Heat Mass Transfer DOI 10.1007/s00231-016-1917-z

ORIGINAL

Thermophoresis on boundary layer heat and mass transfer flow of Walters‑B fluid past a radiate plate with heat sink/source B. Vasu1 · Rama Subba Reddy Gorla2 · P. V. S. N. Murthy3 

Received: 30 March 2016 / Accepted: 12 September 2016 © Springer-Verlag Berlin Heidelberg 2016

Abstract  The Walters-B liquid model is employed to simulate medical creams and other rheological liquids encountered in biotechnology and chemical engineering. This rheological model introduces supplementary terms into the momentum conservation equation. The combined effects of thermal radiation and heat sink/source on transient free convective, laminar flow and mass transfer in a viscoelastic fluid past a vertical plate are presented by taking thermophoresis effect into account. The transformed conservation equations are solved using a stable, robust finite difference method. A parametric study illustrating the influence of viscoelasticity parameter (Γ), thermophoretic parameter (τ), thermal radiation parameter (F), heat sink/source (φ), Prandtl number (Pr), Schmidt number (Sc), thermal Grashof number (Gr), solutal Grashof number (Gm), temperature and concentration profiles as well as local skin-friction, Nusselt and Sherwood number is conducted. The results of this parametric study are shown graphically and inform of table. The study has applications in polymer materials processing. List of symbols x, y Coordinates along the plate generator and normal to the generator respectively u, v Velocity components along the x- and y-directions respectively

* Rama Subba Reddy Gorla [email protected] 1

Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211004, India

2

Department of Mechanical and Civil Engineering, Purdue University Northwest, Westville, IN 46391, USA

3

Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, India



g Gravitational acceleration t′ Time t Dimensionless time Gr Thermal Grashof number Gm Solutal Grashof number F Radiation parameter k0 Walters-B viscoelasticity parameter NuX Non-dimensional local Nusselt number Pr Prandtl number T′ Temperature T Dimensionless temperature C′ Concentration C Dimensionless concentration D Mass diffusion coefficient N Buoyancy ration number U, V Dimensionless velocity components along the Xand Y-directions respectively X, Y Dimensionless spatial coordinates along the plate generator and normal to the generator respectively Sc Schmidt number ShX Non-dimensional local Sherwood number Greek symbols α Thermal diffusivity β Volumetric thermal expansion coefficient β* Volumetric concentration expansion coefficient Γ Viscoelastic parameter τ Thermophoretic parameter Φ Heat sink/source parameter ν Kinematic viscosity Δt Dimensionless time-step ΔX Dimensionless finite difference grid size in X-direction ΔY Dimensionless finite difference grid size in Y-direction

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τX Dimensionless local skin-friction Subscripts w Condition on the wall ∞ Free stream condition

1 Introduction The role of thermal radiation is a major importance in some industrial applications such as glass production and furnace design and in space technology applications, such as cosmical flight aerodynamics rocket, propulsion systems, plasma physics and space craft reentry aerothermodynamics which operate at high temperatures. In the processes involving high temperatures, the radiation heat transfer in combination with conduction, convection and also mass transfer plays very important role in the design of pertinent equipments in the areas such as nuclear power plants, gas turbines and the various propulsion devices for air craft, missiles, satellites and space vehicles. The interaction of radiation with laminar free convection heat transfer from a vertical plate was investigated by Cess [1] for an absorbing, emitting fluid in the optically thick region, using the singular perturbation technique. Arpaci [2] considered a similar problem in both the optically thin and optically thick regions and used approximate integral technique and first order profiles to solve the energy equation. Cheng et al. [3] studied a related problem for an absorbing, emitting and isotropically scattering fluid, and treated the radiation part of the problem exactly with the normal mode expansion technique. Raptis [4] analyzed both the thermal radiation and free convection flow through a porous medium by using a perturbation technique. Hossain and Takhar [5] studied the radiation effects on mixed convection along a vertical plate with the uniform surface temperature using the Keller Box finite difference method. In all these papers, the flow taken steady, Mansour [6] studied the radiative and free convection effects on the oscillatory flow past a vertical plate. Raptis and Perdikis [7] considered the problem of thermal radiation and free convection flow past moving plate. Das et al. [8] analyzed the radiation effects on the flow past an impulsively started infinite isothermal vertical plate, the governing equations are solved by using Laplace transform technique. Chang et al. [9] analyzed the unsteady buoyancy-driven flow and species diffusion in a Walters-B viscoelastic flow along a vertical plate with transpiration effects. They showed that the flow is accelerated with a rise in viscoelasticity parameter with both time and locations close to the plate surface and increasing Schmidt number (Sc) suppresses both velocity and concentration in time whereas increasing species Grashof number (buoyancy parameter) accelerates flow through time. Prasad et al. [10]

