Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary

ICHMT-03177; No of Pages 12 International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

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Yan Zhao a, Lin Chen a, Xin-Rong Zhang a,b,⁎ a

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a r t i c l e

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Available online xxxx

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Keywords: Fluid dynamics Navier–Stokes equations Traveling wave method Heat transfer Laminar flow

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Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China Beijing Key Laboratory for Solid Waste Utilization and Management, Peking University, Beijing 100871, China

a b s t r a c t

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Analytical solutions play important roles in the understanding of fluid dynamics and heat transfer related problems. Some analytical solutions for incompressible steady/unsteady 2-D problems have been obtained in literature, but only a few of those are found under heat transfer conditions (which brings more complexities into the problem). This paper is focused on the analytical solutions to the basic problem of incompressible unsteady 2-D laminar flows with heat transfer. By using the traveling wave method, fluid dynamic governing equations are developed based on classical Navier–Stokes equations and can be reduced to ordinary differential equations, which provide reliable explanations to the 2-D fluid flows. In this study, a set of analytical solutions to incompressible unsteady 2-D laminar flows with heat transfer are obtained. The results show that both the velocity field and the temperature field take an exponential function form, or a polynomial function form, when traveling wave kind solution is assumed and compared in such fluid flow systems. In addition to heat transfer problem, the effects of boundary input parameters and their categorization and generalization of field forming or field evolutions are also obtained in this study. The current results are also compared with the results of Cai et al. (R. X. Cai, N. Zhang. International Journal of Heat and Mass Transfer, 2002, 45: 2623-2627) and others using different methods. It is found that the current method can cover the results and will also extend the fluid dynamic model into a much wider parameter ranges (and flow situations). © 2015 Published by Elsevier Ltd.

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Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary☆

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1. Introduction

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Analytical solutions have always played important roles in the development of various fluid flow systems, which usually serve as fundamental basis for comparison of fluid dynamic nature and system evolution trends. For example, the analytical solutions of incompressible flow and constant coefficient heat conduction in early days have been the bases of fluid dynamics and heat transfer [1,2]. However, it is difficult to derive the analytical solutions of the governing equations with nonlinear terms, especially the complex governing equations (for fluid systems usually the Navier–Stokes equations are considered) with given initial and boundary conditions. Although with the rapid development of computers and numerical methods, much research has focused on formulating efficient numerical methods to solve fluid dynamics and coupled heat transfer problems, the accuracy of these numerical solutions can only be ascertained by comparison with exact solutions or empirical/half-empirical correlations

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☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author at: Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing 100871, China. E-mail addresses: [email protected] (Y. Zhao), [email protected] (L. Chen), [email protected] (X.-R. Zhang).

from experiments. Therefore, it is meaningful to find out some analytical solutions not only for the reason that they are found be able to describe the detailed behavior of the concerning system, but also that they can be used as benchmark solutions to check the accuracy, convergence and effectiveness of various numerical methods and solutions, and to improve various numerical methods such as their differencing schemes and grid generation skills [3–6]. For fluid dynamic systems, the well-known Navier–Stokes (N–S) equations, first introduced by Navier in 1821, and developed by Stokes in 1845, are the fundamental governing equations. For those two hundred years, many groups have tried to solve this problem. The work on the exact solutions of the Navier–Stokes equations has also accumulated in literature. However, due to the nonlinearity and complexity of Navier–Stokes equations, one can only give the solutions to very limited/simplified cases. Indeed there only exist a small number of exact solutions in literature. In the paper of Wang [7,8], one can find the historical reviews of the trials and solutions up to year 1991. And in the most recent years, super computers have made it possible to numerically solve the Navier–Stokes equations and the accuracy of the results can be compared with an exact solution. Thus, the exact solutions are very important as a test to verify numerical or empirical methods for complex flow problems. In the recent twenty or thirty years, major developments of the exact solutions of the Navier–Stokes equations can be

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006 0735-1933/© 2015 Published by Elsevier Ltd.

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

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Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

α υ Cp ρ g θ p u,w t u0 p0 x z ξ a,b c mi, Mj

2. Basic model description and governing equations

157

2.1. Physical model and basic Navier–Stokes equations

158

In this paper, the general governing equations of the incompressible unsteady 2-D laminar flow with heat transfer is considered. The governing equations consisted of continuity equation, incompressible fluid Navier–Stokes equation and thermal conservation equation, which are written as follows (neglecting radiation and internal heat source; z-direction is opposite to gravitation) [20]:

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thermal diffusivity, [m2/s] kinematic viscosity, [m2/s] specific heat, [J/(kg·K)] density, [kg/m3] gravitational acceleration, [m/s2] temperature, [K] pressure, [Pa] velocity components in x, z directions, [m/s] time, [s] initial velocity, [m/s] initial pressure, [Pa] abscissa, [m] third coordinate, [m] transformation ξ = ax + bz + ct, [m] constants, a2 + b2 ≠ 0 wave speed, [m/s] constants used during the integration of equations

117 118

74

found in Chandna and Oku-Ukpong [9], Profilo et al. [10], Siddique [11], Venkatalaxmi et al. [12], Warsi [13], Naeem and Jamil [14], Naeem and Younus [15] and others [16]. 77 The major methods used in solving the Navier–Stokes equations are 78 variable separation and equation transformation methods; the equation 79 transformation method then includes potential function assumption 80 Q13 and parameter transformation. The basic process is to change/separate 81 the equations to reduce the problem to the class of nonlinear ordinary 82 differential equations for the first step [17]. Indeed, variable separation 83 Q14 has been widely used in solving complex equations. Potential functions 84 Q15 are also often used in similar methods [16,18]. This method has to 85 assume the independence relations between the variables, making it 86 difficult to justify the fundamentals of the solving process and the re87 sults obtained. Besides the results mentioned in review paper of Wang 88 [7,8], in recent years representative ones can be found by many groups. 89 For example, Al-Mdallal [19] utilized the canonical transformation with 90 complex coefficients; the Navier–Stokes equations were reduced to a 91 linear partial differential equations that can be solved by using separa92 tion of variables. According to the type of the canonical transformation 93 constants (real or complex), Al-Mdallal [19] got different types of 94 exact solutions. By using variable separation method with addition, 95 Cai and Zhang. [20] obtained some exact solutions to incompressible 96 unsteady 2-D laminar flow with heat transfer, neglecting gravity, 97 radiation and internal heat source. By this method, Cai discussed a series 98 of problems described by Navier–Stokes equations and found a set of 99 useful solutions to some basic flow situations [21,22]. 100 Variable or equation transformation methods are also generally seen 101 in literatures. For example, Nugroho et al. [17,23] proposed a potential 102 function and transformed coordinate to alter the three-dimensional 103 incompressible Navier–Stokes equations into simpler forms. Further104 more, a special class of solutions to the three-dimensional incompress105 ible Navier–Stokes equations was obtained by dropping the pressure 106 gradient and it was found that constant pressure gradient will produce 107 similar solutions to that of a zero pressure gradient. The authors also 108 proposed another potential function and transformed coordinate, 109 which has a nontrivial relation with respect to time, and a general 110 functional form of static pressure was applied. Fang et al. [24] have 111 investigated the steady momentum and heat transfer of a viscous fluid 112 flow over a stretching/shrinking sheet, and have presented new exact 113 solutions for the Navier–Stokes equations. These solutions provide a 114 more general formulation including the linear stretching and shrinking 115 wall problems as well as the asymptotic suction velocity profiles over a 116 variety of situations.

