^[5]).

(iii) When 0 < ε < 1,i.e.,U_w > 0 and U_∞ > 0,both the sheet and fluid are in motion,and they are moving in the same direction.

In this study,we take that both the stretching sheet velocity U ^w(x) and the free stream velocity U_∞(x) are moving in the same direction. Take ε = 0.5 throughout the computation corresponding to the parallel flow solution.

From the boundary conditions,it can be obtained that f_η = F,i.e.,

where f_w is given by the relation

where the mass transfer velocity is v_w with v_w <0 for suction and v_w >0 for injection or blowing. Here,we study the non-uniform single and double slot suctions/injections into a boundary layer flow over a vertical stretching sheet given in Sections 2.1 and 2.2,respectively.

In the single slot suction/injection,the mass transfer velocity vw is taken as the sinusoidal function given by

Using the above v_w,f_w can be written as

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where ω* and ξ₀ are the two free parameters which determine the slot length and slot location,respectively. The function v_w is continuous for all 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 neighborhood of the leading and trailing edges of the slot. The surface mass transfer parameter A > 0 or A < 0 indicates the suction or injection,respectively.

In the case of double slot,v_w is chosen as

Hence,the expression for f_w is

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where the function

Here,ω*,ξ₁,and ξ₂ are three free parameters which determine the starting points of the first and second slot locations,respectively. The continuous function v_w has a non-zero value only in the intervals [ξ₁,ξ₁ *] and [ξ₂, ξ₂ *].

We assume that the flow is steady at the time τ = 0 and becomes unsteady for τ > 0 due to the time dependent free stream velocity

where φ(τ) = 1 + α(τ)₂, α > 0 or α < 0. Hence,the initial conditions (i.e.,the conditions at τ = 0) are given by the steady state equations obtained from (9)-(11) by substituting

when τ = 0.

The physical quantities are given by the local skin friction coefficient

i.e.,

the heat transfer coefficient in terms of the local Nusselt number

i.e.,

and the mass transfer coefficient in terms of the local Sherwood number

i.e.,

The coupled nonlinear partial differential equations (9)-(11) along with the boundary conditions (12) and (13) are solved numerically by an implicit finite difference scheme in combination with the quasi-linearization technique^[30]. With the help of quasi-linearization,the nonlinear coupled partial differential equations (9)-(11) are replaced by the following set of linear partial differential equations:

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The coefficient functions with the iterative index (i) are known,and the functions with the iterative index (i + 1) are to be determined.

The corresponding boundary conditions become

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The coefficients in Eqs. (17)-(19) are

The finite difference scheme is used in Eqs. (17)-(19). The central difference formulae are used in the η-direction,and the backward difference formulae are used in the ξ- and τ-directions with constant step sizes Δη,Δξ,and Δτ in the η-,ξ- and τ-directions,respectively. The resulting system of linear algebraic equations with a block tri-diagonal matrix is solved by Varga’s algorithm^[31]. The step sizes in the η-,ξ- and τ-directions are chosen as Δη = 0.01,Δξ = 0.01,and Δτ = 0.1,respectively. A convergence criterion based on the relative difference between the current and previous iteration values is used. The iteration procedure is repeated until the convergence criterion

is achieved.

Computations are carried out for different values of P r (0.0 ≤ P r ≤ 100.0),A (−0.5 ≤ A ≤ 1.0),λ (−0.6 ≤ λ ≤ 3.0),S (−6.0 ≤ S ≤ 6.0),Δ (−5.5 ≤ Δ ≤ 4.5),Sc (0.22 ≤ Sc ≤ 2.57),m (0.0 ≤ m ≤ 1.0),n (−2.0 ≤ n ≤ 2.0),ε (0.0 ≤ ε ≤ 1.0),N (0.0 ≤ N ≤ 0.5),and τ (0 ≤ τ ≤ 1). In all the numerical computations,the edge of the boundary layer η_∞ is taken between 5 and 8,which depends upon the values of the parameters. The results are obtained for both the accelerating (φ(τ) = 1 + α(τ)²,α > 0,0 ≤ τ ≤ 1) and decelerating (φ(τ) = 1 + α(τ)²,α < 0,0 ≤ τ ≤ 1) free stream velocity distributions of the fluid. In order to validate the accuracy of our numerical method,the steady state results of the heat transfer parameter −G_η(ξ,0),velocity,and temperature profiles (F and G) are compared with those of Tsou et al.^[2],Soundalgekar and Murty^[4],Chen^[6],Ali^[7],Jacobi^[8],Ali^[9],Chen and Strobel^[10],Moutsoglou and Chen^[11],Afzal^[12],Grubka and Bobba^[13],Patil^[16],and Patil et al.^[26]. The results are found in excellent agreement,and the comparisons are shown in Tables 1-2 and Figs.2-3.

