Falkner-Skan Flow of a Maxwell Fluid with Heat Transfer and Magnetic Field

Abstract

This investigation deals with the Falkner-Skan flow of a Maxwell fluid in the presence of nonuniform applied magnetic fi…eld with heat transfer. Governing problems of flow and heat transfer are solved analytically by employing the homotopy analysis method (HAM). Effects of the involved parameters, namely, the Deborah number, Hartman number, and the Prandtl number, are examined carefully. A comparative study is made with the known numerical solution in a limiting sense and an excellent agreement is noted.

1. Introduction

The Falkner-Skan problem under various aspects has attracted the attention of several researchers [1]. This problem under various aspects has been discussed extensively for viscous fluid. The interested readers may consult the studies in [2–11] for detailed information in viscous fluids. There are several materials which do not obey the Newton's law of viscosity, for example, biological products like blood and vaccines, foodstuffs like honey, ketchup, butter, and mayonnaise, certain paints, cosmetic products, pharmaceutical chemicals and so forth. These fluids are characterized as the non-Newtonian fluids. Investigation of such fluids is very useful in industrial, engineering, and biological applications. However, such fluids cannot be studied by employing a single constitutive relationship. This is due to diverse properties of non-Newtonian fluids in nature. These non-Newtonian fluid models are discussed in view of three main categories, namely, the differential, the rate, and the integral types. The simplest subclass of rate type fluids is called Maxwell. The Maxwell fluid allows for the relaxation effects which cannot be predicted in differential type fluids, namely, second, third, and fourth grades. Recently, there has been an increasing interest in the theory of rate type fluids and, in particular, a Maxwell fluid model has been accorded much attention. The Falkner-Skan wedge flow of a non-Newtonian fluid was firstly investigated by Rajagopal et al. [12]. Massoudi and Ramezan [13] discussed the effect of injection or suction on the Falkner-Skan flows of second grade fluids. The Falkner-Skan wedge flow of power-law fluids embedded in a porous medium is investigated by Kim [14]. Olagunju [15] studied this flow problem for viscoelastic fluid. In [10–15], the attention has been given to the differential type fluids. To the best of our knowledge, no one investigated the Falkner-Skan flow problem for rate type fluids.

In [10], Yao has examined the Falkner-Skan wedge flow. He established series solution for the velocity and temperature by using homotopy analysis method [16–25]. The purpose of the present contribution is to extend the flow analysis of study [10] in two directions. The first generalization is concerned with the consideration of electrically conducting fluid. Such analysis has inserted in power generators the cooling of reactors, MHD accelerators, the design of heat exchanges and electrostatic filters. Secondly, we consider the Maxwell fluid instead of viscous fluid. The rest of the paper is arranged as follows. The description of the problem is presented in Section 2. Section 3 develops the homotopy solutions for velocity and temperature. Convergence of the derived solution is examined in Section 4. Further, the variations of embedded parameters have been discussed in this section.

2. Problem Development

We study the steady two-dimensional flow of a Maxwell fluid in the presence of a magnetic field. The magnetic Reynolds number is small so that induced magnetic field is neglected. The stream velocity 𝑈(𝑥) varies according to 𝑥^𝑛. The constant temperature of surface and free stream is 𝑇_𝑤 and 𝑇_∞, respectively. The boundary layer equations for the considered problem are 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢𝑢𝜕𝑦+𝜆²𝜕²𝑢𝜕𝑥²+𝑣²𝜕²𝑢𝜕𝑦²𝜕+2𝑢𝑣²𝑢𝜕𝜕𝑥𝜕𝑦=𝜈²𝑢𝜕𝑦²+𝑈𝜕𝑈𝜕𝑥+𝜆𝑈²𝜕²𝑈𝜕𝑥²−𝜎𝐵²𝜌𝑢−𝑈+𝜆𝑣𝜕𝑢,𝑢𝜕𝑦𝜕𝑇𝜕𝑥+𝑣𝜕𝑇=𝑘𝜕𝑦𝜌𝑐_𝑝𝜕²𝑇𝜕𝑦², (1)(2)(3) where 𝑢 and 𝑣 are the velocity components in the 𝑥- and 𝑦-directions, respectively, 𝜈 is kinematic viscosity, 𝜎 is the electrical conductivity, 𝑘 is the thermal conductivity, 𝜌 is the fluid density, 𝑇 is the fluid temperature, 𝜆 is relaxation time, 𝐵 is the magnetic field, and 𝑐_𝑝 is specific heat.