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B:2 B:3 B:4 B:5 B:6 B:7 B:8 B:9 B:10 B:11 B:12 B:13 B:14 B:15 B:16 B:17 B:18 B:19

As discussed, one case was the transformation of the Navier–Stokes equations to the Schrödinger equation, performed by application of the Riccati equation [25] and to achieve much simpler forms. This has good prospects since the Schrödinger equation is linear and has well defined solutions. The method of Lie group theory was also applied in order to transform the original partial differential equations into ordinary differential systems [26]. The same route was followed by Meleshko [27] and by Thailert [28], in transforming the Navier–Stokes equations to solvable linear systems. The current study is one continued trial of solving channel flow with heat transfer by using the transformation method. The current study is focused on the transforming of partial differential equations into tractable ordinary differential equations or some particular partial differential equations. Traveling wave method belongs to this family; due to the application of the transform ξ = ∑ai xi, partial differential equations can be reduced to tractable ordinary differential equations, where ξ is a variable of ordinary differential equations and it is called a phase of the wave, x1, …, xn are independent variables of the partial differential equations, and a1, …, an are arbitrary constants. In this paper, transformation method and traveling wave solution are used to solve the incompressible unsteady 2-D laminar flow with boundary heat transfer problem. The current study takes the simplified 2-D laminar flow conditions, where the instability shear flow with wave transportation happens. Such phenomena indeed are fundamental and critical during the formation of shear flow and the establishment of viscous and thermal boundary layers. The basic mathematical model are carefully established and tested with several general cases. It is found that the current model is capable of covering such benchmark cases and can provide new information on the parameter evolutions of Navier–Stokes governed fluid dynamic systems. The following parts of this paper are arranged as follows: In Section 2, the government equations of the incompressible unsteady 2-D laminar flow with heat transfer are presented. In Section 3, two kinds of traveling wave solutions are obtained under various conditions. For the first one, the velocity and temperature fields mainly depend on exponential functions, while for the second one, they depend on polynomial functions. In Section 4, the benchmark case results are analyzed and explained in detail. Further, the physical descriptions of these solutions are explained and some solutions with certain conditions are shown. A short conclusion is given in Section 5.

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Nomenclature

D

B:1

E

2

∂u ∂w þ ¼0 ∂x ∂z

ð1Þ

! 2 2 ∂u ∂u ∂u 1 ∂p ∂ u ∂ u þ þu þw ¼− þυ ρ ∂x ∂t ∂x ∂z ∂x2 ∂z2 ! 2 2 ∂w ∂w ∂w 1 ∂p ∂ w ∂ w −g þ þu þw ¼− þυ ρ ∂z ∂t ∂x ∂z ∂x2 ∂z2

119 120 121 122 123 124 125 126 127 128 129 130 Q16 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 Q17 153 Q18 154 155 156

160 161 162 163 164

166 167

ð2Þ 169 170

ð3Þ

!

" 2  2  2 # 2 2 ∂θ ∂θ ∂θ ∂ θ ∂ θ υ ∂u ∂w ∂u ∂w þ þu þw ¼α þ þ þ 2 þ 2 Cp ∂t ∂x ∂z ∂x2 ∂z2 ∂z ∂x ∂x ∂z

ð4Þ

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

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Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

175

the basic steps of solving its traveling wave solutions are listed as 202 follows.

where u and w are velocity components in x, and z directions, respectively, the velocities u, and w, the fluid pressure p, and the temperature 176 θ are functions of spatial variables x, and z and time variable t, the den177 sity ρ, the thermal diffusivity α, the kinematic viscosity υ, and the specif178 Q19 ic heat Cp are constants, and g is the acceleration of gravity. It is easily 179 seen that to solve Eqs. (1)–(4), Eqs. (1)–(3) should be solved first to ob180 tain u, and w, and then Eq. (4) is solved to obtain θ. In the following sec181 tions, the traveling waves of Eqs. (1)–(4) will be solved, and the physical 182 descriptions of the solutions to Eqs. (1)–(4) are explained. 183

Step 1: Substituting v(x1, …, xn, t) = V(ξ), where ξ = ∑ai xi + ct, and a1, …, an are arbitrary constants, and c is the wave speed, into Eq. (5), then it can be transformed into the following ordinary differential equation:   Q V; V 0 ; V ″ ; … ¼ 0

2.2. Traveling wave method and its basic procedures

184

2.2.1. Traveling wave method and the transformation of equations As has been discussed in the Introduction part, traveling wave method is utilized in many fluid dynamic systems. Indeed, in fluid systems, 187 traveling wave method can well capture various types of hydrodynamic 188 and thermal instabilities [29,30]. In the early developments of traveling 189 wave phenomena and traveling wave method, Nagata [31] found the 190 first traveling wave in plane Couette flow. With the discovery of travel191 ing waves in channel flows [29], and traveling waves in pipes [29,30], 192 parallel theoretical advances have been made in channel and pipe 193 flows. Such solutions can simplify the Navier–Stokes equation in the 194 model development and help transform the equations into ordinary dif195 ferential ones. A more detailed review can be found in ref [32] and it has 196 been proved to be a well-established method to treat fluid flow systems. 197 In detail, traveling wave solutions is a kind of solutions, whose shape 198 and speed keep unchanged when parallel shifting on ξ-axis. Here, ξ is 199 Q20 a linear form of the independent variables xi (i = 1, …, n), and t. 200 Q21 Considering a general nonlinear partial differential equation

