Thermal Jump Effects on Boundary Layer Flow of a Jeffrey Fluid Near the Stagnation Point on a Stretching/Shrinking Sheet with Variable Thermal Conductivity

Abstract

A mathematical model will be analyzed in order to study the effects of thermal jump and variable thermal conductivity on flow and heat transfer near the stagnation point on a stretching/shrinking sheet in a Jeffrey fluid. The highly nonlinear partial differential equation of Jeffrey fluid flow along with the energy equation are transformed to an ordinary system using nondimensional transformations. The arising equations are solved for temperature, velocity, shear stress, and heat flux using finite difference method. The effect of the influences parameters is discussed. For nonradiation regular viscous fluid our results are as that by Nazar et al. (2002).

1. Introduction

It is well known that the thermophysical properties of a fluid play an important role in the engineering applications in aerodynamics, geothermal systems, crude oil extractions, ground water pollution, thermal insulation, heat exchanger, storage of nuclear waste, and so forth, convective flows over bodies. The change in the thermal conductivity with temperature is an important property [1–5]. Prasad and Vajravelu [6] investigated the effect of variable thermal conductivity in a nonisothermal sheet stretching through power law fluids while Prasad et al. [4] reported similar studies for viscoelastic fluids. Abel et al. [7] studied combined effects of thermal buoyancy and variable thermal conductivity on a magnetohydrodynamic flow and the associated heat transfer in a power-law fluid past a vertical stretching sheet in the presence of a nonuniform heat source. The general findings of these studies were that the effects of variable thermal conductivity increase the shear stress. The temperature at wall increase with an increase in variable thermal conductivity by Seddeek et al. [8]. Prasad et al. [9] found that the variable thermal conductivity has an impact in enhancing the skin friction coefficient; hence, fluids with less thermal conductivity may be opted for effective cooling. Abel et al. [10] concluded that the variable thermal conductivity increases the temperature distribution in both prescribed surface temperature and prescribed heat flux cases. Mahanti and Gaur [11] investigated the effects of linearly varying viscosity and thermal conductivity on steady free convective flow of a viscous incompressible fluid along an isothermal vertical plate in the presence of heat sink. Deissler [12] obtained that the effects of second-order terms on the velocity and temperature jumps at a wall are by a physical derivation. The analysis used the concepts of effective mean free paths for momentum and energy transfer; the effective mean free paths are obtained from known viscosities and thermal conductivities. Rahman and Eltayeb [13] studied numerically the convective slip flow of slightly rarefied fluids over a wedge with thermal jump and temperature dependent transport properties such as fluid viscosity and thermal conductivity. Cipolla Jr. [14] studied the temperature jump in polyatomic gas, also Kao [15] and Latyshev and Yushkanov [16] studied the temperature jump. The flow and heat transfer of Jeffry fluid near stagnation point on a stretching/shrinking sheet with parallel external flow was investigated by Turkyilmazoglu and Pop [17]; Akram and Nadeem [18] discussed the peristaltic motion of a two-dimensional Jeffry fluid. Authors in [19–22] studied more properties in Jeffrey fluid. Different non-Newtonian fluids were considered in studies by Pandey and Tripathi [23–25] and Tripathi [26].

Interest in boundary layer flow and heat transfer over a stretching sheet has gained considerable attention because of its application in industry and manufacturing processes. Such applications include polymer extrusion drawing of copper wires, continuous stretching of plastic films and artificial fibers, hot rolling, wire drawing, glass fiber, metal extrusion, and metal spinning. For example, Liu and Andersson [27] studied the heat transfer in a liquid film on an unsteady stretching sheet. The effects of variable fluid properties and thermocapillarity on the flow of a thin film on an unsteady stretching sheet were studied by Dandapat et al. [28]. Hayat et al. [29] investigated the peristaltic mechanism of a Jeffrey fluid in a circular tube. Nadeem et al. [30] analyzed the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface. The effects of thermal radiation are carried out for two cases of heat transfer analysis known as (1) prescribed exponential order surface temperature (PEST) and (2) prescribed exponential order heat flux (PEHF). Hamad [31] studied the convective flow and heat transfer of an incompressible viscous nanofluid past a semi-infinite vertical stretching sheet in the presence of a magnetic field. Hamad and Pop [32] studied theoretically the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption.