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Heat Mass Transfer

studied Radiation and mass transfer effects on two-dimensional flow past an impulsively started infinite vertical plate. Observed when radiation parameter increases, velocity, temperature decreases and time taken to reach steady state increases. The natural convection process in the presence of heat source/sink is present in various physical phenomena such as fire engineering, combustion modelling, nuclear energy, heat exchangers, petroleum reservoir etc. The inducement of free convection fluid motion is due to buoyancy effects. Such flows have been studied more extensively because they are found frequently in nature as well as in engineering and environmental applications. Ghoshdastidar [11] gave various areas of typical applications of free convection such as those found in heat transfer from pipes and transmission lines as well as from various electronic devices, dissipation of heat from the coil of a refrigerator unit to the surrounding air, heat transfer from a heater to room air, heat transfer from nuclear fuel rods to the surrounding coolant, heated and cooled enclosures, quenching, wiredrawing and extrusion, atmospheric and oceanic circulation. These are encountered in a wide range of thermal engineering applications such as in geothermal systems, oil extraction, ground water pollution, thermal insulation, heat exchangers, storage of nuclear wastes, packed bed catalytic reactors and many more Bird et al. [12], Cheng [13] and Cheng [14]. Merkin [15] has studied the natural-convection boundary-layer flow on a vertical surface with Newtonian heating. The free convection boundary layer flow along a vertical surface in a porous medium with Newtonian heating has been presented by Lesnic et al. [16], Chaudhary and Jain [17] have investigated the problem of unsteady free convection boundary layer flow past an impulsively started vertical surface with Newtonian heating. Their study highlighted the usefulness of unsteady boundary layer flows, and discussed the problem of Newtonian heating and its applications. An exact solution to the unsteady free-convection boundary-layer flow past an impulsively started vertical surface with Newtonian heating has been studied by Chaudhary and Jain [18], Mebine and Adigio [19] have analyzed the unsteady free convection flow with thermal radiation past a vertical porous plate with Newtonian heating. Narahari and Ishak [20] have investigated the radiation effects on free convection flow past a moving vertical plate with Newtonian heating. Recently, Das et al. [21] have presented the radiation effects on unsteady free convection flow past a vertical plate with Newtonian heating. The study of heat generation or absorption effects in moving fluids is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reactions. Possible heat generation effects may alter the temperature distribution and consequently, the particle deposition rate in nuclear reactors, electric chips and

Heat Mass Transfer

semiconductor wafers. Seddeek [22] studied the effects of chemical reaction, thermophoresis and variable viscosity on steady hydromagnetic flow with heat and mass transfer over a flat plate in the presence of heat generation/absorption. Patil and Kulkarni [23] studied the effects of chemical reaction on free convective flow of a polar fluid through porous medium in the presence of internal heat generation. Double-diffusive convection–radiation interaction on unsteady MHD flow over a vertical moving porous plate with heat generation and soret effects was studied by Mohamed [24]. Radiation effects on an unsteady MHD convective heat and mass transfer flow past a semi-infinite vertical permeable moving plate embedded in a porous medium was studied by Prasad et al. [25]. Satyanarayana and Venkataramana [26] studied Hall current effect on magneto hydrodynamics free-convection flow past a semi-infinite vertical porous plate with mass transfer. Effects of the chemical reaction and radiation absorption on free convection flow through porous medium with variable suction in the presence of uniform magnetic field were studied by Sudheer Babu and Satyanarayana [27]. Pal et al. [28] studied Perturbation analysis of unsteady magnetohydrodynamic convective heat and mass transfer in a boundary layer slip flow past a vertical permeable plate with thermal radiation and chemical reaction. Recently, Reddy et al. [29] have studied the mass transfer and radiation effects of unsteady MHD free convective fluid flow embedded in porous medium with heat generation/absorption. Also, Vasu et al. [30] have examined the effect of the radiation and mass transfer on transient free convection flow of a dissipative fluid past semi-infinite plate with uniform heat and mass flux numerically using implicit finite difference method. Thermophoresis is a radiometric force by temperature gradient that enhances small particles moving toward a cold surface and away from a hot one. It plays a significant role on particle transport in laminar boundary layer flow. Generally, the mainly effect of thermophoresis on small particle size is especially effective in a range of dp = 0.01– 1.0 μm. Particle deposition from a moving air stream onto a surface caused by thermophoresis is widely seen in a lot of engineering applications, such as particle deposition onto a wafer surface in the modern semiconductor industry, electronic component cooling using a fan, filtration process in gas-cleaning, problems for nuclear reactor safety, clean room and human healthy topics, etc. It has also been proved that thermophoresis is the dominant mass transport mechanism in the chemical vapor deposition process used in the fabrication of optical fibers. Commonly, the deposition mechanisms for particles include Brownian diffusion, convection, thermophoresis and other mechanisms, e.g. electrophoresis. Thermophoresis on particle deposition onto a surface in laminar boundary layer flow is now rather well understood theoretically. Goren [31] developed the

thermophoretic deposition of particles in a laminar compressible boundary layer flow past a flat plate. Peters and Cooper [32], Opiolka et al. [33], and Tsai [34] dealt with the coupled of thermophoresis, forced convection and other effects on the predicted deposition rates for a stagnation point flow. Nazaroff and Cass [35] calculated the particle deposition rates due to combined effects of thermophoresis and natural convection. Therefore the objective of the present paper is to investigate the effect of thermophoresis on an unsteady free convective heat and mass transfer flow past a radiate impulsively started vertical plate using the robust Walters-B viscoelastic rheological material model. A Crank–Nicolson finite difference scheme is utilized to solve the unsteady dimensionless, transformed velocity, thermal and concentration boundary layer equations in the vicinity of the vertical plate.