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vðx1 ; x2 ; …xn ; t Þ ¼ X 1 ðx1 Þ þ X 2 ðx2 Þ þ … þ X n ðxn Þ þ T ðt Þ

C

E

R R

ð8Þ

2

232 233 235 236 238 239

Substituting Eq. (8) into Eqs. (1)–(4) yields au0 þ bw0 ¼ 0

ð9Þ

  a 2 ðc þ au þ bwÞu0 ¼ − p0 þ υ a2 þ b u″ ρ

ð10Þ

  b 2 ðc þ au þ bwÞw0 ¼ − p0 þ υ a2 þ b w″ −g ρ

ð11Þ

  i υ h 0 2 2 2 2 ðc þ au þ bwÞθ0 ¼ α a2 þ b θ″ þ ðbu þ aw0 Þ þ 2ðau0 Þ þ 2ðbw0 Þ Cp

ð12Þ

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230 Q29

N C O

229 Q28 where ξ = ax + bz + ct is the wave parameter, a and b are arbitrary constants, and c is the constant phase velocity, a + b ≠ 0.

3.1. When b ≠ 0

Q31241 Q30 where the denotation ‘′’ denotes d / dξ.

Integrating Eq. (9) once with respect to ξ leads to au þ bw ¼ m0

210 211 212 213 Q24 214

216 217 218 219 Q25 220 221 Q26 222

224 Q27

then convert Eq. (1) to n + 1 ordinary differential equations.

2

209

ð7Þ

The governing Eqs. (1)–(4) are assumed to have traveling wave solutions of the form: 8 uðx; z; t Þ ¼ uðξÞ > > < wðx; z; t Þ ¼ wðξÞ pðx; z; t Þ ¼ pðξÞ > > : θðx; z; t Þ ¼ θðξÞ

206 Q23

2.2.2. Basic characteristics of traveling wave method From Step 1, it can be seen that in the process of seeking traveling wave solutions of Eq. (1), the spatial variable x1, …, xn and time variable t are transformed into a single variable ξ through the linear transformation ξ = ∑ ai xi + ct. Therefore, the original problem is converted to solving an ordinary differential equation. The method of separating variables with addition is the following, i.e.,

225 226 Q82 3. Mathematical developments of traveling wave solutions and explanations 227

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T

ð5Þ

204 Q22

where the prime denotes the derivative with respect to ξ. Step 2: If Eq. (6) is a second order ordinary differential equation with constant coefficients, then it can be solved directly. Otherwise, other method is sought for solving this equation. As far as we know, many effective methods have been presented for solving this equation, such as the inverse scattering method, homogeneous balance method, (G′ / G)-expansion method, F-expansion method, undetermined coefficients method and others [33,34].

E

 P v; vt ; vx1 ; …vxn ; vtt ; vtx1 ; …; vtxn ; vx1 x1 ; …; vxn xn ; … ¼ 0

203

ð6Þ

F

185 186



3

ð13Þ

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

4

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

243 Q32 where m0 is an arbitrary constant. Furthermore, by using Eq. (13), Eqs. (10)–(12) are reduced to

244 245 247 248

  a 2 ðc þ m0 Þu0 ¼ − p0 þ υ a2 þ b u″ ρ

ð14Þ

  b 2 ðc þ m0 Þw0 ¼ − p0 þ υ a2 þ b w″ −g ρ

ð15Þ

 2 2   υ a2 þ b 2 2 ″ 2 ðc þ m0 Þθ ¼ α a þ b θ þ u0 : 2 Cpb 0

ð16Þ

250

In the following, Eqs. (1)–(4) are then transformed and solved. Case I. c + m0 ≠ 0

252 Q34

In this case, by eliminating p from Eqs. (14) and (15), a second order nonhomogeneous ordinary differential equation

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251 Q33

R O

is obtained. This equation can be solved easily and its solution is uðξÞ ¼ m1 þ m2 eλ1 ξ þ

abg  ξ 2 ðc þ m0 Þ a2 þ b

ð17Þ

ð18Þ

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254

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 2   2 2 υ a2 þ b u″ −ðc þ m0 Þ a2 þ b u0 þ abg ¼ 0

where λ1 = (c + m0) / (υ(a2 + b2)), and m1 and m2 are arbitrary constants. Furthermore, from Eqs. (13) and (18), it is deduced that ð19Þ

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D

2 3 m0 a 4 abg λ1 ξ   ξ5: m1 þ m2 e þ − wðξÞ ¼ 2 b b ðc þ m0 Þ a2 þ b 256

2

ξ þ m3

ð20Þ

where m3 is an arbitrary constant. To obtain the temperature θ, the following second order ordinary differential equation 0

θ −λ2 θ þ

1

  2 αC p a2 þ b

" m22 ðc þ m0 Þ2 2

b υ

2λ1 ξ

e

# 2agm2 λ1 ξ a2 g 2 υ e þ þ ¼0 b ðc þ m0 Þ2

E

″

ð21Þ

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258

bρg a2 þ b

C

pðξÞ ¼ −

T

Combining Eq. (18) with Eq. (14), and integrating once with respect to ξ, it is easily obtained that

R

is solved, which is derived from Eqs. (16) and (18). Here, λ2 = (c + m0) / (α(a2 + b2)). The solution to Eq. (21) is expressed as follows

ð22Þ

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8 2λ1 ξ 2λ1 ξ > eλ1 ξ þ M4 ξ þ m5 ; υ ¼ 2α > < M1 ξeλ ξ þ M 2 eλ ξ þ M32λ 1 1 1ξ M ξe þ M e þ M e þ M4 ξ þ m5 ; υ ¼ α 5 7 6 θðξÞ ¼ m > 4 2λ ξ λ ξ λ ξ > M8 e 1 þ M9 e 1 þ e 2 þ M 4 ξ þ m5 ; other : λ2

where M1 = −m22(c + m0) / (2αCpb2), M2 = (a2 + b2)(m22 / (2Cpb2) + m4α / (c + m0)), M3 = 8agm2α(a2 + b2) / (Cpb(c + m0)2), M4 = a2g2υ / (Cp(c + m0)3), M5 = −2agm2 / (Cpb(c + m0)), M6 = (2agm2 / (Cpb(c + m0)) + m4) / λ2, M7 = −m22(a2 + b2) / (2Cpb2), M8 = −m22υ(a2 + b2) / 2 2 2 2 2 261 Q35 (2Cpb (2α − υ)), M9 = −2agm2υ (a + b ) / (Cpb(α − υ)(c + m0) ), mi (i = 4,5) are arbitrary constants. 262 Hence, the velocity components, pressure and temperature in the original variables are