The relevant boundary conditions are prescribed as follows: 𝑢=0,𝑣=0,𝑇=𝑇_𝑤(𝑥)at𝑦=0,𝑢⟶𝑈,𝑇⟶𝑇_∞as𝑦⟶∞, (4) with [1] 𝑈(𝑥)=𝑎𝑥^𝑛, (5) and [7, 8] 𝐵(𝑥)=𝐵₀𝑥^(𝑛−1)/2. (6) Putting 𝜓=2√𝑛+1𝜈𝑥𝑈𝑓𝜂,𝜂=𝑛+12𝑈𝑥𝜈𝑦,𝑢=𝑈𝑓^′𝜂,𝑣=−𝑛+12𝑈𝑓𝜂+𝑥𝜈𝑛−1𝑛+1𝜂𝑓^′𝜂,𝜃𝜂=𝑇_𝑤−𝑇𝑇_𝑤−𝑇_∞,𝑢=𝜕𝜓𝜕𝑦,𝑣=−𝜕𝜓,𝜕𝑥 (7)(8) (1) is satisfied identically and (2)–(4) give 𝑓^′′′+𝑓𝑓^′′+2𝑛𝑛+11−𝑓^′2−𝑀²𝑓^′−1+𝑀²𝛽+1𝑓𝑓^′′+𝛽2𝑛𝑛−1𝑛+11−𝑓^′3+(3𝑛−1)𝑓𝑓^′𝑓^′′−𝑛+12𝑓²𝑓^′′′+𝑛−12𝜂𝑓^′2𝑓^′′𝜃=0,_′′+Pr𝑓𝜃^′𝑓(0)=0,=0,𝑓^′(0)(0)𝑓=0,𝜃=0,_′(∞)=1,𝜃(∞)=1. (9)(10)(11)

Here 𝜓 is the stream function, prime denotes the differentiation with respect to 𝜂, 𝛾 is the local Deborah number, 𝑛 is the constant parameter, 𝑀 is Hartmann number, and Pr is the Prandtl number. The values of 𝛽, 𝑀, and Pr are 𝛽=𝜆𝑈𝑥,𝑀=2𝜎𝐵²₀𝜌𝑎(𝑛+1),Pr=𝜇𝑐_𝑝𝑘. (12)

The local Nusselt number 𝑁𝑢_𝑥 and heat transfer from the plate 𝑞_𝑤 are given by 𝑁𝑢_𝑥=𝑥𝑞_𝑤𝑘𝑇_𝑤−𝑇_∞,𝑞_𝑤=−𝑘𝜕𝑇𝜕𝑦_𝑦=0. (13) Invoking (7) one obtains Re_𝑥^−1/2𝑛+12𝑁𝑢_𝑥=−𝜃^′(0) (14) in which the local Reynolds number Re_𝑥=𝑈𝑥/𝜈. For 𝑀=0=𝛽, from (9), we have 𝑓^′′′+𝑓𝑓^′′+𝛾1−𝑓^′2=0, (15) where 𝛾=2𝑛/(𝑛+1).