2 Mathematical formulation An unsteady two-dimensional laminar free convective flow of a viscoelastic fluid past an impulsively started semi-infinite vertical plate is considered. The x-axis is taken along the plate in the upward direction and the y-axis is taken normal to it. The physical model is shown in Fig. 1a. Initially, it is assumed that the plate and the fluid are at the same temperature T∞′ and concentration level C∞′ everywhere in the fluid. At time, t′ > 0, the plate starts moving impulsively in the vertical direction with constant velocity u0 against the gravitational field. It is assumed that the concentration C′ of the diffusing species in the binary mixture is very less in comparison to the other chemical species, which are present, and hence the Soret and Dufour effects are negligible. It is also assumed that there is no chemical reaction between the diffusing species and the fluid. Then, under the above assumptions, the governing boundary layer equations with Boussinesq’s approximation are:

∂u ∂v + =0 ∂x ∂y

(1)

  ∂u ∂u ∂u ′ +v = gβ T ′ − T∞ +u ′ ∂t ∂x ∂y   ∂ 3u ∂ 2u ′ + gβ ∗ C ′ − C∞ + ν 2 − k0 2 ′ ∂y ∂y ∂t

(2)

∂T ′ k ∂ 2T ′ Q  ∂T ′ ∂T ′ 1 ∂qr ′  +v = + 0 T ′ − T∞ +u − ′ 2 ∂t ∂x ∂y ρcp ∂y ρcp ∂y ρcp

(3)

∂C ′ ∂ 2C′ ∂C ′ ∂  ′  ∂C ′ + v = D c vt + u − ′ 2 ∂t ∂x ∂y ∂y ∂y

(4)

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Heat Mass Transfer

(a)

X Impulsively started upward moving plate

Velocity, temperature and concentration boundary layers

g Y

(b)

(c)

Fig.  1  a Flow configuration coordinate system. b Comparisons of steady state local skin friction coefficient values. c Comparisons of steady state local Nusselt number values

The initial and boundary conditions are ′ , ′ t ′ ≤ 0 : u = 0, v = 0, T ′ = T∞ C ′ = C∞ ′ ′ ′ ′ t > 0 : u = u0 , v = 0, T = Tw , C = Cw′ at y = 0 at x = 0 ′ , ′ u = 0, T ′ = T∞ C ′ = C∞

u → 0,

′ , T ′ → T∞

′ C ′ → C∞

as

y→∞

(5)

where u, v are velocity components in x and y directions respectively, t′—the time, g—the acceleration due to gravity, β—the volumetric coefficient of thermal expansion, β*—the volumetric coefficient of expansion with concentration, T′—the temperature of the fluid in the boundary layer, C′—the species concentration in the boundary layer, Tw′—the wall temperature, T∞′—the free stream temperature far away from the plate, Cw—the concentration at the plate, C∞′—the free stream concentration in fluid far away from the plate, ν—the kinematic viscosity, α—the thermal

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diffusivity, ρ—the density of the fluid and D—the species diffusion coefficient. By using the Rosseland approximation the radiative heat flux qr is given by

qr = −

4σ ∂T ′4 3k ∗ ∂y

(6)

where σ is the Stefan–Boltzmann constant and k* is the mean absorption coefficient. It should be noted that by using the Rosseland approximation, the present analysis is limited to optically thick fluids. If temperature differences within the flow are significantly small, then Eq. (6) can be linearized by expanding T′4 into the Taylor series about T∞′, which after neglect higher order terms takes the form: ′3 ′ ′4 T ′4 ∼ T − 3T∞ = 4T∞

In view of Eqs. (6) and (7), (3) reduces to:

(7)

Heat Mass Transfer

∂T ′ ∂ 2T ′ ∂T ′ ∂T ′ +v =α 2 +u ′ ∂t ∂x ∂y ∂y 2 ′ ′3  Q0  ′ 16σ T∞ ∂ T ′ + + T − T∞ 3k × ρcp ∂y2 ρcp

∂C ∂C ∂C 1 ∂ 2C + +U +V = ∂t ∂X ∂Y Sc ∂Y 2

(8)

(9)

where T0 is some reference temperature, the value of Kν represents the thermophoretic diffusivity, and K is the thermophoretic coefficient which ranges in value from 0.2 to 1.2 as indicated by Batchelor and Shen [36] and is defined from the theory of Talbot et al. [37] by

    2Cs g /p + Ct Kn 1 + Kn C1 + C2 e−Cs /Kn   K= (10) (1 + 3Cm Kn ) 1 + 2g /p + 2Ct Kn where C1,  C2,  C3,  Cm,  Cs,  Ct are constants, λg and λp are the thermal conductivities of the fluid and diffused particles respectively and Kn is the Knudsen number. A thermophoretic parameter τ can be defined (Tsai [33]) as follows;

K(Tw − T∞ ) τ= T0

(11)

Typical values of τ are 0.01, 0.05 and 0.1 corresponding to approximate values of K(Tw  −  T∞) Equal to 3, 15 and 30 k for a reference temperature of T0 = 300 k. On introducing the following non-dimensional quantities

X= t= T= Γ = Gr =

xu0 yu0 u v , Y= , U= , V= , ν ν u0 u0 t ′ u02 ν ν , Pr = , Sc = , ν α D ′ ′ C ′ − C∞ T ′ − T∞ , C = , ′ ′ Tw′ − T∞ Cw′ − C∞   ′ kν Tw′ − T∞ k0 u02 , τ = , ν2 Tw u 0     ′ ′ νgβ Tw′ − T∞ νgβ ∗ Cw′ − C∞ , Gm = u03 u03