264 265

267

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uðx; z; t Þ ¼ m1 þ m2 eλ1 ðaxþbzþct Þ þ

2 3 m0 a 4 abg λ1 ðaxþbzþct Þ   ðax þ bz þ ct Þ5 m1 þ m2 e − wðx; z; t Þ ¼ þ 2 b b ðc þ m0 Þ a2 þ b

268

pðx; z; t Þ ¼ − 270 271

273

abg   ðax þ bz þ ct Þ 2 ðc þ m0 Þ a2 þ b

bρg 2

a2 þ b

ðax þ bz þ ct Þ þ m3

8 2λ1 ðaxþbzþct Þ 2λ1 ðaxþbzþct Þ > eλ1 ðaxþbzþct Þ þ M 4 ðax þ bz þ ct Þ þ m5 ; υ ¼ 2α > < M1 ðax þ bz þ ct Þeλ ðaxþbzþct Þ þ M 2 eλ ðaxþbzþct Þ þ M32λ 1 1 þ M6 e þ M 7 e 1 ðaxþbzþct Þ þ M4 ðax þ bz þ ct Þ þ m5 ; υ ¼ α θðx; z; t Þ ¼ M5 ðax þ bz þ ct Þe m4 λ2 ðaxþbzþctÞ > 2λ1 ðaxþbzþct Þ λ1 ðaxþbzþct Þ > : M8 e þ M9 e þ e þ M4 ðax þ bz þ ct Þ þ m5 ; other: λ2

ð23Þ

ð24Þ

ð25Þ

ð26Þ

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

274 275 276

5

Eqs. (23)–(24) indicate that the velocity components u and w in x and y directions change in exponential form. Eq. (25) indicates that the pressure p changes in linear form. The values of kinematic viscosity υ and thermal diffusivity α determine the change of temperature θ. When υ is twice as large as α, the temperature is changed based on the first formula of Eq. (26). When υ is equal to α, the temperature is changed based on the second formula of Eq. (26). In other case, the temperature is changed based on the third formula of Eq. (26).

277

Case II. c + m0 = 0

278

In this case, Eqs. (14)–(16) can be simplified as

  a 0 2 p ¼ υ a2 þ b u″ 280 Q36 ρ

″

αθ þ

  2 υ a2 þ b 2

Cp b

ð28Þ

F

283 284

  b 0 2 p ¼ υ a2 þ b w″ −g ρ

2

u0 ¼ 0:

O

281

ð27Þ

   2  υ a2 þ b bu″ −aw″ þ ag ¼ 0: 288

Utilizing Eq. (9), and integrating twice with respect to ξ, it is obtained that abg   ξ2 þ m6 ξ þ m7 2 2υ a2 þ b

ð30Þ

ð31Þ

D

uðξÞ ¼ −

P

Eliminating p from Eqs. (27) and (28) yields

R O

286

ð29Þ

Q38290 Q37 where m6, and m7 are arbitrary constants. Furthermore, in Eq. (31) together with Eqs. (13), (27) and (29), the pressure p(ξ) with the same formula as

293

4

3

T

2 3 m0 a 4 abg 2  ξ þ m6 ξ þ m7 5 −  − wðξÞ ¼ 2 b b 2υ a2 þ b

C

292

E

Eq. (20) is obtained,

2

ð33Þ

E

θðξÞ ¼ M10 ξ þ M11 ξ þ M 12 ξ þ m8 ξ þ m9

ð32Þ

respectively, where M10 = −a 2g 2 / (12Cp α υ(a2 + b2)3), M11 = agm6 / (3αCpb(a2 + b2)), M12 = −υm62(a2 + b2) / (2αCpb2), and mi (i = 8,9) are arbitrary constants. 296 Q39 Consequently, the solutions to Eqs. (1)–(4) are Eq. (25),

298 299

301 302

abg   ðax þ bz þ ct Þ2 þ m6 ðax þ bz þ ct Þ þ m7 2 2υ a2 þ b

N C O

uðx; z; t Þ ¼ −

R

R

295

ð34Þ

2 3 m0 a 4 abg 2  ðax þ bz þ ct Þ þ m6 ðax þ bz þ ct Þ þ m7 5 −  − wðx; z; t Þ ¼ 2 b b 2υ a2 þ b 4

3

2

θðx; z; t Þ ¼ M 10 ðax þ bz þ ct Þ þ M11 ðax þ bz þ ct Þ þ M 12 ðax þ bz þ ct Þ þ m8 ðax þ bz þ ct Þ þ m9 : 304

ð35Þ

ð36Þ

U

Q40 Eq. (34) indicates that when ab N 0, u becomes faster with the increase of υ; when ab b 0, u becomes slower with the increase of υ. Eq. (35) inQ42305 Q41 dicates that w becomes faster with the decrease of υ. Eq. (36) indicates that the θ becomes higher with the decrease of the Prandtl number Pr = υ / α.

306

3.2. When b = 0

307 308

Similar to the procedure mentioned above, it is easy to obtain the solutions to Eqs. (1)–(4) and they are summarized as follows. When c + au0 ≠ 0, the velocity components, pressure and temperature are

310

uðx; t Þ ¼ u0

311 313 314 316

wðx; t Þ ¼

m10 λ3 ðaxþctÞ g e − ðax þ ct Þ þ m11 c þ au0 λ3

pðx; t Þ ¼ p0

ð37Þ ð38Þ ð39Þ

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

6

317 Q43

319

320

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

8 < M 13 ðax þ ct Þe2λ3 ðaxþctÞ þ M 14 e2λ3 ðaxþct Þ þ M15 eλ3 ðaxþct Þ þ M16 ðax þ ct Þ þ m13 ; υ ¼ 2α θðx; t Þ ¼ M 17 ðax þ ct Þeλ3 ðaxþct Þ þ M 18 e2λ3 ðaxþctÞ þ M19 eλ3 ðaxþct Þ þ M16 ðax þ ct Þ þ m13 ; υ ¼ α : M 20 e2λ3 ðaxþct Þ þ M21 eλ3 ðaxþct Þ þ M22 eλ4 ðaxþct Þ þ M16 ðax þ ct Þ þ m13 ; other