The objective of the present study is to investigate the dynamics of the thermal boundary layer flow of a viscous incompressible Jeffrey fluid near the stagnation point on a stretching sheet taking into account the thermal jump condition at the surface. Thus, the main focus of the analysis is to investigate how the flow field, temperature field, shear stress, and heat flux vary within the boundary layer with thermal jump at the wall when the thermal conductivity is temperature dependent. The similarity equations are derived and solved numerically with the widely used and robust computer algebra software. Graphs and tables are presented to illustrate and discuss important hydrodynamic and thermal features of the flow.

2. Problem Formulation

Consider a steady two dimensional flow of an incompressible Jeffrey fluid near the stagnation point on a stretching/shrinking sheet. The thermal conductivity is assumed to be functions of temperature. A thermal jump condition is assumed to occur at the wall. We are considering Cartesian coordinate system in such a way that 𝑥-axis is taken along the stretching sheet in the direction of the motion and 𝑦-axis is normal to it. The plate is stretched in the 𝑥-direction with a velocity 𝑢_𝑤=𝑐𝑥 defined at 𝑦=0. The flow and heat transfer characteristics under the boundary layer approximations are governed by the following equations: 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝜕𝑦=𝑢_𝑒𝑑𝑢_𝑒+𝜈𝑑𝑥1+𝛾₁𝜕²𝑢𝜕𝑦²+𝛾₂𝑢𝜕³𝑢𝜕𝑥𝜕𝑦²𝜕+𝑣³𝑢𝜕𝑦³−𝜕𝑢𝜕𝜕𝑥²𝑢𝜕𝑦²+𝜕𝑢𝜕𝜕𝑦²𝑢,𝜕𝑥𝜕𝑦𝜌𝐶_𝑝𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=𝜕𝜕𝑦𝜅(𝑇)𝜕𝑦𝜕𝑇,𝜕𝑦 (1)(2)(3) with the boundary conditions (see Rahman and Eltayeb [13]): 𝑣=𝑣_𝑤(𝑥),𝑢=𝑢_𝑤(𝑥)𝑇=𝑐𝑥,_jump=𝑇_𝑓−𝑇_𝑤=𝜆₁2𝜎_𝑇−12𝛾𝜅(𝑇)𝛾+1𝜇𝐶_𝑝𝜕𝑇𝜕𝑦at𝑦=0,𝑢=𝑢_𝑒(𝑥)=𝑎𝑥,𝜕𝑢𝜕𝑦=0,𝑇=𝑇_∞as𝑦⟶∞. (4) Here 𝑥 and 𝑦 are the Cartesian coordinates along the plate and normal to it, respectively, 𝑢 and 𝑣 are the velocity components along 𝑥- and 𝑦-axes, 𝑣_𝑤(𝑥) is the mass transfer velocity with 𝑣_𝑤(𝑥)<0 for suction and 𝑣_𝑤(𝑥)>0 for injection or withdrawal, 𝑇 is the fluid temperature, 𝛼 is thermal diffusivity, 𝜈 is the kinematic viscosity, 𝛾₁ is the ratio of relaxation and retardation times, 𝛾₂ is the relaxation time, 𝛾 is the ratio of specific heats, 𝜎_𝑇 is the thermal accommodation coefficient, 𝜆₁ is the mean free path, 𝜇 is the dynamic viscosity, and 𝜅(𝑇) is the thermal conductivity which can be, following Chiam [1], written as 𝜅=𝜅_∞1+𝜀𝑇−𝑇_∞𝑇_𝑤−𝑇_∞, (5) where 𝜀 is the thermal conductivity parameter.