(12)

Equations (6), (7), (8), (9) and (10) are reduced to the following non-dimensional form

∂U ∂U + =0 ∂X ∂Y

(16) The corresponding initial and boundary conditions are

In the Eq. (4), the thermophoretic velocity νt was given by Talbot et al. [35] as

Kv ∂T ′ ∇T ′ =− vt = −Kv T0 T0 ∂y

  ∂C ∂T τ ∂ 2T +C 2 1 ∂Y ∂Y ∂Y Gr / 4

(13)

∂U ∂U ∂ 2U ∂ 3U ∂U +U +V = + GrT + GmC − Ŵ ∂t ∂X ∂Y ∂Y 2 ∂Y 2 ∂t (14)   ∂T ∂T 1 4 ∂ 2T ∂T +U +V = 1+ + φT ∂t ∂X ∂Y Pr 3F ∂Y 2 (15)

t ≤ 0 : U = 0,

V = 0,

T = 0,

C=0

t > 0 : U = 1,

V = 0,

T = 1,

C=1

U = 0, U → 0,

T = 0, T → 0,

C=0 C→0

at

Y =0

at

X=0

as

Y →∞

(17) where Gr is the thermal Grashof number, Gm is the solutal Grashof number, Pr is the fluid Prandtl number, Sc is the Schmidt number, Γ is the viscoelastic parameter and τ is the thermophoretic parameter. To obtain an estimate of flow dynamics at the barrier boundary, we also define several important rate functions at Y = 0. These are the dimensionless wall shear stress function, i.e. local skin friction function, the local Nusselt number (dimensionless temperature gradient) and the local Sherwood number (dimensionless species, i.e. contaminant transfer gradient) are computed with the following mathematical expressions.

1     −XGr / 4 ∂T 3/ ∂U ∂Y Y =0 4 , Nux = , τX = Gr ∂Y Y =0 TY =0 1   −XGr / 4 ∂C ∂Y Y =0 ShX Z = (18) CY =0 We note that the dimensionless model defined by Eqs. (13)–(16) under conditions (17) reduces to Newtonian flow in the case of vanishing viscoelasticity i.e. when Γ → 0

3 Numerical solution In order to solve these unsteady, non-linear coupled Eqs. (13)–(16) under the conditions (17), an implicit finite difference scheme of Crank–Nicolson type has been employed. This method was originally developed for heat conduction problems [38]. Prasad et al. [10], Vasu et al. [30] described the solution procedure in detail. The region of integration is considered as a rectangle with and where boundary layer corresponds to which lies well outside the momentum, thermal and concentration. After some preliminary numerical experiments the mesh sizes have been fixed as, with time step. The computations are executed initially by reducing the spatial mesh sizes by 50 % in one direction, and later in both directions by 50 % and the results are compared. It is observed that, in all the cases, the results differ only in the fifth decimal place. Hence

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Table 1  Comparisons of non-dimensional time to reach steady-state when Γ = 0, φ = 0, τ = 0, Pr = 0.71

Heat Mass Transfer S no.

Gr

Gm

F

Steady-state values Velocity

Temperature

Concentration

1 2 3 4

2 4 2 2

2 2 4 2

0.6 0.6 0.6 2.0

3 3 3 3

8.65 (8.65) 8.47 (8.48) 8.21 (8.22) 7.58 (7.58)

8.65 (8.65) 8.47 (8.48) 8.21 (8.22) 7.58 (7.58)

8.65 (8.65) 8.47 (8.48) 8.21 (8.22) 7.58 (7.58)

5

2

2

0.6

5

9.55 (9.54)

9.55 (9.54)

9.55 (9.54)

these mesh sizes are considered to be appropriate mesh sizes for present calculations. The local truncation error is O(Δt2 + ΔX2 + ΔY2) and it tends to zero as Δt, ΔX, and ΔY tend to zero. It follows that the CNM scheme is compatible. Stability and compatibility ensure the convergence.

4 Results and discussion A representative set of numerical results is presented graphically to illustrate the influence of viscoelastic parameter Γ, thermophoretic parameter τ, Grashof number Gr, mass Grashof number Gm, Schmidt number Sc, radiation parameter F and heat sink/source parameter φ on the velocity, temperature, concentration, skin-friction, Nusselt number and Sherwood number. The value of the Prandtl number Pr is chosen to be 0.71 (i.e. for air), the values of thermophoretic parameter is taken as 0.0, 0.5, 1.0 and 1.5, the value of Sc is chosen such that they represent water vapor (0.6). The other parameters are chosen arbitrarily. In ordered to validate the present study we made comparisons with the available study by Prasad et al. [10]. The results are shown graphically in Fig. 1b, c for the steady state local skin friction values and Nusselt numbers. The figure depicts that, the local shear stress τX decreases with the decreasing value of Sc and increasing values of Gr, Gm. Also the local heat transfer rate NuX decreases with increasing values of Sc and increases with increasing Gr or Gm. Moreover, we calculated the time to reach the steady states of the velocity, temperature and concentration for different values of governing parameters is shown in Table 1 and compared with the results obtained by Prasad et al. [10] noted in parenthesis. From Table 1, it is observed that the time taken for the velocity, temperature and concentration to reach the steady-state decreases with the increase in Gr or Gm or Sc, whereas it increases with the increase in radiation parameter F. It is observed that the present results are in good agreement with that of Prasad et al. [10]. In Fig. 2a–c we have presented the variation of the velocity, temperature and concentration with collective effects of thermal Grashof number Gr, solutal Grashof number Gm and thermophoretic parameter (τ) at X = 1.0. From Fig. 2a, it is seen that an increase in the values of

