Analytic Approximations for the Flows and Heat Transfer in Microchannels between Two Parallel Plates

Abstract

We consider the nonlinear problem for the flow of Newtonian fluid in a microchannel between two parallel plates with the effects of velocity slip, viscous dissipation, and temperature jump at the wall. This problem is modelled by both the Navier-Stokes equation and energy equation with two thermal boundary conditions related to the two cases: the constant wall temperature (CWT) and the constant heat flux (CHF). The homotopy analysis method is applied via a polynomial exponential basis to obtain analytic approximations for this problem. A rarefaction effects on the velocity profile and the flow friction are investigated. Also, as a result of the application, the effects, on the Nusselt number Nu, with variation in Brinkman number Br and Knudsen number Kn for both (CWT) case and (CHF) case are discussed.

1. Introduction

Research on flow and heat transfer in microchannels has increased, recently, due to the developments in specific areas such as microfabrication technology, microdevices, Microelectromechanical Systems (MEMSs), the electronic industry, and the biomedical engineering. Microscale fluid flow and heat transfer has a different behaviour from that of macroscale case. At macroscale case, classical conservation equations are successfully coupled with the corresponding wall boundary conditions, usual no-slip for hydrodynamic boundary condition and no-temperature jump for thermal boundary condition. For a rarefied fluid flow at microscale a slip condition for the velocity and a jump condition for the temperature should be adopted. Viscous dissipation is another parameter that should be taking into account at microscale. Viscous dissipation changes the temperature by behaving like an energy source due to a power generation induced by the shear stress.

The Knudsen number Kn is the ratio of the fluid mean free path to the characteristic dimension in the flow field, and it determines the degree of rarefaction and the degree of the validity of continuum approach. For flows in continuum and slip regions, Eckert and Drake [1] indicated that there is a strong evidence to use the Navier-Stokes equations modified by boundary conditions. Beskok and Karniadakis [2] defined four different flow regimes based on the value of the Knudsen number: continuum flow (Kn<0.001), slip flow (0.001<Kn<0.1), transition flow (0.1<Kn<10), and free molecular flow (Kn>10). Liu et al. [3] and Arkilic et al. [4] pointed that the Navier-Stokes equations, when they are combined with slip-flow boundary conditions, give results for pressure drop and friction factor which are in agreement with the experimental data for some microchannel flows.

Gad-el-Hak [5] treated analysis for microchannel flows through study of microdevices fluid mechanics. Guo and Li [6] studied the size effects on microscale single-phase fluid flow and heat transfer. Wu and Cheng [7] studied the friction factor and convective heat transfer in smooth silicon microchannels of trapezoidal cross-section. Zhang et al. [8] validated the Navier-Stokes equations for slip flow with transition region. For recent studies in microchannel flows and heat transfer, see, for example, [9–15]. The homotopy analysis method presented by Liao [16–19] is a powerful tool for such kind of problems which have complicated and strong nonlinearity. This method does not depend on existence of small or large parameters in studied problems such as the perturbation methods [20–22], it does not use discretization of numerical methods, which have difficulties in case of the existence of singularities, and unlike other methods, such as Lyapunov's small parameter method [23] and the Adomian decomposition method [24–27], it has the ability to control the convergence for obtained approximations. Liao and other authors applied this method in a successful manner to various nonlinear applications in science and engineering. See, for example, [28–34]. To show the basic ideas and aspects of this method, let us consider the following:𝑁𝑔(𝑟,𝑡)=0, (1.1) where 𝑁 is a nonlinear operator, 𝑔 is unknown function, 𝑟 is a vector of spatial variables, and 𝑡 is the time. By means of the traditional concept of homotopy, Liao constructs the so-called zero-order deformation equation:1−𝑞𝐿𝜙𝑟,𝑡,𝑞−𝑔₀(𝑟,𝑡)=𝑞ℎ𝑁𝜙𝑟,𝑡,𝑞, (1.2) where 𝑞𝜀[0,1] is an embedding parameter, 𝐿 is an auxiliary linear operator, 𝑔₀ is an initial approximation for the unknown function 𝑔, and ℎ is an auxiliary parameter. As 𝑞 varies from 0 to 1, the solution of (1.2) varies from the initial approximation 𝑔₀ to the exact solution 𝑔 of the nonlinear equation (1.1): 𝜙(𝑟,𝑡,0)=𝑔₀(𝑟,𝑡),𝜙(𝑟,𝑡,1)=𝑔(𝑟,𝑡). (1.3) Expanding 𝜙(𝑟,𝑡,𝑞) in Taylor series, with respect to 𝑞, one has𝜙𝑟,𝑡,𝑞=𝑔₀(𝑟,𝑡)+^∞_𝑚=1𝑔_𝑚(𝑟,𝑡)𝑞^𝑚,𝑔_𝑚()=1𝑟,𝑡𝜕𝑚!^𝑚𝜙𝑟,𝑡,𝑞𝜕𝑞^𝑚∣_𝑞=0. (1.4)(1.5) If the parameter ℎ is properly chosen such that the series (1.4) is convergent at 𝑞=1, then from (1.3) 𝑔(𝑟,𝑡)=𝑔₀(𝑟,𝑡)+^∞_𝑚=1𝑔_𝑚(𝑟,𝑡). (1.6) Differentiating the zero-order deformation equation (1.2) 𝑚 times with respect to 𝑞, setting 𝑞=0, and finally dividing by 𝑚!, we obtain the 𝑚th order deformation equation:𝐿𝑔_𝑚(𝑟,𝑡)−𝜒_𝑚𝑔_𝑚−1(𝑟,𝑡)=ℎ𝑃_𝑚()𝑃𝑟,𝑡,𝑚≥1,_𝑚()=1𝑟,𝑡()!𝜕𝑚−1^𝑚−1𝑁𝜙𝑟,𝑡,𝑞𝜕𝑞^𝑚−1∣_𝑞=0𝜒_𝑚,=0,𝑚=1,1,𝑚>1. (1.7)(1.8)(1.9) The solutions 𝑔_𝑚(𝑟,𝑡),𝑚≥1, of (1.7) are called the 𝑚th deformations and can be computed by any symbolic software. The homotopy analysis method provides a freedom to choose 𝐿, 𝑔₀ and control the convergence, by the aid of the parameter ℎ, to obtain analytic approximations in terms of suitable basis of functions.

