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

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.


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.

Mathematical Problems in Engineering
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  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: 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 − q L φ r, t, q − g 0 r, t qhN φ r, t, q , 1.2 where q ε 0, 1 is an embedding parameter, L is an auxiliary linear operator, g 0 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 g 0 to the exact solution g of the nonlinear equation 1.1 : φ r, t, 0 g 0 r, t , φ r, t, 1 g r, t .

1.3
Expanding φ r, t, q in Taylor series, with respect to q, one has φ r, t, q g 0 r, t ∞ m 1 g m r, t q m , 1.4 If the parameter h is properly chosen such that the series 1.4 is convergent at q 1, then from 1.3 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: 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, g 0 and control the convergence, by the aid of the parameter h, to obtain analytic approximations in terms of suitable basis of functions.

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: The Navier-Stokes equations:

2.3
The energy equation 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.Equations 2.1 -2.4 are subject to the following boundary conditions: where 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 form where ψ is a stream function and U is the inlet velocity, 2.3 can be reduced to and hence from 2.7 we can write where C is a constant.The mean velocity u m has the following form: Also, the mean temperature T m has following the form: By the aid of 2.6 , 2.8 , and the following transformations: equations 2.1 -2.4 are transformed to the system of two equations: with the following boundary conditions: where Kn / 2H is the Knudsen number, Br μ U 2 / λ T w −T m is the Brinkman number, μ is the dynamic viscosity, Re UH/υ is the Reynolds number, ε CH 3 / 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:

Mathematical Problems in Engineering
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.

Application of the Homotopy Analysis Method
Applying the homotopy analysis method on 2.12 -2.13 , we use the basis Also, we express the solutions by the rule of the following expressions: C n,m η m e nη for CHF case.

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 have

3.14
By Taylor's series, at q 0, we have

3.15
where 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

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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:

3.19
The solutions of 3.18 are referred as the mth deformations, m ≥ 1, and these deformations may satisfy the following conditions:

3.20
The solutions of 3.18 can be obtained by a symbolic software such as Mathematica, and these solutions can be taken as c k m,n η k e nη for CHF case.For a best choice of h f and h θ , the Mth order approximations for the solutions of 2.12 -2.13 are then c k m,n η k e nη for CHF case.

3.22
The best values of parameters h f and h 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.

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 .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.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.

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 Mathematical Problems in Engineering 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.
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.

10 Figure 8 :
Figure 8: Effects on Nu with Br at Kn 0.05 thin line and 0.1 solid line for CWT case.

20 Figure 9 :
Figure 9: Effects on Nu with Br at Kn 0 for CHF case.
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 .

10 Figure 10 :
Figure 10: Effects on Nu with Br at Kn 0.05 thin line and 0.1 solid line for CHF case.