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Sc

Gr or Gm from 2.0 to 4.0 leads to an increase in the velocity for different values of F, φ, Sc, Γ and for fluid Prandtl number Pr (=0.71). An increase in the values of τ from 0.0 through 0.5, 1.0 to the maximum value of 1.5, clearly enhances the velocity which ascends sharply and peaks in close vicinity to the plate (Y = 0). With increasing distance from the plate however the velocity is adversely affected by increasing thermophoretic effect, i.e. the flow is decelerated. Therefore close to the plate the flow velocity is maximized for the case of τ  = 1.5. But this trend is reversed as we progress further into the boundary layer regime. The switchover in behavior corresponds to approximately Y  = 3.5, with increasing velocity profiles decay smoothly to zero in the free stream at the edge of the boundary layer. From Fig. 2b, it is observed that the temperature decreases with an increase in the values of Gr or Gm. With increasing thermophoretic parameter (τ), the temperature decreases both in the near-wall regime and the far-field regime of the boundary layer. As we approach the free stream the effects of thermophoretic parameter are negligible since the profiles are all merged together. Temperature is therefore maximized when τ  = 0.0 and minimized for the largest value of τ = 1.5. From Fig. 2c, it is observed that the concentration decreases with an increase in the values of Gr or Gm from 2.0 to 4.0. It is also observed that the concentration increases throughout the boundary layer due to an increase in τ. All profiles decay from the maximum at the wall to zero in the free stream. Figure  3a–c illustrate the effect of radiation parameter F, thermal Grashof number Gr, solutal Grashof number Gm on the velocity, temperature and concentration. From Fig. 3a, it is noticed that an increase in the values of F causes a decrease in the velocity both in the near-wall regime and far-field regime of the boundary layer. Velocity is therefore maximized when F = 0.1 (minimum) and minimized for the largest value of F = 100.0. It is also observed that the velocity increases due to an increase in the values of Gr or Gm from 2.0 to 4.0. The graphs show therefore that increasing Gr or Gm cools the flow. From Fig. 3b, it is noticed that an increase in the values of F causes a decrease in the temperature throughout the boundary layer. All profiles decay from the maximum at the wall to zero in the free stream. It is also observed that the temperature

Heat Mass Transfer

Fig.  2  a Steady state velocity profiles at X = 1.0 for different Gr, Gm and τ. b Steady state temperature profiles at X = 1.0 for different Gr, Gm and τ. c Steady state concentration profiles at X = 1.0 for different Gr, Gm and τ

profiles decreases due to an increase in Gr or Gm from 2.0 to 4.0. From Fig. 3c, an increase in the values of F leads to an increase in the concentration throughout the boundary layer. All profiles decay from the maximum at the wall to zero in the free stream. It is also observed that the temperature profiles decreases due to an increase in the values of Gr or Gm from 2.0 to 4.0. Figure  4a–c illustrate the effect of viscoelastic parameter (Γ) thermal Grashof number Gr, mass Grashof number Gm on the velocity, temperature and concentration at X = 1.0. From Fig. 4a, an increase in the values of Γ from 0 through 0.001, 0.003, 0.005, to the maximum value of 0.007 leads to an increase in the velocity, which ascends

sharply and peaks in close vicinity to the plate surface (Y  = 0). With increasing distance from the plate however the velocity U is adversely affected by increasing viscoelasticity, i.e. the flow is decelerated. Therefore close to the plate the flow velocity is maximized for the case of a Newtonian fluid (vanishing viscoelastic effect i.e. Γ  = 0). But this trend is reversed as we progress further into the boundary layer regime. The switchover in behavior corresponds to approximately Y  = 1, with increasing (Y) velocity profiles decay smoothly to zero in the free stream at the edge of the boundary layer. It is also observed that the velocity, U increases due to an increase in the values of Gr or Gm from 2.0 to 4.0. From Fig. 4b, an increase in the values of Γ

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Heat Mass Transfer

Fig.  3  a Steady state velocity profiles at X = 1.0 for different Gr, Gm and F. b Steady state temperature profiles at X = 1.0 for different Gr, Gm and F. c Steady state concentration profiles at X = 1.0 for different Gr, Gm and F

from 0 through 0.001, 0.003, 0.005, to the maximum value of 0.007 causes a decrease in the temperature throughout the boundary layer. All profiles decay from the maximum at the wall to zero in the free stream. It is also observed that the temperature profiles decreases due to an increase in the values of Gr or Gm from 2.0 to 4.0. From Fig. 4c, an increase in the values of Γ from 0 through 0.001, 0.003 and 0.005 to the maximum value of 0.007, causes a decrease in the concentration throughout the boundary layer. All profiles decay from the maximum at the wall to zero in the free stream. It is also observed that the concentration profiles