2. Mathematical Formulation

The mathematical modelling for two-dimensional steady laminar slip flow in a microchannel between two parallel plates, with the consideration of Viscous dissipation, has the following form.

The continuity equation:𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝑦=0. (2.1) The Navier-Stokes equations:𝑢𝜕𝑢𝜕𝑥+𝑣𝜕𝑢1𝜕𝑦=−𝜌𝜕𝑝𝜕𝜕𝑥+𝜐²𝑢𝜕𝑥²+𝜕²𝑢𝜕𝑦²,𝑢𝜕𝑣𝜕𝑥+𝑣𝜕𝑣1𝜕𝑦=−𝜌𝜕𝑝𝜕𝜕𝑦+𝜐²𝑣𝜕𝑥²+𝜕²𝑣𝜕𝑦². (2.2)(2.3) The energy equation𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝜕𝑦=𝛼²𝑇𝜕𝑦²+𝜐𝐶𝑝𝜕𝑢𝜕𝑦², (2.4) where in the previous equations, it is assumed that the inlet velocity and the inlet temperature are uniform, the distance between the two parallel plates is 2H (see Figure 1), and the tangential and thermal accommodation coefficients are taken to be unity. Also, 𝑢 and 𝑣 are velocity components, 𝜌 is the density, 𝑝 is the pressure, 𝜐 is the kinematic viscosity, 𝑇 is the temperature, 𝛼 is the thermal diffusivity, and 𝐶_𝑝 is the specific heat.

Figure 1 

Microchannel between two parallel plates.

Equations (2.1)–(2.4) are subject to the following boundary conditions: 𝑢=ℓ𝜕𝑢𝜕𝑦,𝑣=0at𝑦=0,𝜕𝑢𝜕𝑦=0at𝑦=𝐻,𝑇−𝑇_𝑤=ℓ𝜕𝑇()𝜕𝑦forCWT,𝑇−𝑇_𝑤𝑞=−ℓ^𝑤𝜆(forCHF)at𝑦=0,𝜕𝑇𝜕𝑦=0at𝑦=𝐻, (2.5) where ℓ is the molecular free path, 𝑇_𝑤 is the temperature at the wall, 𝑞_𝑤 is the heat flux at the wall, and 𝜆 is the thermal conductivity.

Using transformations of the form1𝜂=𝐻𝑦,𝜓=𝑓𝜂𝑈𝑥, (2.6) where 𝜓 is a stream function and 𝑈 is the inlet velocity, (2.3) can be reduced to𝜕𝜕𝑦𝜕𝑝𝜕𝑥=0, (2.7) and hence from (2.7) we can write−1𝜌𝜕𝑝𝜕𝑥=𝐶𝑥, (2.8) where 𝐶 is a constant.

