# Analytic Approximations for the Flows and Heat Transfer

Authors: A. El-Nahhas
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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  numberKn  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:(1.1)N[g(r,t)]=0,
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:(1.2)(1-q)L[ϕ(r,t,q)-g0(r,t)]=qhN[ϕ(r,t,q)],
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): (1.3)ϕ(r,t,0)=g0(r,t),ϕ(r,t,1)=g(r,t).
Expanding ϕ(r,t,q) in Taylor series, with  respect  to  q, one has(1.4)ϕ(r,t,q)=g0(r,t)+∑m=1∞gm(r,t)qm,(1.5)gm(r,t)=1m!∂mϕ(r,t,q)∂qm∣q=0.
If the parameter h  is properly chosen such that  the series (1.4) is convergent at q=1, then from (1.3) (1.6)g(r,t)=g0(r,t)+∑m=1∞gm(r,t).
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:(1.7)L[gm(r,t)-χmgm-1(r,t)]=hPm(r,t),m≥1,(1.8)Pm(r,t)=1(m-1)!∂m-1N[ϕ(r,t,q)]∂qm-1∣q=0,(1.9)χm={0,m=1,1,m>1.
The  solutions gm(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:(2.1)∂u∂x+∂v∂y=0.
The Navier-Stokes  equations:(2.2)u∂u∂x+v∂u∂y=-1ρ∂p∂x+υ(∂2u∂x2+∂2u∂y2),(2.3)u∂v∂x+v∂v∂y=-1ρ∂p∂y+υ(∂2v∂x2+∂2v∂y2).
The energy equation(2.4)u∂T∂x+v∂T∂y=α∂2T∂y2+υCp(∂u∂y)2,
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  Cp  is  the  specific  heat.Figure 1
Microchannel between two parallel plates.Equations (2.1)–(2.4) are subject to the following boundary conditions: (2.5)u=l∂u∂y,v=0aty=0,∂u∂y=0aty=H,T-Tw=l∂T∂y(forCWT),T-Tw=-lqwλ(forCHF)aty=0,∂T∂y=0aty=H,
where ℓ is the molecular free path, Tw is the temperature at the wall, qw is the heat flux at the wall, and λ is the thermal conductivity.Using transformations of the form(2.6)η=1Hy,ψ=f(η)Ux,
where ψ is a stream function and  U is the inlet velocity,  (2.3) can be reduced to(2.7)∂∂y[∂p∂x]=0,
and hence from (2.7) we can write(2.8)-1ρ∂p∂x=Cx,
where C is a constant.The mean velocityum has the following form:(2.9)um=1H∫0Hudy=1Hf(1)Ux.
Also, the mean temperature Tm has following the form:(2.10)Tm=1H∫0H(uum)Tdy=1f(1)∫01f′(η)Tdη.
By the aid of  (2.6), (2.8), and the following  transformations: (2.11)θ=[H2x2][T-TwTm-Tw](forCWT),θ=[H2x2][T-Tw+lqw/λTm-Tw+lqw/λ](forCHF),
equations (2.1)–(2.4) are transformed to the system of two equations:(2.12)f′′′(η)-(Re)f′2(η)+(Re)f(η)f′′(η)+ε=0,θ′′(η)-(Br)f′′2(η)+(Pe)f(η)θ′(η)-2(Pe)f′(η)θ(η)=0,
with the following boundary conditions:(2.13)f′(0)-(Kn)f′′(0)=0,f(0)=0,f′′(1)=0,θ(0)-(Kn)θ′(0)=0(forCWT),θ(0)=0(forCHF),θ′(1)=0,
where  Kn=ℓ/[2H] is  the  Knudsen  number, Br=μU2/[λ(Tw-Tm)] 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 (2.14)Po=4f′′(0)f(1).
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:(2.15)Nu=2f(1)θ′(0)∫01f′(η)θ(η)dηfor(CWT)case,Nu=2f(1)θ′(0)∫01f′(η)θ(η)dη-2(Kn)f(1)θ′(0)for(CHF)case.
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 basis(3.1)ηmenη,m,n≥0.
Also, we express the solutions by the rule of  the following expressions:(3.2)f(η)=∑n=1∞∑m=1∞An,mηmenη,(3.3)θ(η)=∑n=0∞∑m=0∞Bn,mηmenηfor(CWT)case,(3.4)θ(η)=∑n=1∞∑m=1∞Cn,mηmenηfor(CHF)case.Viewing (2.12)–(2.13)  and  (3.1)–(3.4), we choose the initial approximations and the auxiliary linear operators as follows:(3.5)f0(η)=2(Kn)ηeη+(1-2Kn)η2eη+113(8Kn-7)η3eη,(3.6)θ0(η)=Kn+ηeη-23η2eηfor(CWT)case,(3.7)θ0(η)=ηeη-23η2eηfor(CHF)case,(3.8)Lf=13e-η∂3∂η3,(3.9)Lθ=12e-η∂2∂η2.