Table 1 Comparison of steady state results −G_η(ξ, 0) for λ = 0, τ = 0, ξ = 0, ε = 0, m = 0, n = 0,N = 0, S = 0, Δ = 0, Sc = 0, A = 0, and selected values of P r with previously published works

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Table 2 Comparison of steady state results −G_η(ξ, 0) for m = 1.0, λ = 0, τ = 0, ξ = 0, ε = 0,N = 0, S = 0, Δ = 0, Sc = 0, and A = 0 for various values of P r and n

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It is noticed from Table 1 that the surface heat transfer parameter −G_η(ξ,0) increases significantly with Pr as the higher Prandtl number fluid has a lower thermal boundary layer. Hence,a higher surface heat transfer rate occurs. It is observed from Table 2 that the increase in the temperature exponent parameter n(> 0) causes an increase in the temperature gradient while it decreases for n = −2.

The comparison of velocity profiles (F) with those of Afzal^[12] when λ = 0,n = 1.0,ξ = 0,A = 0,m = 1.0,N = 0,S = 0,Sc = 0,Pr = 0,Δ = 0,and τ = 0 for different values of ε (the ratio of the free stream velocity to the composite reference velocity) is displayed in Fig. 2. The results indicate that there is a gradual variation in the velocity profile when the velocity ratio ε increases from 0 to 1. The effect of n on the temperature profile (G) compared with that of Grubka and Bobba^[13] when λ = 0,ε = 0,ξ = 0,A = 0,m = 0,N = 0,S = 0,Sc = 0,Pr = 0.72,Δ = 0,and τ = 0 is shown in Fig. 3. It is observed that the temperature profile (G) decreases monotonically with an increase in the temperature exponent parameter (n).

Fig. 2 Comparison of velocity profiles with those of Afzal[12]

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Fig. 3 Comparison of temperature profiles with those of Grubka and Bobba[13]

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The effects of the surface mass transfer A and the velocity exponent parameter m on the velocity profiles (F) are shown in Fig. 4 for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5,when λ = 1.0,n = 1.0,ω* = 2π,ε = 0.5,Pr = 0.7,S = 0.5,Δ = 0.5,Sc = 0.22,N = 0.5, τ = 1.0,and ξ = 0.75. The results indicate that the high velocity overshoot is observed near the wall of the stretching sheet within the boundary layer for injection (A < 0),and the overshoot is reduced for suction (A > 0). The magnitude of the velocity profile increases with the uniform motion (m = 0),while it decreases with the linear stretched surface velocity (m = 1).

Fig. 4 Effects of A and m on velocity profiles with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Figures 5-6 depict the effects of the surface mass transfer parameter A and the temperature exponent parameter n on the temperature and concentration profiles (G and H) for the accelerating flow case φ(τ) = 1+ α(τ)² with α = 0.5,when λ = 1.0,m = 1.0, ω* = 2π,ε = 0.5,Pr = 0.7,S = 0.5,Δ = 0.5,Sc = 0.22,N = 0.5,τ = 1.0,and ξ = 0.75. The injection (A < 0) causes a decrease in the steepness of the temperature and concentration profiles near the wall of the stretching sheet within the boundary layer. However,the steepness of the temperature and concentration profiles increases with the suction (A > 0). Further,the thermal and concentration boundary layer thicknesses increase for the uniform surface temperature (n = 0),while they decrease for the linear stretched surface temperature (n = 1). Moreover,the thermal and concentration boundary layer thicknesses increase with injection while they decrease with suction.