The mean velocity 𝑢_𝑚 has the following form:𝑢_𝑚=1𝐻^𝐻₀1𝑢𝑑𝑦=𝐻𝑓(1)𝑈𝑥. (2.9) Also, the mean temperature 𝑇_𝑚 has following the form:𝑇_𝑚=1𝐻^𝐻₀𝑢𝑢_𝑚1𝑇𝑑𝑦=𝑓(1)¹₀𝑓^′𝜂𝑇𝑑𝜂. (2.10) By the aid of (2.6), (2.8), and the following transformations: 𝐻𝜃=²𝑥²𝑇−𝑇_𝑤𝑇_𝑚−𝑇_𝑤),𝐻(forCWT𝜃=²𝑥²𝑇−𝑇_𝑤+ℓ𝑞_𝑤/𝜆𝑇_𝑚−𝑇_𝑤+ℓ𝑞_𝑤),/𝜆(forCHF (2.11) equations (2.1)–(2.4) are transformed to the system of two equations:𝑓^′′′𝜂−(Re)𝑓^′2𝜂+(Re)𝑓𝜂𝑓^′′𝜂𝜃+𝜀=0,_′′𝜂−(Br)𝑓^′′2𝜂+(Pe)𝑓𝜂𝜃^′𝜂−2(Pe)𝑓^′𝜂𝜃𝜂=0, (2.12) with the following boundary conditions:𝑓^′(0)−(Kn)𝑓^′′(0)=0,𝑓(0)=0,𝑓^′′(1)𝜃(0)−()𝜃=0,Kn^′(0)=0(forCWT),𝜃(0)=0(forCHF),𝜃^′(1)=0, (2.13) where Kn=ℓ/[2𝐻] is the Knudsen number, Br=𝜇𝑈²/[𝜆(𝑇_𝑤−𝑇_𝑚)] is the Brinkman number, 𝜇 is the dynamic viscosity,Re=𝑈𝐻/𝜐 is the Reynolds number, 𝜀=𝐶𝐻³/[𝑈𝜐] and Pe=𝑈𝐻/𝛼 is the Peclet number.

Not only (2.12) are strongly nonlinear but also they have linear boundary conditions. Poiseuille number is one of the most important parameters in fluid flow which is the product of friction factor and local Reynolds number. This number is denoted by Po and, based on the mean velocity [15], it can be given by Po=4𝑓^′′(0)𝑓(1). (2.14) The most important parameter in heat transfer is the ratio of convective to conductive heat transfer across the boundary which is called the Nusselt number. This number is denoted by Nu and, as in [15], analytic expressions for Nu can be drived for for both the (CWT) and (CHF) cases as follows:(1)𝜃Nu=2𝑓^′(0)∫¹₀𝑓^′𝜂𝜃𝜂()(1)𝜃𝑑𝜂forCWTcase,Nu=2𝑓^′(0)∫¹₀𝑓^′𝜂𝜃𝜂𝑑𝜂−2(Kn)𝑓(1)𝜃^′(0)for(CHF)case. (2.15) Here, we apply the homotopy analysis method via a polynomial exponential basis to get analytic approximations for solutions of (2.12)–(2.13), and then we use these approximations to discuss the rarefaction effects on velocity profile and flow friction (effects on Poiseuille number). Also, we use the expressions (2.15) to study the effects on the Nusselt number with the variations in Brinkman and Knudsen numbers.

3. Application of the Homotopy Analysis Method

Applying the homotopy analysis method on (2.12)–(2.13), we use the basis𝜂^𝑚𝑒^𝑛𝜂,𝑚,𝑛≥0. (3.1) Also, we express the solutions by the rule of the following expressions:𝑓𝜂=^∞∞_𝑛=1𝑚=1𝐴_𝑛,𝑚𝜂^𝑚𝑒^𝑛𝜂,𝜃𝜂=^∞∞_𝑛=0𝑚=0𝐵_𝑛,𝑚𝜂^𝑚𝑒^𝑛𝜂()𝜃𝜂=forCWTcase,_{∞∞𝑛=1𝑚=1}𝐶_𝑛,𝑚𝜂^𝑚𝑒^𝑛𝜂for(CHF)case. (3.2)(3.3)(3.4)