ð40Þ

respectively, whereas when c + au0 = 0, they are Eqs. (37) and (39), wðx; t Þ ¼

g ðax þ ct Þ2 þ m14 ðax þ ct Þ þ m15 2υa2

321

θðx; t Þ ¼ −

ð41Þ

 m214 υ g2 m14 g 4 3 2 ð ax þ ct Þ ð ax þ ct Þ þ ð ax þ ct Þ þ þ m ð ax þ ct Þ þ m17 16 C p α 12υ2 a4 2 3υa2

ð42Þ

respectively. Here, u0 is initial velocity, p0 is the initial pressure, λ3 = (c + au0) / (υa2), λ4 = (c + au0) / (αa2), M13 = −m210 / (Cpλ3), M14 = (m210 / (Cpλ3) + m12) / (2λ3), M15 = − 4m10g / (Cpλ23(c + au0)), M16 = g2 / (Cpλ3(c + au0)2), M17 = 2m10g / (Cpλ3(c + au0)), M18 = −m210 / (2Cpλ23), 324 M19 = (m12 − 2m10g / (Cpλ3(c + au0))) / λ3, M20 = −m210υa2 / (2Cpλ23(2α − υ)), M21 = 2m10gυ / (Cpλ23(c + au0)(α − υ)), M22 = m14 / λ4, mi 325 Q44 (i = 10, …, 17) are arbitrary constants. 326 Q45 Eq. (38) indicates that the velocity component w in y directions changes in exponential form. The values of kinematic viscosity υ and thermal dif327 fusivity α determine the change of temperature θ. When υ is twice as large as α, the temperature is changed based on the first formula of Eq. (40). 328 When υ is equal to α, the temperature is changed based on the second formula of Eq. (40). In other cases, the temperature is changed based on the Q47329 Q46 third formula of Eq. (40). Eq. (41) indicates that w becomes slower with the increase of υ. Eq. (42) indicates that the θ becomes higher with the de330 crease of the Prandtl number Pr = υ / α.

R O

O

F

323

331

4.1. Basic results and comparisons

pðz; t Þ ¼ −

334 335

C

E

● a, and c in Eqs. (23)–(25), and (26) with υ ≠ 2α and υ ≠ α as well as C0 in Eqs. (16a) and (16c)–(16d), C4 in Eq. (16d), and T(t) in Eq. 347 Q50 (16c) are equal to zero, C1 = m0 / b, C5 = m4b / C1; 348 Q51 ● a, and c in Eqs. (23)–(25), and (26) with υ ≠ 2α and υ ≠ α as well as 349 Q52 C4 in Eqs. (18a) and (18c)–(18d), C5 and C6 in Eq. (18d), and T(t) in 350 Eq. (18c) are equal to zero, C1 = m0 / b, C8 = αm4b / C1.

R

R

345 Q49 346

!

ð46Þ 376

which are derived from Eqs. (23)–(26), and Eqs. (44) and (45) uðz; t Þ ¼ m6 ðbz þ ct Þ þ m7 θðz; t Þ ¼ −

υm26 2 ðbz þ ct Þ þ m8 ðbz þ ct Þ þ m9 2αC p

ð47Þ 377 378

ð48Þ

which are derived from Eq. (25) and Eqs. (34)–(36). When c = 0, the flow is steady. The physical features of Eqs. (43)–(48) can be described as follows. There are two infinite plates parallel to x abscissa moving along the abscissa direction with different speeds given by Eqs. (43) or (47). The velocity of fluid flow, u, is an exponential function as

Q53 Eqs. (23)–(25) and (26) with υ ≠ 2α and υ ≠ α are just Eqs. 352 353 (16a)–(16d) or Eqs. (18a)–(18d). When m6 in Eqs. (33)–(35), C1 in 354 355

362

4.2. Physical aspect of the problem

363 364

Next, the physical phenomenon of traveling wave solutions Eqs. (23)–(26) and Eqs. (34)–(36) is explained. The simplest case of Eqs. (23)–(26) and Eqs. (34)–(36) with a = 0 is studied. That is,

359 360

365

367 368

N

357 358

U

356

C

361

Eqs. (19a)–(19d), C4 in Eqs. (19a)–(19c), and T(t) in Eq. (19c) are equal to zero, b = −a, C2 = m7, C3 = m7 + m0 / b, C6 = am8, Eq. (25) and Eqs. (34)–(36) are just Eqs. (19a)–(19d). From Eq. (16d) and Eq. (18d) in Cai's result [20], it is obvious that the temperature is obtained on the assumption of υ ≠ 2α or υ ≠ α. However, in this paper, no matter what the relationship between υ and α is, the temperature can always be obtained, i.e. Eq. (26). Hence, some new solutions to Eqs. (1a)–(1d) are presented in this paper.

λ1 ðbzþct Þ

uðz; t Þ ¼ m1 þ m2 e wðz; t Þ ¼

m0 b

ð43Þ ð44Þ

373 374 Q54

υ ¼ 2α

O

351

ð45Þ

8 m22 m0 m22 m4 αb2 2λ1 ðbzþct Þ > > > ðbz þ ct Þe2λ1 ðbzþct Þ þ e þ þ m5 ; <− 2 2C m0 p 2αC p b θðz; t Þ ¼ 2 > m2 υ m4 λ2 ðbzþctÞ > > :− e þ m5 ; other e2λ1 ðbzþctÞ þ λ2 2C p ð2α−υÞ

T

The governing Eqs. (1a)–(1d) presented in ref [20] is a special case of the governing Eqs. (1)–(4) discussed in this paper. Setting g in 336 Q48 Eqs. (23)–(26), and Eqs. (34)–(42) to zero, and replacing z by y, the so337 lutions obtained here are the ones to Eqs. (1a)–(1d) in Cai's result [20]. 338 Next, the solutions obtained here are compared with those obtained in 339 Cai's result [20]. 340 By using the method of separating variables with addition, Cai and 341 Zhang. [20] obtained some exact solutions to Eqs. (1a)–(1d), for 342 example, Eqs. (16a)–(16d), (18a)–(18d), and (19a)–(19d). It is easily 343 seen that when the parameters satisfy one of the following conditions 344 (in comparison with Cai's result [20]):

ρg ðbz þ ct Þ þ m3 b

P

333

370 371

D

4. Results and discussion

E

332

Fig. 1. This figure describes the velocity of fluid flow in the case c = 0, m0b N 0, m2 N 0, m1 + m2 N 0, λ1b N 0. There are two infinite porous plates parallel to x abscissa moving along the abscissa direction with different speeds given by Eq. (43). The velocity component u (Eq. (43)) is an exponential function. The velocity component w (Eq. (44)) indicates that fluid is injected from the lower porous plate into the flow field between two porous plates with a uniform constant, and ejected to the upper porous plate with the some velocity.