Fig. 5 Effects of A and n on temperature profiles with slot position [ξ0 =0.5, ξ0 * = 1.0]

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Fig. 6 Effects of A and n on concentration profiles with slot position [ξ0 =0.5, ξ0 * = 1.0]

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Figure 7 illustrates the effects of ε (the ratio of the free stream velocity to the composite reference velocity) and the time τ on the velocity profiles (F) for both the accelerating and decelerating free stream flows (φ(τ) = 1+α(τ)² with α = 0.5 and α=−0.5) for λ=1.0,m=1.0,ω*=2π,n = 1.0,Pr = 0.7,S = 0.5,Δ = 1.0,Sc = 0.22,N = 0.5,A = 1.0,and ξ = 0.75. It is clear that the magnitude of the velocity profile in the boundary layer increases with the increase of ε,while it decreases as τ increases for α > 0. The reason is that assisting the buoyancy ratio forces N > 0,when ε is increased,it acts like a favorable pressure gradient,which accelerates the fluid and hence the velocity overshoot occurs. The velocity overshoot reduces significantly when τ = 1. Further,the velocity profile increases within the stretching boundary layer for the decelerating flow (α < 0). However,it decreases with the accelerating flow (α > 0). When η = 1.0 and at a given time τ = 1.0,the velocity profile increases approximately by 15% when α decreases from 0.5 to −0.5 at ε = 1.0. This reveals that the effect of stretching boundary on decelerating flow is prominent.

Fig. 7 Effects of ε, α, and τ on velocity profiles with slot position [ξ0 = 0.5, ξ0 * =1.0]

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The effect of the chemical reaction parameter Δ on the concentration profiles (H) is displayed in Fig. 8 for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5,when λ = 1.0,m = 1.0,ω* = 2π,ε = 0.5,Pr = 0.7,S = 0.5,A = 0.5,Sc = 0.22,n = 0.0,N = 0.5,τ = 1.0,and ξ = 0.75. The strength of species consumption (Δ < 0) is relatively large,and the overshoot is observed in the concentration profiles within the boundary layer,while the overshoot is not observed for the species generation (Δ > 0). Moreover,the concentration boundary layer thickness reduces with the species generation (Δ > 0). However,it is opposite for the species consumption (Δ < 0). The physical reason is that the presence of species generation (Δ > 0) effect has the tendency to increase the concentration state of the fluid causing its concentration and concentration boundary layer to decrease. The variations in the velocity and temperature profiles (F and G) due to the chemical reaction parameter (Δ) are very small,and the profiles are not displayed here for the sake of brevity.

Fig. 8 Effect of Δ on concentration profiles with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Figure 9 reveals the effects of the buoyancy parameter λ,the Prandtl number Pr,and the time τ on the velocity profiles (F) for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5,when Δ = 0.5,m = 1.0,ω* = 2π,ε = 0.5,S = 0.5,A = 1.0,Sc = 0.22,n = 1.0,N = 0.5,and ξ = 0.75. In the case of buoyancy assisting flow (λ > 0),the velocity overshoot is obtained near the surface for a lower Prandtl number (Pr = 0.7,air),while the overshoot is not observed for a higher Prandtl number (Pr = 7.0,water). The physical reason is that the assisting buoyancy force is larger for the lower Prandtl number (Pr = 0.7) due to the low viscosity of the fluid which enhances the velocity profile within the moving boundary layer. Hence,the velocity overshoot occurs. It is evident that for the higher values of Pr,the overshoot is not presented because the higher Prandtl number fluid (Pr = 7.0,water) means the more viscous fluid which makes it less sensitive to the buoyancy parameter λ. For high Prandtl numbers,the magnitude of the velocity profile decreases. In addition,the velocity overshoot increases for τ = 0 while it decreases for τ = 1. Moreover,the buoyancy opposing flow (λ < 0) reduces the magnitude of the velocity profile near the surface for lower values of (Pr = 0.7) as well as for higher values of (Pr = 7.0).