Viewing (2.12)–(2.13) and (3.1)–(3.4), we choose the initial approximations and the auxiliary linear operators as follows:𝑓₀𝜂=2(Kn)𝜂𝑒^𝜂+(1−2Kn)𝜂²𝑒^𝜂+1()𝜂138Kn−7³𝑒^𝜂,𝜃₀𝜂=Kn+𝜂𝑒^𝜂−23𝜂²𝑒^𝜂()𝜃forCWTcase,₀𝜂=𝜂𝑒^𝜂−23𝜂²𝑒^𝜂()𝐿forCHFcase,_𝑓=13𝑒^−𝜂𝜕³𝜕𝜂³𝐿_𝜃,=12𝑒^−𝜂𝜕²𝜕𝜂². (3.5)(3.6)(3.7)(3.8)(3.9) We construct the so-called zero-order deformation equations:1−𝑞𝐿_𝑓𝜙𝜂,𝑞−𝑓₀𝜂=ℎ_𝑓𝑞𝑁_𝑓,𝐿𝜙,𝜓,𝑞1−𝑞_𝜃𝜓𝜂,𝑞−𝜃₀𝜂=ℎ_𝜃𝑞𝑁_𝜃𝜙,𝜓,𝑞, (3.10)(3.11) with the following conditions:𝜕𝜙𝜂,𝑞∣𝜕𝜂^𝜂=0−()𝜕Kn²𝜙𝜂,𝑞𝜕𝜂²∣_𝜂=0𝜕=0,𝜙0,𝑞=0,²𝜙𝜂,𝑞𝜕𝜂²∣_𝜂=1𝜓−()=0,0,𝑞Kn𝜕𝜓𝜂,𝑞∣𝜕𝜂^𝜂=0=0(forCWT),𝜓0,𝑞=0(forCHF),𝜕𝜓𝜂,𝑞∣𝜕𝜂^𝜂=1=0, (3.12) where ℎ_𝑓 and ℎ_𝜃 are auxiliary parameters and 𝑞 is an embedding parameter that varies from 0 to 1. As 𝑞 varies from 0 to, 1 the solution of (3.10)–(3.12) varies from the initial approximations to the exact solutions of (2.12)–(2.13). Thus we have𝜙𝜂,0=𝑓₀𝜂,𝜙𝜂,1=𝑓𝜂,𝜓𝜂,0=𝜃₀𝜂,𝜓𝜂,1=𝜃𝜂. (3.13) The nonlinear operators 𝑁_𝑓 and 𝑁_𝜃 in (3.10) and (3.11) are defined as𝑁_𝑓=𝜕𝜙,𝜓,𝑞³𝜙𝜕𝜂³−(Re)𝜕𝜙𝜕𝜂²+()𝜙𝜕Re²𝜙𝜕𝜂²𝑁+𝜀,_𝜃=𝜕𝜙,𝜓,𝑞²𝜓𝜕𝜂²−()𝜕Br²𝜙𝜕𝜂²²+(Pe)𝜙𝜕𝜓()𝜕𝜂−2Pe𝜕𝜙𝜕𝜂𝜓. (3.14) By Taylor's series, at 𝑞=0, we have𝜙𝜂,𝑞=𝜙𝜂,0+^∞_𝑚=1𝑓_𝑚𝜂𝑞^𝑚,𝜓𝜂,𝑞=𝜓𝜂,0+^∞_𝑚=1𝜃_𝑚𝜂𝑞^𝑚, (3.15) where𝑓_𝑚𝜂=1𝜕𝑚!^𝑚𝜙𝜂,𝑞𝜕𝑞^𝑚∣_𝑞=0,𝜃_𝑚𝜂=1𝜕𝑚!^𝑚𝜓𝜂,𝑞𝜕𝑞^𝑚∣_𝑞=0. (3.16) If the parameters ℎ_𝑓 and ℎ_𝜃 are properly chosen such that the series (3.15) converge at 𝑞=1, then from  (3.13)𝑓𝜂=𝑓₀𝜂+^∞_𝑚=1𝑓_𝑚𝜂,𝜃𝜂=𝜃₀𝜂+^∞_𝑚=1𝜃_𝑚𝜂. (3.17) Differentiating the zero-order deformation equations (3.10) and (3.11) 𝑚 times