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decreases due to an increase in the values of Gr or Gm from 2.0 to 4.0. Figure  5a–c illustrate the effect of heat sink/source parameter (Φ), thermal Grashof number Gr, mass Grashof number Gm on the velocity, temperature and concentration at X = 1.0. From Fig. 5a, it is observed that an increase in the values of Gr or Gm leads to an increase in the velocity. It is also observed that an increase in the values of Φ from −1.0 through −0.5, 0.0 and 0.5 to the maximum value of 1.0 causes an increase in the velocity both in the near-wall regime and the far-field regime of the boundary

Heat Mass Transfer

Fig.  4  a Steady state velocity profiles at X = 1.0 for different Gr, Gm and Γ. b Steady state temperature profiles at X = 1.0 for different Gr, Gm and Γ. c Steady state concentration profiles at X = 1.0 for different Gr, Gm and Γ

layer. Velocity is therefore maximized when Φ  = 1.0 and minimized for Φ = −1.0. From Fig. 5b, it is observed that the temperature decreases with an increase in the values of Gr or Gm. It is also observed that the temperature increases due to increase in heat sink/source parameter (Φ). From Fig. 5c, it is observed that the concentration decreases with the increase in Gr or Gm. It is also observed that the concentration decreases due to increase in heat sink/source parameter (Φ).

Figure 6a–c depict the distributions of velocity, temperature and concentration for different values of thermophoretic parameter τ and various radiation parameter (F) in case of φ  = 1.0 at X  = 1.0. From Fig. 6a, it is observed that an increase in the values of thermophoretic parameter causes an increase in the velocity both in the near-wall regime and the far-field regime of the boundary layer. It is also observed that an increase in F from 0.1 through 1.0 to the maximum value of 10 causes a decrease in the velocity

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Heat Mass Transfer

Fig.  5  a Steady state velocity profiles at X = 1.0 for different Gr, Gm and Φ. b Steady state temperature profiles at X = 1.0 for different Gr, Gm and Φ. c Steady state concentration profiles at X = 1.0 for different Gr, Gm and Φ

throughout the boundary layer. Figure 6b shows an increase in the values of τ causes a decrease in the temperature. It is noted that an increase in the radiation parameter (F) leads to fall in temperature. From Fig. 6c, an increase in the values τ from 0.0 through 0.5 to the maximum value of 1.0 leads to an increase in the concentration. It can also be seen that an increase in F leads to an increase in the concentration.

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Figure 7a–c depict the distributions of velocity, temperature and concentration for different values of thermophoretic parameter τ and various radiation parameter (F) in case of Φ = −1.0 at X = 1.0. From Fig. 7a, it is observed that an increase in the values of thermophoretic parameter causes an increase in the velocity both in the near-wall regime and the far-field regime of the boundary layer. It is also observed that an increase in F from 0.1 through 1.0 to

Heat Mass Transfer

Fig.  6  a Steady state velocity profiles at X = 1.0 for different τ and F, when Φ = 1.0. b Steady state temperature profiles at X = 1.0 for different τ and F, when Φ = 1.0. c Steady state concentration profiles at X = 1.0 for different τ and F, when Φ = 1.0

the maximum value of 10.0 causes a decrease in the velocity throughout the boundary layer. Figure 7b depicts an increase in the values of τ causes a decrease in the temperature. It is noted that an increase in the radiation parameter (F) leads to fall in temperature. From Fig. 7c, an increase in the values τ from 0.0 through 0.5 to the maximum value of 1.0 leads to an increase in the concentration. It can also be seen that an increase in F leads to an increase in the concentration.

In Fig. 8a–c, the variation of dimensionless local skin friction (surface shear stress), τX, Nusselt number (surface heat transfer gradient), NuX and the Sherwood number (surface concentration gradient), ShX, for various viscoelasticity parameters (Γ) at steady case are illustrated also shown in Table 1. Shear stress is clearly enhanced with increasing viscoelasticity (i.e. stronger elastic effects) i.e. the flow is accelerated (Fig. 8a). Figure 8b shows that the local Nusselt number, NuX values are decreased in case of Newtonian

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Heat Mass Transfer

Fig.  7  a Steady state velocity profiles at X = 1.0 for different τ and F, when Φ = −1.0. b Steady state temperature profiles at X = 1.0 for different τ and F, when Φ = −1.0. c Steady state concentration profiles at X = 1.0 for different τ and F, when Φ = −1.0

fluids and with increase in the values of Γ, whereas it increases in the free stream at the edge of the boundary layer. Figure 8c shows that the local Sherwood number (surface concentration gradient), ShX values are decreased in case of Newtonian fluids and with increase in Γ, whereas it increases in the free stream at the edge of the boundary layer.

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In Fig. 9a–c, the variation of dimensionless local skin friction (surface shear stress), τX, Nusselt number (surface heat transfer gradient), NuX and the Sherwood number (surface concentration gradient), ShX, for various heat sink/ source parameters (Φ) are illustrated. Shear stress is clearly decreased with increasing the values of Φ from −1.0, through −0.5, 0.0, 0.5 to the maximum value of 1.0 i.e.