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

380 381 382 383

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

F

Fig. 4. This figure describes the temperature θ (Eq. (46)) in two infinite porous plates parallel to x abscissa as c = 0, m0 ≠ 0, m4 = 0, υ N 2α, −m22υ / (2Cp(2α − υ)) + m5 N 0, Q1 λ1b N 0.

393 394 395 396 Q58

C

391 392

E

R O

P

389 Q56 390 Q57

400 401

cg þ bm0 ≠0 cg þ bm0 ¼ 0

Fig. 3. This figure describes the pressure p (Eq. (45)) in two infinite plates parallel to x abscissa as c = 0. The pressure is independent of m0, and varies along the z direction with linear relation. The dotted lines are isobaric lines.

409 410 411 412 413 414 415 416 Q61

ð49Þ

where m17 is an arbitrary constant. 418 For traveling wave solutions Eqs. (37)–(42), let m10 = m12 = 0, then Eqs. (37)–(40) can be rewritten as 419 ð50Þ 421

uðx; t Þ ¼ u0 wðx; t Þ ¼ −

g ðax þ ct Þ þ m11 c þ au0

ð52Þ 427 g 2 υa2

C p ðc þ au0 Þ3

428

ðax þ ct Þ þ m13

ð53Þ 430

while Eqs. (37), (39), (41), (42) become Eqs. (50) and (52), wðx; t Þ ¼

424 425

pðx; t Þ ¼ p0 θðx; t Þ ¼

422

ð51Þ

g ðax þ ct Þ2 þ m14 ðax þ ct Þ þ m15 2υa2

ð54Þ

U

N C O

R

R

4.3.1. Basic discussion on solutions and its implications The results shown in Eqs. (43)–(48) under certain parameter condi402 tions are shown in Figs. 1–5 and discussed in this section. For the un403 steady case, i.e., c ≠ 0, the physical descriptions of Eqs. (43)–(48) are 404 similar to those presented above. Here, only the difference is given. 405 The pressure changes along z direction with a constant pressure 406 Q59 gradient −ρg to maintain the acceleration. If it is assumed that there 407 are two porous plates parallel to x abscissa, Eqs. (43) and (47) mean 408 Q60 that both two porous plates and the fluid between the plates accelerate

8m g m2 g λ1 ðbzþct Þ 1 > zþ e þ m17 ; < m0 λ1 bm0 x¼ m b > : 6 z2 þ ðm6 ct þ m7 Þz þ m17 ; 2

D

4.3. Physical explanations of the fluid flow and heat transfer

387 388

E

399

385 Q55 386

in x direction with different accelerations. When m0 ≠ 0, the acceleration is a function of t or (t,z). When m0 = 0, the acceleration is constant. If it is assumed that there are two porous plates parallel to z abscissa, Eqs. (43) and (47) indicate that the fluid flow in and out the porous plates with acceleration m2λ1cexp(λ1(bz + ct)) as m0 ≠ 0, and m6c as m0 = 0. The expression of absolute coordinates streamlines in a fixed time can be obtained by integrating dz / dx = w / u, i.e.,

T

397 398

m0 ≠ 0, and a polynomial function as m0 = 0. When m0b N 0, the velocity component w = m0 / b N 0 indicates that fluid is injected from the lower porous plate into the flow field between two porous plates with a uniform constant, and ejected to the upper porous plate with the some velocity. When m0b b 0, the fluid flows in the opposite direction along z-axis. When m0 = 0, the plates are solid. It is assumed that m0b, m2, m1 + m2, and λ1b in Fig. 1 as well as m6b and m7 in Fig. 2 are positive, and m0 in Fig. 2 is equal to zero. The pressure is independent of m0, and varies along the z direction with linear relation. Fig. 3 shows its physical feature. When m0 ≠ 0, the temperature mainly depend on exponential function, and when m0 = 0, it is a quadratic polynomial. Figs. 4 and 5 illustrate the physical features of Eqs. (46) and (48) with m0 ≠ 0, m4 = 0, υ N 2α, −m22υ / (2Cp(2α − υ)) + m5 N 0, and λ1b N 0, and m0 = 0, m8 = 0, and m9 N 0, respectively. H in the following figures indicates the distance of two plates.

O

Fig. 2. This figure describes the velocity of fluid flow in the case c = 0, m0 = 0, m6b N 0, m7 N 0. There are two infinite solid plates parallel to x abscissa moving along the abscissa direction with different speeds given by Eq. (47). The velocity component u (Eq. (47)) is a linear function.

384

7

Fig. 5. This figure describes the temperature θ (Eq. (48)) in two infinite solid plates parallel to x abscissa as c = 0, m0 = 0, m8 = 0, m9 N 0.

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

431

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

θðx; t Þ ¼ −

ð55Þ 434

The physical phenomenon of Eqs. (50)–(55) can be explained as

w is a quadratic function and the temperature is fourth order polynomi- 448 al function. Similarly, the physical features of Eqs. (50)–(55) under 449 other parameters conditions can be obtained. 450 4.3.2. Effects of Reynolds number and Prandtl number In the previous section, the phenomenon of traveling wave solutions Eqs. (43) and (46) is explained. In this section, the effects of Reynolds number Re = m0H / (bυ) and Prandtl number Pr = υ / α will be studied. The velocity profiles are shown in Figs. 10 and 11 for given parameter conditions c, H, m1, m2, and υ at different Reynolds number at time t = 0.05. It is found that the velocity becomes stronger with the increase of Reynolds number at a certain time. In addition, under the same parameter conditions, the velocity with b b 0 is stronger than the one with b N 0 at a certain time. The temperature profiles for given parameter conditions c, H, m0, m2, m4, m5, Cp, and υ at different Prandtl number

T

435 Q62 follows. There is a fluid flow between two plates which is perpendicular

P

 m2 υ g2 m14 g ðax þ ct Þ4 þ ðax þ ct Þ3 þ 14 ðax þ ct Þ2 þ m16 ðax þ ct Þ þ m17 : 2 4 2 2 C p α 12υ a 3υa

Fig. 8. This figure describes the velocity of fluid flow in the case c = 0, u0 = 0, m14 = 0, m15 N 0. The infinite solid plates move along the z direction with different speeds given by Eq. (54). The velocity component w (Eq. (54)) is a quadratic function.