Fig. 9 Effects of λ, P r, and τ on velocity profiles with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Figure 10 shows the effects of the heat generation or absorption parameter S and the time τ on the temperature profiles (G) for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5,when Δ = 1.0,m = 1.0,ω* = 2π,ε = 0.5,N = 0.5,A = 1.0,λ = 0.5,Sc = 0.22,n = 0.0,Pr = 0.7,and ξ = 1.0. Increasing the value of S produces an increase in the temperature distribution of the fluid. This can be expected because the heat generation (S > 0) causes the thermal boundary layer to become thicker. On the contrary,the heat absorption (S < 0) has the opposite effect,namely,cooling of the fluid and reducing the thermal buoyancy effect. Also,the thermal boundary layer thickness decreases with the heat absorption (S < 0). However,it is opposite for the heat generation (S > 0). Moreover,the thermal boundary layer thickness reduces as the time τ increases from 0 to 1.

Fig. 10 Effects of S and τ on temperature profiles with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Figure 11 shows the effects of the Schmidt number Sc and the time τ on the concentration profiles (H) for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5,when Δ = 0.5,m = 1.0,ω* = 2π,ε = 0.5,N = 0.5,A = 1.0,λ = 1.0,S = 0.5,n = 1.0,Pr = 0.7,and ξ = 0.75. It is observed from Fig. 11 that the values of Schmidt number are chosen to be realistic,helium (Sc = 0.22),water vapour (Sc = 0.60),and propyl benzene (Sc = 2.57) at 298.15 K at one atmospheric pressure. It is evident that the concentration boundary layer becomes thin as Sc increases. The physical reason is that higher values of Sc have a low mass diffusivity which leads to a thinning of the concentration boundary layer. The effect of Sc on the velocity and temperature profiles is very small because the physical parameter appears only in the concentration equation. Hence,the effect of Sc on those profiles is not presented here. The time effect is crucial for the thickness of the boundary layer. Hence,the concentration boundary layer thickness reduces with the increase in the time τ.

Fig. 11 Effects of Sc and τ on concentration profiles with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Figure 12 displays the effects of the non-uniform suction (A > 0) and the injection (A < 0) of a single slot located at ξ0 = 0.5 on the local Nusselt and Sherwood numbers (Nu_x(Re_x)^−1/2 and Sh _x(Re_x)^−1/2) for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5 and τ = 1.0,when Δ = 1.0,m = 1.0,ω* = 2π,ε = 0.5,N = 0.5,λ = 1.0,S = 0.5,n = 1.0,Sc = 0.22,Pr = 0.7,and ξ = 0.75. In the case of non-uniform slot suction,the local Nusselt and Sherwood numbers gradually increase from the leading edge of the slots,attain a maximum,and then start decreasing at the rear end of the slots. Also,heat and mass transfer coefficients increase with the increase in mass transfer rates,i.e.,A(A > 0). Non-uniform slot injection has the reverse effect. However,these are not a mirror reflection of each other. Hence,non-uniform slot injection helps to reduce the heat and mass transfer coefficients at a particular streamwise location on the surface of the stretching sheet.

Fig. 12 Effects of suction and injection on Nux(Rex)−1/2 and Shx(Rex)−1/2 with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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The effect of the chemical reaction parameter Δ on the local Sherwood number (Sh_x(Re_x)^−1/2) is shown in Fig. 13 for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5 and τ = 1.0,when A = 0.5,m = 1.0,ω* = 2π,ε = 0.6,N = 0.5,λ = 1.0,S = 0.5,n = 1.0,Sc = 0.60,Pr = 0.7,and ξ = 0.75. It is found that the local Sherwood number increases with the species generation (Δ > 0),while it decreases with the species consumption (Δ < 0). The physical reason is that the presence of species generation (Δ > 0) effect has the tendency to decrease the concentration state of the fluid causing its concentration and concentration boundary layer to decrease and consequently the negative concentration gradient. Hence,the local Sherwood number increases with the species generation (Δ > 0). Moreover,when the species consumption (Δ < 0) effects are present,the reverse trends are observed that both the fluid concentration and its concentration boundary layer increase,and consequently the negative concentration gradient occurs. Hence,the local Sherwood number decreases with the species consumption (Δ < 0). Therefore,the negative values in the mass transfer rate are due to the relatively large species consumption effects.