We introduce now the following similarity variables: 𝜓=√𝑎𝜈𝑥𝑓𝜂,𝜃𝜂=𝑇−𝑇_∞𝑇_𝑤−𝑇_∞,𝜂=𝑎𝜈𝑦, (6) where 𝜓 is the stream function which is defined in the usual way as 𝑢=𝜕𝜓/𝜕𝑦 and 𝑣=−𝜕𝜓/𝜕𝑥. Thus, 𝑣_𝑤(𝑥)=−√𝑎𝜈𝑠, where 𝑠 is the mass transfer parameter with 𝑠>0 for suction and 𝑠<0 for injection, respectively. Substituting (5) and (6) into (2) and (3), the following set of ordinary differential equations results in 𝑓^′′′+1+𝛾₁𝑓𝑓^′′−𝑓^′2+𝛽𝑓^′′2−𝑓𝑓^′′′′+1+𝛾₁()𝜃=0,1+𝜀𝜃^′′+𝜀𝜃^′2+Pr𝑓𝜃^′=0, (7) and the boundary conditions (4) become 𝑓(0)=𝑠,𝑓^′(0)𝜃(0)=𝜆,=1+2𝛾𝑇_𝑠((0𝜃𝛾+1Pr1+𝜀𝜃))^′(0),𝑓^′(∞)=1,𝑓^′′(∞)=0,𝜃(∞)=0, (8) where Pr=𝜇𝐶_𝑝/𝜅_∞ is the Prandtl number, 𝜆=𝑐/𝑎 is the stretching (𝜆>0) or shrinking (𝜆<0) parameter, 𝛽=𝑐𝛾₂ is the Deborah number, 𝑇_𝑠=𝜆₁(2/(𝜎_𝑇−1))𝑎/𝜈 is the slip parameter, and primes denote differentiation with respect to 𝜂.

2.1. Particular Case

It is worth mentioning that for a regular viscous fluid (𝛽=𝛾₁=0), (7) reduce to the steady state equations from the paper by Nazar et al. [33] when we neglect the radiation effect.

2.2. Physical Quantities

The physical quantities of interest are the skin friction coefficient 𝐶_𝑓 and the local Nusselt number Nu_𝑥, which are defined as 𝐶_𝑓=𝜏_𝑤𝜌𝑢²_𝑒(𝑥),Nu_𝑥=𝑥𝑞_𝑤𝜅𝑇_𝑤−𝑇_∞, (9) where 𝜏_𝑤 is the skin friction or shear stress along the stretching surface and 𝑞_𝑤 is the heat flux from the surface, which are given by 𝜏_𝑤=𝜇𝜕𝑢𝜕𝑦_𝑦=0,𝑞_𝑤=−𝜅𝜕𝑇_jump𝜕𝑦_𝑦=0. (10) Using (6), we get Re_𝑥^1/2𝐶_𝑓=𝑓^′′(0),Re_𝑥^−1/2Nu_𝑥=−2𝛾𝑇𝛾+1^𝑠Pr(1+𝜀𝜃)𝜃^′′(0)+𝜀𝜃^′(0)², (11) where Re_𝑥=𝑢_𝑒(𝑥)𝑥/𝜈 is the local Reynolds number.