D

432

Fig. 6. This figure describes the velocity of fluid flow in the case c = 0, u0 N 0, m11 N 0, m11 − gL / u0 N 0. The infinite porous plates move along the z direction with different speeds given by Eq. (51). The velocity component u (Eq. (50)) indicates that the fluid is injected from the left porous plate into the flow field between two porous plates with a uniform constant, and ejected to the right porous plate with the some velocity. The velocity component w (Eq. (51)) is a linear function.

E

Q2

R O

O

F

8

436 437

451 452 453 454 455 456 457 458 459 460 461

U

N

C

O

R

R

E

C

to x-axis, and the infinite plates move along the z direction with different speeds which are given by Eqs. (51) or (54). The related results are 438 also plotted in Figs. 6–9. When c = 0, these plates make a constant 439 motion. When c ≠ 0, these plates make a variable motion with acceler2 440 Q63 ation −gc / (c + au0) or (ax + ct)gc / (υa ) + m14c (x = 0,L). The flow is 441 Q64 driven by the moving plates and has no need of pressure drop in the 442 flow direction. When u0 = 0, the plates are solid. When u0 N 0, the 443 fluid is injected from the left porous plate into the flow field between 444 two porous plates with a uniform constant, and ejected to the right po445 rous plate with the some velocity. When u0 b 0, the fluid flows in the op446 posite direction parallel to x-axis. When c + au0 ≠ 0, both the velocity 447 component w and temperature are linear functions. When c + au0 = 0,

Q3

Fig. 7. This figure describes the temperature θ (Eq. (53)) in two porous plates moving along the z direction as c = 0, u0 N 0, m13 N 0.

Fig. 9. This figure describes the temperature θ (Eq. (55)) in two solid plates moving along Q4 the z direction as c = 0, u0 = 0, m14 = 0, m16 = 0, m17 N 0.

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

Fig. 12. The graph describes the temperature θ (Eq. (46)) for given parameter conditions b = 106, c = −3, m0 = −1, m2 = 1.4, m4 = m5 = 0, Cp = 103, υ = 2 × 106, and H = 1 at different Prandtl number Pr = υ / α, that is α = 4 × 107, 4 × 106, 2 × 106, 106, 0.5 × 106 at Q7 time t = 0.05.

F

Fig. 10. The graph describes the velocity component u (Eq. (43)) for given parameter conditions b = 106, c = −3, m1 = 0.3, m2 = 1.4, υ = 2 × 106, and H = 1 at different Rayleigh number Re = m0H / (bυ), that is m0 = −4, −2, −1, −0,1, 0.1, 1, 2, 4 at time t = 0.05.

O

Q5

P

D

E

T

C

E

4.4. Further comments on the traveling wave solutions

480 481

488

In the current development of simplified Navier–Stokes equations (coupled with heat transfer equations), the transformation is made by traveling wave method, considering the incompressible 2-D flow under gravity. Compared with the results of Cai group [20–22] (in the absence of gravity) using parameter separation method, the current results show a different form and can cover those results shown in Cai's equations. As discussed formerly, new behaviors and situations are found in the fluid convective flow system in this paper. Indeed, in the transformation of Navier–Stokes equations, parameter separation,

Q6

Fig. 11. The graph describes the velocity component u (Eq. (43)) for given parameter conditions b = −106, c = −3, m1 = 0.3, m2 = 1.4, υ = 2 × 106, and H = 1 at different Rayleigh number Re = m0H / (bυ), that is m0 = −4, −2, −1, −0.1, 0.1, 1, 2, 4 at time t = 0.05.

489 490 491 492 493 494 495 496 497 498 499 500 Q66 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517

U

486 487

R

484 485

N C O

482 483

R

479

potential flow assumption and other parameter assumptions can be used and many groups have made trials through different ways [11, 12,16,17,19,23]. However, the results obtained are only applicable in a limited set of conditions. For more details, for example, the solutions obtained by Al-Mdallal [19] involve either exp, sin, cos, sinh or cosh under certain conditions depending on the type of the constants in the canonical transformation. Comparing the solutions only in the exp form with the solutions shown in Eqs. (15)–(16), it is found that under some certain conditions, they are the same. The solutions shown in Eqs. (27)–(28) are polynomial functions, which are not involved in Al-Mdallal's work [19]. Fang et al. [24], Wu et al. [35] and others also presented some solutions in different forms (but with the same physical meaning) from the ones obtained in this paper as well as in Cai's work [20–22]. Further, the development of 3-D flow description, compressible flows, or under more general situations of fluid flow (or coupled heat transfer conditions) should be encouraged by many groups. It is also hoped that more studies will be conducted in the future to reveal the complex but interesting problems in this field. Fang [36] considered an incompressible viscous Couette flow problem, in which the bottom wall is fixed and subjected to a mass injection velocity vw, and there is a mass suction velocity vw at the top wall. The top plate is stationary when t b 0, there is only mass transfer in the transverse direction, say y-direction. At t = 0, the top wall is started impulsively to a constant velocity U0. Furthermore, Fang [37] considered an incompressible pressure-driven flow in a channel, in which the bottom wall is subjected to a mass injection velocity vw and a mass suction velocity vw at the top wall. The fluid is stationary along the plate direction when t b 0, there is only mass transfer in the transverse direction, namely y-direction. At t = 0, a pressure gradient along the plate

R O

462 463

at time t = 0.05 are plotted in Figs. 12 and 13. It is seen that there is a critical Prandtl number Pr corresponding to υ = 2α. The profiles of the 464 temperature corresponding to the Prandtl number, which is smaller 465 than the critical Prandtl number, lie on the left hand side of the profile 466 of the temperature corresponding to the critical Prandtl number. The 467 profiles of the temperature corresponding to the Prandtl number, 468 which is larger than the critical Prandtl number, lie on the right hand 469 side of the profile of the temperature corresponding to the critical 470 Prandtl number. No matter what the Prandtl number is smaller/larger 471 Q65 the critical Prandtl number, the temperature becomes higher with the 472 decrease of Prandtl number, and it is close to the temperature 473 corresponding to the critical Prandtl number. When b N 0, the tempera474 ture is increasing along the positive direction of z-axis. When b b 0, the 475 temperature is decreasing along the positive direction of z-axis. For 476 other traveling wave solutions with different parameter conditions, 477 the effects of Reynolds number Re = m0H / (bυ) and Prandtl number 478 Pr = υ / α can be studied similarly.