We construct the so-called zero-order deformation equations:(3.10)(1-q)Lf[ϕ(η,q)-f0(η)]=hfqNf[ϕ,ψ,q],(3.11)(1-q)Lθ[ψ(η,q)-θ0(η)]=hθqNθ[ϕ,ψ,q],
with the following conditions:(3.12)∂ϕ(η,q)∂η∣η=0-(Kn)∂2ϕ(η,q)∂η2∣η=0=0,ϕ(0,q)=0,∂2ϕ(η,q)∂η2∣η=1=0,ψ(0,q)-(Kn)∂ψ(η,q)∂η∣η=0=0(forCWT),ψ(0,q)=0(forCHF),∂ψ(η,q)∂η∣η=1=0,
where hf and hθ  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(3.13)ϕ(η,0)=f0(η),ϕ(η,1)=f(η),ψ(η,0)=θ0(η),ψ(η,1)=θ(η).
The nonlinear operators Nf and Nθ in (3.10) and (3.11)  are defined as(3.14)Nf[ϕ,ψ,q]=∂3ϕ∂η3-(Re)(∂ϕ∂η)2+(Re)ϕ∂2ϕ∂η2+ε,Nθ[ϕ,ψ,q]=∂2ψ∂η2-(Br)(∂2ϕ∂η2)2+(Pe)ϕ∂ψ∂η-2(Pe)∂ϕ∂ηψ.
By Taylor's series, at q=0, we have(3.15)ϕ(η,q)=ϕ(η,0)+∑m=1∞fm(η)qm,ψ(η,q)=ψ(η,0)+∑m=1∞θm(η)qm,
where(3.16)fm(η)=1m!∂mϕ(η,q)∂qm∣q=0,θm(η)=1m!∂mψ(η,q)∂qm∣q=0.
If the parameters hf and hθ are properly chosen such that the series  (3.15) converge at  q=1,  then from  (3.13)(3.17)f(η)=f0(η)+∑m=1∞fm(η),θ(η)=θ0(η)+∑m=1∞θm(η).
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:(3.18)Lf[fm(η)-χmfm-1(η)]=hfPm(η),m≥1,Lθ[θm(η)-χmθm-1(η)]=hθQm(η),m≥1,
where(3.19)Pm(η)=fm-1′′′(η)-[Re]∑n=0m-1fn′(η)fm-1-n′(η)+[Re]∑n=0m-1fn(η)fm-1-n′′(η)+ε(1-χm),Qm(η)=θm-1′′(η)-[Br]∑n=0m-1fn′′(η)fm-1-n′′(η)+[Pe]∑n=0m-1fn(η)θm-1-n′(η)-2[Pe]∑n=0m-1fn′(η)θm-1-n(η),χm={0,m=1,1,m>1.
The solutions of  (3.18)  are  referred  as the mth deformations, m≥1, and these deformations may satisfy the following conditions:(3.20)∂fm(η)∂η∣η=0-(Kn)∂2fm(η)∂η2∣η=0=0,fm(0)=0,∂2fm(η)∂η2∣η=1=0,θm(0)-(Kn)∂θm(η)∂η∣η=0=0(forCWT),θm(0)=0(forCHF),∂θm(η)∂η∣η=1=0.
The solutions of (3.18) can be obtained by a symbolic software such as Mathematica, and these solutions can be taken as(3.21)fm(η)=∑n=12m+1∑k=12m+1am,nkηkenη,θm(η)=∑n=02m+1∑k=02m+1bm,nkηkenηfor(CWT)case,θm(η)=∑n=12m+1∑k=12m+1cm,nkηkenηfor(CHF)case.
For a best  choice  of  hfandhθ, the Mth  order  approximations  for  the solutions  of  (2.12)–(2.13)  are  then(3.22)f(η)=f0(η)+∑m=1M∑n=12m+1∑k=12m+1am,nkηkenη,θ(η)=θ0(η)+∑m=1M∑n=02m+1∑k=02m+1bm,nkηkenηfor(CWT)case,θ(η)=θ0(η)+∑m=1M∑n=12m+1∑k=12m+1cm,nkηkenηfor(CHF)case.
The best values of parameters hfandhθ 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  hf and hθ such that  hf=hθ=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
Theh-curve for f′′(0) at Kn=0.05, Br=5, Pe=0.5, and Re=10.Figure 3
Theh-curve for θ′′(0) for CHF case at Kn=0.05, Br=5, Pe=0.5, and Re=10.Figure 4
Theh-curve for θ′′(0) for CWT case at Kn=0.05, Br=5, Pe=0.5, and Re=10.Also, it is noted thatKn=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.Figure5 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 ofKn=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, thatPo=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 atKn=0 for CWT case.Figure 8
Effects on Nu with Br atKn=0.05 (thin line) and 0.1 (solid line) for CWT case.Figure 9
Effects on Nu with Br atKn=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.



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