Fig. 13 Effect of Δ on Shx(Rex)−1/2 with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Figure 14 shows the effects of the heat generation and the absorption parameter S on the local Nusselt number (Nu_x(Re_x)^−1/2) for the accelerating flow case φ(τ) = 1 + α(τ)² with α = 0.5 and τ = 1.0,when A = 0.5,m = 1.0,ω* = 2π,ε = 0.5,N = 0.5,λ = 1.0,Δ = 0.5,n = 1.0,Sc = 0.22,Pr = 0.7,and ξ = 0.75. It can be seen that the local Nusselt number (Nu_x(Re_x)^−1/2) decreases with the heat generation (S > 0),while it increases with the heat absorption (S < 0). The physical reason is that the presence of heat absorption (S < 0) effect has the tendency to increase the thermal state of the fluid causing its temperature and thermal boundary layer to increase and consequently the negative temperature gradient. Hence,the heat transfer rate decreases with the heat generation (S > 0). The negative values in the heat transfer rate are due to the relatively large heat generation effects. Moreover,when the heat absorption (S < 0) effects are present,the reverse trends are observed where both the temperature and its thermal boundary layer decrease and consequently the negative temperature gradient,and hence the heat transfer rate increases with heat absorption (S < 0).

Fig. 14 Effects of heat generation and absorption on Nux(Rex)−1/2 with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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The effects of the non-uniform double slot suction (A > 0) and the injection (A < 0) on the local skin friction coefficient and the local Nusselt and Sherwood numbers (Cf_x(Re_x)^1/2,Nu ^x(Re_x)^−1/2,and Sh_x(Re_x)^−1/2) are shown in Fig. 15. In the case of double slot suction,the local skin friction coefficient and the local Nusselt and Sherwood numbers increase and attain their maximum values before the trailing edge of the first slot. Next,Cfx(Re_x)^1/2,Nu _x(Re_x)^−1/2,and Sh_x(Re_x)^−1/2 decrease from their maximum values at the trailing edge of the first slot. Similar variations are observed in the second slot.

Fig. 15 Effects of double slot suction and injection on Cfx(Rex)1/2, Nux(Rex)−1/2, and Shx(Rex)−1/2 with slot positions [ξ1 = 0.3, ξ1 * = 0.8] and [ξ2 = 1.0, ξ2 * = 1.5]

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The effects of the buoyancy parameter λ and the ratio of the free stream velocity to the composite reference velocity ε for both the accelerating and decelerating free stream flows (φ(τ) = 1 + α(τ)² with α = 1.0 and α = −1.0) on the local skin friction coefficient are shown in Fig. 16. The results indicate that the skin friction coefficient increases with the buoyancy parameter (λ) and the ratio of the free stream velocity to the composite reference velocity (ε). It is also observed that the effect of λ is more pronounced on Cfx(Re_x)^1/2. In particular,when Δ = 1.0,S = 0.5,N = 0.5,Sc = 0.22,ξ = 0.5,m = 1.0,n = 1.0,Pr = 0.7,A = 1.0,α = −1.0,and ε = 1.0 at the time τ = 0.2,the local skin friction coefficient is increased approximately by 56% as λ increases from −0.3 to 1.0. Moreover,Cfx(Re_x)^1/2 increases with the increase in the time τ for the accelerating flow α > 0,while it decreases for the decelerating flow α < 0.

Fig. 16 Effects of λ and ε on Cfx(Rex)1/2 with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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The effects of the Prandtl number Pr and the velocity exponent parameter m for the accelerating and decelerating free stream flows (φ(τ) = 1 + α(τ)² with α = 0.5 and α = −0.5) on the local skin friction coefficient (Cf_x(Re_x)^1/2) are displayed in Fig. 17. The skin friction coefficient decreases with the increase in the Prandtl number for both accelerating and decelerating flows. The reason is that the higher Prandtl number fluid implies the more viscous fluid which increases the boundary layer thickness. Consequently,the skin friction coefficient is reduced as Pr increases. In particular,when Δ = 1.0,S = 0.5,N = 0.5,Sc = 0.22,ξ = 0.5,ε = 0.5,n = 1.0,λ = 1.0,A = 0.5,m = 1.0,and α = −0.5 at the time τ = 0.5,the skin friction coefficient decreases approximately by 3.2% with the increase in the Prandtl number Pr = 0.7 to 7.0. Further,the velocity exponent parameter (m) is more effective on the skin friction coefficient. It is observed that the local skin friction coefficient increases with the uniform motion (m = 0),while it decreases with the linear stretched surface velocity (m = 1).