3. Results and Discussion

The transformed equations (7) with boundary conditions (8) are solved numerically by using a finite difference method. The asymptotic boundary conditions at 𝜂=∞ are replaced by 𝜂=6. In Table 1, we have shown the variation of wall temperature and heat transfer rates with the Prandtl numbers for three different values of Deborah numbers. It is observed that the wall temperature decreases whereas the heat transfer rates increase with an increase in Prandtl number. However, the wall temperature increases slightly and heat transfer values decrease slightly with an increase in Deborah number, whereas both decrease with an increase in the ratio of specific heats. This is shown in Tables 1(a) and 1(b). Tables 1(c) and 1(d) show the effects of slip temperature on the wall temperature and heat transfer rates for the constant thermal conductivity. Table 1(e) shows the effects of thermal slip on the wall temperature and heat transfer rates when the thermal conductivity varies with temperature. It can be seen that both the wall temperature and heat transfer rates decrease with an increase in the Prandtl number due to decrease in the thermal conductivity. The effects of the ratio of relaxation and retardation times, Deborah number, and suction and stretching parameters on the 𝑥-component of velocity are shown in Figures 1 and 2. Figure 1 shows the effects in the absence of stretching. It shows that the velocity increases as the ratio of relaxation and retardation times increases but decreases with an increase in Deborah number. The velocity boundary layer converges quickly for small Deborah numbers. In fact, small Deborah numbers correspond to situations where the material has time to relax (and behaves in a viscous manner), while high Deborah numbers correspond to situations where the material behaves rather elastically. Figure 2 shows the effects of stretching parameter on the velocity for different values of Deborah number. It is observed that the velocity becomes constant when 𝜆=0, increases when 𝜆<1, and decreases when 𝜆>0. Accordingly, the velocity decreases or increases with Deborah number when 𝜆<1 or 𝜆>0.

Table 1 

Wall temperature and heat transfer values when one has the following.

Figure 1 

Effect of 𝛽 and 𝛾1 on the velocity profiles.

Figure 2 

Effect of 𝛽 and 𝜆 on the velocity profiles.

The effect of thermal conductivity parameter on temperature profiles is shown in Figure 3 for two different Prandtl numbers. It is observed that the thermal boundary layer thickness decreases with an increase in Prandtl number. As the thermal conductivity parameter increases, the temperature in the thermal boundary layer increases. The variaton of skin friction with the ratio of relaxation and retardation times for different parameters is shown in Figures 4(a) and 4(b). When there is no stretching, the skin friction increases with the ratio of relaxation and retardation times and decreases with an increase in Deborah number. As expected, the skin friction reduces with an increase in the suction parameter in both cases. Comparing Figures 4(a) and 4(b), it can be seen that the skin friction decreases with an increase in the stretching parameter. The variation in heat transfer rates with the ratio of relaxation and retardation times is shown in Figures 5 and 6 for different values of suction and thermal conductivity parameters and Prandtl and Deborah numbers. The other parameters are kept constant. As the ratio of relaxation and retardation times increases, the heat transfer rate increases. For higher values of the suction parameter, the heat transfer rates are found to be higher. It is also observed that the heat transfer rates decrease with an increase in Deborah number. This is shown in Figure 5. As evident from Figure 6, the heat transfer rates increase with an increase in the Prandtl number. Figure 6 also shows that the heat transfer rates decrease with an increase in the thermal conductivity parameter.

Figure 3 

Effect of Pr and 𝜀 on the temperature profiles.

(a)

(b)

Figure 4 

Effect of 𝛽 and 𝜆 on the skin friction profiles.

Figure 5 

Effect of 𝛽 and 𝑠 on the heat transfer rate.

Figure 6 

Effect of Pr and 𝜀 on the heat transfer rate.

4. Conclusions

The effects of thermal jump and variable thermal conductivity on flow and heat transfer near the stagnation point on a stretching/shrinking sheet are investigated numerically in a Jeffrey fluid. The effects of governing parameters including ratio of relaxation and retardation times 𝛾₁, Deborah number 𝛽, Prandtl number, stretching parameter 𝜆, suction parameter 𝑠, and thermal conductivity parameter 𝜀 on the dimensionless velocity, temperature, skin friction, and heat transfer rates are investigated and are presented graphically and in tabular form. We conclude the following.(a)The wall temperature increases slightly while heat transfer values decrease slightly with an increase in Deborah number.(b)The wall temperature and heat transfer decrease with an increase in the ratio of specific heats.(c)The decreases of Deborah number and the increases of relaxation and retardation times leads to increases in the velocity.(d)The skin friction decreases with an increase in the stretching parameter.(e)As the ratio of relaxation and retardation times increases, the heat transfer rate increases.(f)For higher values of the suction parameter, the heat transfer rates are found to be higher.(g)The heat transfer rates decrease with an increase in Deborah number.