3. Homotopy Analysis Solutions

For an interest in homotopy solutions, we express 𝑓(𝜂) and 𝜃(𝜂) by a set of base functions 𝜂^𝑚,𝜂exp−2𝑛𝜂,𝑚,𝑛≥0_𝑚exp−𝑛𝜂,𝑚,𝑛≥0, (16) as follows: 𝑓𝜂=^∞∞_𝑚=0𝑛=0𝑎_𝑚,𝑛𝜂^𝑚,𝜃𝜂=exp−2𝑛𝜂_{∞∞𝑚=0𝑛=0}𝑏_𝑚,𝑛𝜂^𝑚exp−2𝑛𝜂, (17) where 𝑎_𝑚,𝑛 and 𝑏_𝑚,𝑛 are the coefficients. The initial approximations of 𝑓(𝜂) and 𝜃(𝜂) and the auxiliary linear operators ℒ_𝑓 and ℒ_𝜃 are 𝑓₀𝜂=𝜂−1−exp−2𝜂2,𝜃₀𝜂,ℒ=1−exp−𝜂_𝑓𝑓=𝑑³𝑓𝑑𝜂³−4𝑑𝑓𝑑𝜂,ℒ_𝜃(𝜃)=𝑑²𝜃𝑑𝜂²+𝑑𝜃ℒ𝑑𝜂_𝑓𝐶₁+𝐶₂exp2𝜂+𝐶₃ℒexp−2𝜂=0,_𝜃𝐶₄exp𝜂+𝐶₅exp−𝜂=0 (18)(19)(20)(21) in which 𝐶_𝑖  (𝑖=1−5) are the arbitrary constants. If 𝑝∈[0,1] is the embedding parameter and ℏ_𝑓,  ℏ_𝜃 and ℏ_𝜙, are the nonzero auxiliary parameters, then the problems at zeroth order give 1−𝑝ℒ_𝑓𝑓𝜂;𝑝−𝑓₀𝜂=𝑝ℏ_𝑓𝒩_𝑓𝑓,ℒ𝜂;𝑝1−𝑝_𝜃𝜃𝜂;𝑝−𝜃₀𝜂=𝑝ℏ_𝜃𝒩_𝜃𝜃,𝑓,𝑓𝑓𝜂;𝑝𝜂;𝑝0;𝑝=0,^′0;𝑝=0,𝑓^′𝜃𝜃𝒩∞;𝑝=1,0;𝑝=0,∞;𝑝=1,_𝑓𝑓=𝜕𝜂;𝑝₃𝑓𝜂;𝑝𝜕𝜂³+𝑓𝜕𝜂;𝑝²𝑓𝜂;𝑝𝜕𝜂²+2𝑛𝜕𝑛+11−𝑓(𝜂,𝑝)𝜕𝜂²−𝑀²𝜕𝑓𝜂,𝑝+𝑀𝜕𝜂−1₂𝜕𝑓𝛽+1𝜂,𝑝𝜕𝜕𝜂²𝑓𝜂,𝑝𝜕𝜂²+𝛽2𝑛𝑛−1𝜕𝑓𝑛+11−𝜂,𝑝𝜕𝜂³+()𝑓𝜕𝑓3𝑛−1𝜂;𝑝𝜂,𝑝𝜕𝜕𝜂²𝑓𝜂;𝑝𝜕𝜂²−𝑛+12𝑓²𝑓𝜂;𝑝𝜕3𝜂;𝑝+𝜕𝜂3𝑛−12𝜕𝑓(𝜂,𝑝)𝜕𝜂²𝜕²𝑓𝜂,𝑝𝜕𝜂²,𝒩_𝜃𝜃,𝑓=𝜕𝜂;𝑝𝜂;𝑝²𝜃𝜂,𝑝𝜕𝜂²𝑓𝜕𝜃+Pr𝜂;𝑝𝜂;𝑝.𝜕𝜂 (22)(23)(24)(25)(26)(27) For 𝑝=0 and 𝑝=1, we have 𝑓𝜂;0=𝑓₀𝜂,𝑓𝜂,𝜃𝜂;1=𝑓𝜂;0=𝜃₀𝜂,𝜃𝜂;1=𝜃𝜂. (28)(29) When 𝑝 increases from 0 to 1, 𝑓(𝜂;𝑝) and 𝜃(𝜂;𝑝) vary from 𝑓₀(𝜂) and 𝜃₀(𝜂) to the exact solutions 𝑓(𝜂) and 𝜃(𝜂). In view of Taylors theorem and (20) and (21), one arrives at 𝑓𝜂;𝑝=𝑓₀𝜂+^∞_𝑚=1𝑓_𝑚𝜂𝑝^𝑚,𝜃𝜂;𝑝=𝜃₀𝜂+^∞_𝑚=1𝜃_𝑚𝜂𝑝^𝑚,𝑓_𝑚𝜂=1𝜕𝑚!^𝑚𝑓𝜂;𝑝𝜕𝑝^𝑚_𝑝=0,𝜃_𝑚𝜂=1𝜕𝑚!^𝑚𝜃𝜂;𝑝𝜕𝑝^𝑚_𝑝=0. (30)(31)(32) The auxiliary parameters are so properly chosen that the series (28) and (29) converge at 𝑝=1, and hence 𝑓𝜂=𝑓₀𝜂+^∞_𝑚=1𝑓_𝑚𝜂,𝜃𝜂=𝜃₀𝜂+^∞_𝑚=1𝜃_𝑚𝜂. (33)(34) The problems at 𝑚th-order deformation satisfy the following equations and boundary conditions: ℒ_𝑓𝑓_𝑚𝜂−𝜒_𝑚𝑓_𝑚−1𝜂=ℏ_𝑓ℛ^𝑓_𝑚𝜂,ℒ_𝜃𝜃_𝑚𝜂−𝜒_𝑚𝜃_𝑚−1𝜂=ℏ_𝜃ℛ^𝜃_𝑚𝜂,𝑓_𝑚(0)=𝑓^′_𝑚(0)=𝑓^′_𝑚(∞)=𝑓_𝑚^′′(∞)𝜃=0,_𝑚(0)=𝜃_𝑚(∞)ℛ=0,_𝑓𝑚𝜂=𝑓^′′′_𝑚−1𝜂+^𝑚−1_𝑘=0𝑓_{𝑚−1−𝑘}𝑓_𝑘^′′+2𝑛𝑛+11−𝑓^′_{𝑚−1−𝑘}𝑓^′_𝑘+𝑀²𝛽+1𝑓^′_{𝑚−1−𝑘}𝑓_𝑘^′′+𝛽^𝑚−1_𝑘=02𝑛𝑛−1𝑛+11−𝑓^′𝑘_{𝑚−1−𝑘𝑙=0}𝑓^′_𝑘−𝑙𝑓_𝑙^′′+(3𝑛−1)𝑓^𝑘_{𝑚−1−𝑘𝑙=0}𝑓^′_𝑘−𝑙𝑓_𝑙^′′−𝑛+12𝑓^𝑘′′_{𝑚−1−𝑘𝑙=0}𝑓^′_𝑘−𝑙𝑓_𝑙^′′′+𝑛−12𝑓^′𝑘_{𝑚−1−𝑘𝑙=0}𝑓^′_𝑘−𝑙𝑓_𝑙^′′ℛ^𝜃_𝑚𝜂=𝜃^′′_𝑚−1+Pr^𝑚−1_𝑘=0𝜃^′_{𝑚−1−𝑘}𝑓_𝑘,𝜒_𝑚=0,𝑚≤11,𝑚>1. (35)(36)(37)(38)(39)(40)(41)