Heat Mass Transfer

Fig.  8  a The local skin friction profiles for different Γ at the steady state case. b The local heat transfer rate (Nusselt number) profiles for different Γ at the steady state case. c The local mass transfer rate (Sherwood number) profiles for different Γ at the steady state case

the flow is decelerated (Fig. 9a). An increase in the values of Φ, decreases the local Nusselt number values (Fig. 9b). Similarly from Fig. 9c, it is seen that the local Sherwood number values are also increased with an increase in that heat sink/source parameter. In Fig. 10a–c, the variation of dimensionless local skin friction (surface shear stress), τX Nusselt number (surface heat transfer gradient), NuX and the Sherwood number (surface concentration gradient), ShX, for various thermophoretic parameters (τ) are illustrated. Shear stress is clearly

decreased with increasing thermophoretic parameter, i.e. the flow is decelerated (Fig. 9a). An increase in the values of τ clearly increases the local Nusselt number, (Fig. 9b). In Fig. 9c, it is observed that the local Sherwood number values are decreased with an increase in thermophoretic parameter, i.e. a rise in τ from 0.0, through 0.3, 0.5, 0.7, and 1.0–1.5. Table  2 displays the results for the flow from a vertical plate that the surface values of velocity, temperature and concentration gradient components at various X and at the steady state. These are proportional to the friction

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Heat Mass Transfer

Fig.  9  a. The local skin friction profiles for different Φ at the steady state case. b The local heat transfer rate (Nusselt number) profiles for different Φ at the steady state case. c The local mass transfer rate (Sherwood number) profiles for different Φ at the steady state case

factor, Nusselt number and Sherwood number respectively. This table indicates that as the parameter Gr/Gm and τ increases, this leads to significant changes in the values of skin friction coefficient, local heat transfer coefficient and local mass transfer coefficient. This can be attributed to diffusion-thermal interaction decrease, which decreases the fluid velocity. It is notice that the values of Γ and Nux decreases with decreasing with Gr/Gm increase and Shx increase. While the values of τ increases Γ and

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Shx decrease whereas Nux increase. Also, It is seen that the values of τX and Nux decreases with increasing along the plate whereas the values of Shx increase. The same trend has been observed for both Newtonian and non-Newtonian fluids. It is clear that as Γ increasing in the local skin friction coefficient increase whereas Nux and Shx decrease is shown in Table 3. Tabulated values of the skin friction, local heat transfer and local mass transfer coefficients against X are shown in

Heat Mass Transfer

Fig.  10  a The local skin friction profiles for different τ at the steady state case. b The local heat transfer rate (Nusselt number) profiles for different τ at the steady state case. c The local mass transfer rate (Sherwood number) profiles for different τ at the steady state case

Table  4 for different values of thermal radiation (F) and the Prandtl number (Pr). In this table, we see that the values of the skin friction, the local heat transfer increase with the increasing values of Pr, and the local mass

transfer coefficients are decreased. The same behaviour is observed for increasing F. It is clear that along the plate, the skin friction coefficient is decreased whereas Nux and Shx increased.

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Heat Mass Transfer

Table 2  Values of local skin friction (surface shear stress), τX, Nusselt number (surface heat transfer gradient), NuX and the Sherwood number (surface concentration gradient), ShX for different Gr/Gm and τ at X = 0.1, 0.5 and 1.0 Gr/Gm

0.0

0.2

1.0

5.0

τ

X = 0.1

X = 0.5

X = 1.0

τx

NuX

ShX

τx

NuX

ShX

τx

NuX

ShX

0.0 0.1 0.5 1.0 0.0 0.1 0.5 1.0 0.0 0.1 0.5 1.0

3.48356 3.48356 3.48356 3.48356 3.29112 3.29068 3.28887 3.28651 2.70790 2.70595 2.69802 2.68784

0.00538 0.00538 0.00538 0.00538 0.04439 0.04442 0.04452 0.04466 0.06137 0.06142 0.06161 0.06186

0.12279 0.12181 0.11783 0.11275 0.13842 0.13614 0.12705 0.11577 0.15776 0.15487 0.14339 0.12927

−0.37539 −0.37539 −0.37539 −0.37539 0.02204 0.02212 0.02246 0.02293 0.10592 0.10606 0.10664 0.10742

0.20536 0.21759 0.26919 0.33947 0.27854 0.27721 0.27184 0.26498 0.34787 0.34370 0.32711 0.30666

0.28596 0.31724 0.45392 0.64966 0.42795 0.43043 0.44022 0.45209 0.55120 0.54700 0.53021 0.50928

0.37403 0.36491 0.32819

0.08625 0.08633 0.08666

0.19732 0.19341 0.17791

0.19806 0.19828 0.19920

0.46916 0.46169 0.43223

0.67170 0.67170 0.67170 0.67170 0.09803 0.09775 0.09661 0.09508 −1.50810 −1.51062 −1.52081 −1.53401

−0.96083 −0.96083 −0.96083 −0.96083 −0.08902 −0.08888 −0.08833 −0.08759 0.10836 0.06142 0.10959 0.11092

0.0 0.1 0.5

1.00545 1.00545 1.00545 1.00545 0.58053 0.58005 0.57811 0.57555 −0.66845 −0.67092 −0.68103 −0.69408