Fig. 17 Effects of P r and m on Cfx(Rex)1/2 with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Figure 18 shows the effect of the Prandtl number Pr and the temperature exponent parameter n for accelerating and decelerating free stream flows (φ(τ) = 1 + α(τ)² with α = 0.5 and α = −0.5) on the local Nusselt number (Nu_x(Re_x)^−1/2). It is observed that the local Nusselt number increases with the increase in Pr because the higher Prandtl number fluid implies the more viscous fluid which decreases the thermal boundary layer thickness. Consequently,the Nusselt number increases as Pr increases. The local Nusselt number increases with the increase in the temperature exponent parameter n for the linear stretching surface temperature (n = 1) while it decreases for the uniform surface temperature (n = 0). In particular,when Δ = 1.0,S = 0.5,N = 0.5,Sc = 0.22,ξ = 0.5,ε = 0.5,m = 1.0,λ = 1.0,A = 0.5,α = 0.5,and Pr = 0.7 at the time τ = 0.5,the percentage of increase in Nu_x(Re_x)^−1/2 due to the increase in n from 0 to 1 is about 58% approximately. The local Nusselt number increases for the accelerating flow,while it decreases for the decelerating flow.

Fig. 18 Effects of P r and n on Nux(Rex)−1/2 with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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The effects of the Schmidt number Sc and the temperature exponent parameter n for accelerating and decelerating free stream flows (φ(τ) = 1+α(τ)² with α = 0.5 and α = −0.5) on the local Sherwood number Sh _x(Re_x)^−1/2 are shown in Fig. 19. It is seen from Fig. 19 that when Sc increases from 0.22 to 2.57,the local Sherwood number increases. It is evident that the increase in Sc causes a reduction in the concentration boundary layer thickness,and consequently the concentration gradient at the wall increases with Sc. In particular,when Δ = 1.0,S = 0.5,N = 0.5,ξ = 0.5,ε = 0.5,m = 1.0,λ = 1.0,A = 0.5,α = −0.5,n = 1.0,and Pr = 0.7 at the time τ = 0.5,the Sherwood number increases by 72.4% approximately as Sc increases from 0.22 to 2.57. Hence,the local Sherwood number increases with Sc. Moreover,the local Sherwood number increases significantly with the time τ for the accelerating flow α > 0 while it decreases for the decelerating flow α < 0.

Fig. 19 Effects of Sc and n on Shx(Rex)−1/2 with slot position [ξ0 = 0.5, ξ0 * = 1.0]

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Non-uniform single and double slot suctions (injections) into an unsteady flow over a vertical stretching sheet with chemical reaction and heat generation or absorption effects are investigated numerically. The conclusions may be noted from the present study given as below:

(i) Suction thins the boundary layer and greatly increases the local skin friction coefficient and the local Nusselt and Sherwood numbers. However,injection thickens the boundary layer and makes it reduce the local skin friction and the local Nusselt and Sherwood numbers.

(ii) The skin friction coefficient is strongly affected by the time dependent free stream velocity than the local Nusselt and Sherwood numbers.

(iii) The buoyancy parameter (λ > 0) is found to cause an overshoot in the velocity profiles for the lower Prandtl number fluid (Pr = 0.7,air),and the overshoot is reduced by the increase in the time τ.

(iv) Heat generation causes a thicker thermal boundary layer whereas species generation causes a thinner concentration boundary layer.

(v) The ratio of the composite velocity to the free stream velocity ε causes a overshoot for the increase in the time τ.

(vi) The thermal and concentration boundary layer thicknesses decrease for the linear stretched surface temperature (n = 1),while they increase for the uniform surface temperature (n = 0).

(vii) The magnitude of the velocity increases for the uniform motion (m = 0),while it decreases for the linear stretched surface velocity (m = 1).

(viii) High values of the Schmidt number (Sc = 2.57) cause a thinner concentration boundary layer.

Acknowledgements The authors would like to express their sincere thanks to the reviewers for their valuable comments,which lead to the improvement of the manuscript. One of the authors (N. SAMYUKTHA) is thankful to Dr. M. GANAPATHIRAO of National Institute of Information Technology University for his helpful discussion.