with respect to 𝑞, setting 𝑞=0, and then dividing by 𝑚!, we obtain the 𝑚th order deformation equations:𝐿_𝑓𝑓_𝑚𝜂−𝜒_𝑚𝑓_𝑚−1𝜂=ℎ_𝑓𝑃_𝑚𝜂𝐿,𝑚≥1,_𝜃𝜃_𝑚𝜂−𝜒_𝑚𝜃_𝑚−1𝜂=ℎ_𝜃𝑄_𝑚𝜂,𝑚≥1, (3.18) where𝑃_𝑚𝜂=𝑓^′′′_𝑚−1𝜂−[Re]^𝑚−1_𝑛=0𝑓^′_𝑛𝜂𝑓^′_{𝑚−1−𝑛}𝜂+[Re]^𝑚−1_𝑛=0𝑓_𝑛𝜂𝑓^′′_{𝑚−1−𝑛}𝜂+𝜀1−𝜒_𝑚,𝑄_𝑚𝜂=𝜃^′′_𝑚−1𝜂−[Br]^𝑚−1_𝑛=0𝑓_𝑛^′′𝜂𝑓^′′_{𝑚−1−𝑛}𝜂+[Pe]^𝑚−1_𝑛=0𝑓_𝑛𝜂𝜃^′_{𝑚−1−𝑛}𝜂−2[Pe]^𝑚−1_𝑛=0𝑓^′_𝑛𝜂𝜃_{𝑚−1−𝑛}𝜂,𝜒_𝑚=0,𝑚=1,1,𝑚>1. (3.19) The solutions of (3.18) are referred as the 𝑚th deformations, 𝑚≥1, and these deformations may satisfy the following conditions:𝜕𝑓_𝑚𝜂∣𝜕𝜂^𝜂=0−()𝜕Kn²𝑓_𝑚𝜂𝜕𝜂²∣_𝜂=0=0,𝑓_𝑚(0)𝜕=0,²𝑓_𝑚𝜂𝜕𝜂²∣_𝜂=1𝜃=0,_𝑚(0)−(Kn)𝜕𝜃_𝑚𝜂∣𝜕𝜂^𝜂=0=0(forCWT),𝜃_𝑚(0)=0(forCHF),𝜕𝜃_𝑚𝜂∣𝜕𝜂^𝜂=1=0. (3.20) The solutions of (3.18) can be obtained by a symbolic software such as Mathematica, and these solutions can be taken as𝑓_𝑚𝜂=^2𝑚+1_𝑛=1^2𝑚+1_𝑘=1𝑎^𝑘_𝑚,𝑛𝜂^𝑘𝑒^𝑛𝜂,𝜃_𝑚𝜂=^2𝑚+1_𝑛=0^2𝑚+1_𝑘=0𝑏^𝑘_𝑚,𝑛𝜂^𝑘𝑒^𝑛𝜂()𝜃forCWTcase,_𝑚𝜂=^2𝑚+1_𝑛=1^2𝑚+1_𝑘=1𝑐^𝑘_𝑚,𝑛𝜂^𝑘𝑒^𝑛𝜂for(CHF)case. (3.21) For a best choice of ℎ_𝑓andℎ_𝜃, the Mth order approximations for the solutions of (2.12)–(2.13) are then𝑓𝜂=𝑓₀𝜂+^𝑀_𝑚=1^2𝑚+1_𝑛=1^2𝑚+1_𝑘=1𝑎^𝑘_𝑚,𝑛𝜂^𝑘𝑒^𝑛𝜂,𝜃𝜂=𝜃₀𝜂+^𝑀_𝑚=1^2𝑚+1_𝑛=0^2𝑚+1_𝑘=0𝑏^𝑘_𝑚,𝑛𝜂^𝑘𝑒^𝑛𝜂()𝜃𝜂forCWTcase,=𝜃₀𝜂+^𝑀_𝑚=1^2𝑚+1_𝑛=1^2𝑚+1_𝑘=1𝑐^𝑘_𝑚,𝑛𝜂^𝑘𝑒^𝑛𝜂for(CHF)case. (3.22) The best values of parameters ℎ_𝑓andℎ_𝜃 which control the convergence of the approximations (3.22) can be deduced by plotting the ℎ-curve of 𝑓^′′(0) and 𝜃^′′(0) which takes a horizontal line through the position of convergence.