0.28900 0.28937 0.29092

0.76110 0.75049 0.70851

1.0

0.28171

0.08710

0.15888

−5.55005

0.20042

0.39633

−7.55039

0.29297

0.65714

−5.43892 −5.44982 −5.49389

−7.42887 −7.44076 −7.48890

Table 3  Values of local skin friction (surface shear stress), τX, Nusselt number (surface heat transfer gradient), NuX and the Sherwood number (surface concentration gradient), ShX for different Gr/Gm and Γ at X = 0.1, 0.5 and 1.0 Gr/Gm

0.0

0.2

1.0

5.0

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Γ

X = 0.1

X = 0.5

X = 1.0

τx

NuX

ShX

τx

NuX

ShX

τx

NuX

ShX

0.000 0.003 0.005 0.000 0.003 0.005 0.000 0.003 0.005

2.33434 2.86552 3.48356 2.22122 2.71366 3.28887 1.84585 2.23511 2.69802

0.04241 0.02481 0.00538 0.05590 0.04946 0.04452 0.06826 0.06449 0.06161

0.14392 0.13099 0.11783 0.14770 0.13717 0.12705 0.15715 0.15003 0.14339

−0.19641 −0.29431 −0.37539 0.03630 0.02831 0.02246 0.10812 0.10715 0.10664

0.27069 0.27039 0.26917 0.27783 0.27454 0.27184 0.32187 0.32457 0.32711

0.43961 0.44929 0.45392 0.43451 0.43748 0.44022 0.51075 0.52099 0.53021

0.30726 0.30188

0.08900 0.08765

0.18165 0.17969

0.19313 0.19638

0.41293 0.42319

0.47338 0.56658 0.67170 0.11293 0.10623 0.09661 −0.92500 −1.20051 −1.52081

−0.69093 −0.84508 −0.96083 −0.11293 −0.08002 −0.08833 0.10720 0.10834 0.06161

0.000 0.003

0.70438 0.84489 1.00545 0.44557 0.50806 0.57811 −0.35104 −0.50301 −0.68103

0.27698 0.28455

0.66878 0.68993

0.005

0.32819

0.08666

0.17791

−5.49389

0.19920

0.43223

−7.48890

0.29092

0.70851

−3.40433 −4.36698

−4.74973 −6.01240

Heat Mass Transfer Table 4  Values of local skin friction (surface shear stress), τX, Nusselt number (surface heat transfer gradient), NuX and the Sherwood number (surface concentration gradient), ShX for different F and Pr Pr

0.05

0.70

7.0

F

X = 0.1

X = 0.5

τx

NuX

0.1 0.5 1.0 10.0 0.1 0.5 1.0 10.0

3.33696 3.34047 3.34299 3.34952 3.35075 3.37627 3.38747 3.40315

0.00831 0.00879 0.00952 0.01254 0.01326 0.03802 0.05827 0.11736

0.1 0.5 1.0

3.39862 3.41835 3.42337

0.09364 0.32545 0.48546

10.0

3.43231

0.91236

ShX

X = 1.0

τx

NuX

0.13825 0.13743 0.13678 0.13483 0.13443 0.12341 0.11589 0.09707

0.73697 0.73845 0.73969 0.74357 0.74441 0.76790 0.78123 0.80556

0.03919 0.03191 0.02631 0.01105 0.00823 −0.01919 −0.00712 0.06384

0.10426 0.03751 −0.01052

0.79757 0.83528 0.84314 0.85341

−0.14772

5 Conclusions A two-dimensional, unsteady laminar incompressible boundary layer model has been presented for the external flow, heat and mass transfer in a viscoelastic buoyancydriven flow past a radiate impulsively started vertical plate under the influence of thermophoresis and heat sink/source. The Walters-B viscoelastic model has been employed which is valid for short memory polymeric fluids. The dimensionless conservation equations have been solved with the well-tested, robust, highly efficient, implicit Crank–Nicolson finite difference numerical method. The present computations have shown that • Increasing thermophoretic parameter accelerates the velocity and concentration, but reduces the temperature. • Increasing viscoelastic parameter accelerates the velocity, but reduces the temperature and concentration. • Increasing radiation parameter accelerates the concentration, but reduces the velocity and temperature. • Increasing heat sink/source parameter accelerates the velocity and temperature, but reduces the concentration. • Increasing buoyancy ratio parameter accelerates the concentration but reduces the velocity and temperature. • Increasing heat sink/source parameter decreases the dimensionless wall shear stress function, i.e. local skin friction function and the mass transfer rate (local Sherwood number) at the plate with the opposite effect sustained for the local heat transfer rate (local Nusselt number). Acknowledgments The authors are grateful to reviewers for their constructive comments which have helped to improve the present article. One of the authors (B. Vasu) is thankful to the Motilal Nehru

ShX

τx

NuX

ShX

0.25924 0.26062 0.26158 0.26366 0.26396 0.26052 0.25245 0.22516

0.33666 0.33633 0.33616 0.33610 0.33616 0.34699 0.35820 0.38574

0.37700 0.38132 0.38465 0.39416 0.39605 0.42554 0.42667 0.40102

0.03198 0.32163 0.47495

0.23626 0.15221 0.11123

0.37572 0.42980 0.44162

0.07761 0.06027 0.04626 0.00272 −0.00665 −0.20061 −0.24430 −0.20658

−0.23758 0.12172 0.27783

0.41365 0.31168 0.27409

0.87837

−0.00264

0.45626

0.60188

0.19172

National Institute of Technology Allahabad, India for the necessary support.

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