4. Convergence of the Homotopy Solutions

Obviously the auxiliary parameters ℏ_𝑓 and ℏ_𝜃 in the series solutions (31) and (32) have a definite role in adjusting and controlling the convergence. For the admissible values of ℏ_𝑓 and ℏ_𝜃, the ℏ_𝑓 and ℏ_𝜃 curves are portrayed for 15th-order of approximations. It is noticed that the ranges for the admissible values of ℏ_𝑓 and ℏ_𝜃 are −1.5≤ℏ_𝑓≤−0.2 and −1.2≤ℏ_𝜃≤−0.6 (Figure 1). Moreover, the series given by (31) and (32) converge in the whole region of 𝜂 when ℏ_𝑓=−0.5 and ℏ_𝜃=−0.8.

Figure 1 

ℏ curves for 15th-order approximations.

5. Results and Discussion

The purpose of this section is to investigate the variations of parameters including Deborah number 𝛾, the constant 𝑛, Hartman number 𝑀, and the Prandtl number Pr on the velocity 𝑓^′ and the temperature 𝜃. The variation of the Deborah number 𝛽, Hartman number 𝑀, and the parameter 𝑛 on the velocity 𝑓^′ can be seen in Figures 2–5. It is found from Figure 2 that the boundary layer thickness decreases with an increase in 𝛾. Figure 3 is plotted for the effects of Hartman number 𝑀 on the velocity profile 𝑓^′. The effects of 𝑀 on 𝑓^′ are qualitatively similar to those of 𝛾. Figure 4 depicts the velocity field for different values of parameter 𝑛. Clearly 𝑓^′ is an increasing function of 𝑛 while the boundary layer thickness decreases. Figure 5 shows the effects of Prandtl number Pr on the temperature profile 𝜃. The thermal boundary layer thickness decreases when Pr increases. The variation of Hartman number 𝑀 on the temperature profile is shown in Figure 6. It is noticed that the temperature profile increases when 𝑀 is increased. The effects of parameter 𝑛 are shown in Figure 7. It is observed that an increase in the value of 𝑛 decreases the thermal boundary layer thickness. Figures 8 and 9 are displayed to analyze the comparison of the present analytical results with the existing numerical solutions. An excellent agreement is found between the two solutions for different values of 𝛾. In Table 1, we have computed the numerical values of 𝑓(𝜂),  𝑓^′(𝜂), and 𝑓^′′(𝜂) for the comparison of the present analytical results with the numerical solution [26]. Table 2 is also a comparison between homotopy solution and numerical solution [26] in a special case. An excellent agreement is found between the two solutions.

Table 1 

Comparison of the results 𝜃^′(𝜂) when 𝛽=0=𝑀.

Table 2 

Comparison of the results when 𝑀=0=𝛽.

Figure 2 

Influence of 𝛽 on 𝑓′.

Figure 3 

Influence of 𝑀 on 𝑓′.

Figure 4 

Influence of 𝑛 on 𝑓′.

Figure 5 

Influence of Pr on 𝜃.

Figure 6 

Influence of 𝑀 on 𝜃.

Figure 7 

Influence of 𝑛 on 𝜃.

Figure 8 

Influence of 𝛾 on 𝑓′ when 𝑀=𝛽=0. Solid lines: HAM solution [10]; filled circles: numerical solution [26].

Figure 9 

Influence of 𝛾 on 𝜃′(0) when 𝛽=0. Solid lines: HAM solution [10]; filled circles: numerical solution [26].