4. Results of the Application

Applying the homotopy analysis method for (2.12)–(2.13) up to the 10th order approximation it is found that it is best to take the control parameters ℎ_𝑓 and ℎ_𝜃 such that ℎ_𝑓=ℎ_𝜃=ℎ=−1.2 to get the best approximations for the solutions, as indicated in Figures 2, 3, and 4. In our study, Brinkman number Br and the Knudsen number Kn are the main parameters for heat and fluid flow in a microchannel. We study the interactive effects of these parameters for both the hydrodynamic and the thermal transport in the microchannel. We examine two different thermal boundary conditions at the wall of the microchannel: the constant wall temperature (CWT) and the constant heat flux (CHF).

Figure 2 

The ℎ-curve for 𝑓′′(0) at Kn=0.05, Br=5, Pe=0.5, and Re=10.

Figure 3 

The ℎ-curve for 𝜃′′(0) for CHF case at Kn=0.05, Br=5, Pe=0.5, and Re=10.

Figure 4 

The ℎ-curve for 𝜃′′(0) for CWT case at Kn=0.05, Br=5, Pe=0.5, and Re=10.

Also, it is noted that Kn=0 represents the macroscale case, while Kn>0 holds for the microscale case and Br=0 represents the case without effect of the viscous dissipation.

Figure 5 investigates the dimensionless velocity profile for no-slip and slip microchannel flow. When Kn=0, the velocity increases from zero at wall to a maximum value (1.5) at plate centerline. The dimensionless velocity profile seems near parabolic. The velocity profile for Kn=0.05 and Kn=0.1 are traced in this figure and it is remarked that all velocity profiles for the three cases of Kn intersect at one point. Figure 6 indicates the Poiseuille number Po with the continuous variation of the Knudsen number Kn. In this figure, the values of the Poiseuille number decrease gradually with the increase of Kn.

Figure 5 

The dimensionless velocity profile for different values of Kn=0,0.05,0.1.

Figure 6 

The Poiseuille number Po with continuous variation of Kn.

This is expected because the higher value of Kn implies larger slip and the higher velocity near the wall decreases the shear stress which causes the Po to drop further. This figure shows, also, that Po=12 when Kn=0 which agrees with the macroscale case and goes to an asymptotic value with the increasing of Kn. Figures 7 and 8 illustrate the effects on Nusselt number Nu, with variation in the Brinkman number Br and different values of Knudsen number Kn for the constant wall temperature (CWT) case. These figures seem to be a hyperbola for each Kn. In general, these figures show that the increase of Knudsen number Kn decreases the Nusselt number Nu due to the temperature jump at the wall. If the Brinkman number Br takes positive values, then this corresponds to what is called wall heating case. For this case, the wall temperature is greater than that of the bulk fluid. For this case it is noticed that Nu decreases as Br increases. Also, the value of Br indicates the effects of the viscous dissipation. Viscous dissipation increases the temperature of the bulk fluid especially near the wall, because the highest shear rate occurs in this region. This effect causes the temperature difference between wall and the bulk fluid decrease. The extra increasing of Br yields that the heat supplied by the wall into the fluid is balanced with the internal heat due to the viscous heating and hence the Nu reaches an asymptotic value. Also, for wall cooling case when Br takes negative values, the Nu reaches an asymptotic value with the increasing of Br in the negative direction. Figures 9 and 10 depict the effects on the Nusselt number Nu, with the variation in Brinkman number Br and different values of Knudsen number Kn for the constant heat flux (CHF) case. The effects on Nu are very similar to the case of constant wall temperature (CWT). This similarity can be expected from the similarity of the formulae (2.15) of the Nusselt number Nu and also from the similarity of the initials of the temperature profile in (3.6) and (3.7).

Figure 7 

Effects on Nu with Br at Kn=0 for CWT case.

Figure 8 

Effects on Nu with Br at Kn=0.05 (thin line) and 0.1 (solid line) for CWT case.

Figure 9 

Effects on Nu with Br at Kn=0 for CHF case.

Figure 10 

Effects on Nu with Br at Kn = 0.05 (thin line) and 0.1 (solid line) for CHF case.

5. Conclusion

In this paper the homotopy analysis method was applied via a polynomial exponential basis to obtain analytic approximations for the nonlinear problem of the flow fluid in a microchannel between two parallel plates taking into account the effects of velocity slip, viscous dissipation, and temperature jump at the wall. As a result of this application, the rarefied effects on the velocity profile and the flow friction are studied and also effects on the Nusselt number with the variation in Brinkman and Knudsen numbers are discussed for both the constant wall temperature (CWT) and the constant heat flux (CHF). Also, the application proved the great ability and flexibility of this method to treat the strongly nonlinear problems which have linear boundary conditions via any suitable set of continuous functional basis.

Acknowledgment

The author thanks the anonymous reviewers for their comments and suggestions to enhance this work.