9

Fig. 13. The graph describes the temperature θ (Eq. (46)) for given parameter conditions b = − 106, c = −3, m0 = − 1, m2 = 1.4, m4 = m5 = 0, Cp = 103, υ = 2 × 106, and H = 1 at different Prandtl number Pr = υ / α, that is α = 4 × 107, 4 × 106, 2 × 106, 106, 0.5 × 106 at time t = 0.05. Q8

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

541 542 543 544 545 546 547 548 549 Q70 550 551 Q71

F

O

R O

539 540 Q69

P

537 538

D

535 536

558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577

578

Acknowledgments

The support of National Natural Science Foundation of China 579 Q72 (No. 51276001) and the Common Development Fund of Beijing are 580 Q73 gratefully acknowledged. 581

T

533 534

C

531 532

557

E

529 530

555 556

R

527 528

553 554

Appendix A. Developments on the initial and boundary conditions In Section 3, some traveling wave solutions are obtained for Eqs. (1)–(4) without initial and boundary conditions. Indeed, the initial and boundary conditions can be derived from these solutions. For example, it is assumed that the governing Eqs. (1)–(4) are considered in horizontal parallel plates x ∈ R, z ∈ [0,H]. Substituting t = 0, z = 0 and z = H into Eqs. (23)–(26) and Eqs. (34)–(42), then some initial and boundary conditions are obtained. Q74 Q75 ● Eqs. (23)–(26) satisfy the following initial conditions

R

525 526

This paper presents the analytical solutions to the basic problem of incompressible unsteady two-dimensional (2-D) laminar flows with boundary heat transfer. By using the traveling wave method, fluid dynamic governing equations are developed based on classical Navier–Stokes equations and they can be reduced to ordinary differential equations, which provide reliable explanations to the 2-D fluid flows. In this study, a set of analytical solutions to incompressible unsteady 2-D laminar flows with heat transfer are obtained. The current results are also compared with other groups using different methods. It is found that the current method can cover the results and will also extend the fluid dynamic model into a much wider parameter ranges (and flow situations). The results show that both the velocity field and temperature field take an exponential function form, or a polynomial function form, when traveling wave kind solution is assumed and compared in such fluid flow systems. In detail, the expression of the temperature field will be dependent on the relationship between thermal diffusivity α and the kinematic viscosity υ of the system, which decides the traveling wave system evolution as well as the heat flow inside the boundary layer. It should be noted that by the current method the temperature field can always be obtained whatever kind of α and υ relationship is assumed, making it capable of extended heat transfer boundary analysis. In addition to heat transfer problem, the effects of boundary input parameters and their categorization and generalization of field forming or field evolutions are also obtained in this study.

uðx; z; 0Þ ¼ m1 þ m2 eλ1 ðaxþbzÞ þ

O

524

552

abg   ðax þ bzÞ 2 ðc þ m0 Þ a2 þ b

C

522 Q68 523

5. Concluding remarks

2 3 m0 a 4 abg λ1 ðaxþbzÞ   m1 þ m2 e − ðax þ bzÞ5 þ wðx; z; 0Þ ¼ 2 b b ðc þ m0 Þ a2 þ b

N

520 521 Q67

direction is applied impulsively to the fluid. By solving two separate problems: the steady state and transient problems, an analytical solution of series form for the velocity U in dimensionless was obtained respectively in refs [36,37], where U = u / U0. It is found that Eqs. (43) and (47) in this paper are also solutions to Eq. (1) in refs [36,37], because Eq. (45) in this paper is a function of z. But Eqs. (43) and (47) satisfy different boundary conditions and initial condition. Eq. (43) satisfies the boundary conditions u(0,t) = m1 + m2exp(λ1ct) and u(h,t) = m1 + m2exp(λ1(bh + ct)), and the initial condition u(z,0) = m1 + m2exp(λ1bz), which means that at t = 0, the top wall is started impulsively with the velocity m1 + m2exp(λ1bh) and the bottom wall is started impulsively with a constant velocity m1 + m2. Eq. (47) satisfies the boundary conditions u(0,t) = m6ct + m7 and u(h,t) = m6(bh + ct) + m7, and the initial condition u(z,0) = m6bz + m7, which means that at t = 0, the top wall is started impulsively with a constant velocity m6bh + m7 and the bottom wall is started impulsively with a constant velocity m7. The velocity obtained in ref [36] becomes weaker with the increase of Reynolds number. In ref [37], there is a series of critical values of y with respect to time t. At the same time, when the fluid is lower than some critical value, the flow velocity becomes weaker as Reynolds number increases, and when the fluid is higher than the same critical value, the flow velocity becomes stronger as Reynolds number increases. For the solutions obtained in this paper, the velocity becomes stronger with the increase of Reynolds number. For energy equation, Fang [36,37] studied a steady state energy equation and obtained temperature distributions for different Prandtl numbers. In this paper, temperature distributions for different Prandtl numbers to a transient energy equation are presented. The temperature obtained in refs [36,37] becomes higher with the decrease of Prandtl number. In this paper, there exists a critical Prandtl number. When the Prandtl number is less than this critical value, the temperature becomes higher as Prandtl number decreases. When the Prandtl number is more than this critical value, the temperature becomes higher as Prandtl number increases.

pðx; z; 0Þ ¼ −

U

518 519

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

E

10

bρg

2

a2 þ b

ðax þ bzÞ þ m3

8 2λ1 ðaxþbzÞ 2λ1 ðaxþbzÞ > eλ1 ðaxþbzÞ þ M4 ðax þ bzÞ þ m5 ; υ ¼ 2α > < M1 ðax þ bzÞeλ ðaxþbzÞ þ M2 eλ ðaxþbzÞ þ M32λ 1 1 þ M6 e þ M 7 e 1 ðaxþbzÞ þ M4 ðax þ bzÞ þ m5 ; υ ¼ α : θðx; z; 0Þ ¼ M5 ðax þ bzÞe m4 λ2 ðaxþbzÞ > 2λ1 ðaxþbzÞ λ1 ðaxþbzÞ > : M8 e þ M9 e þ e þ M4 ðax þ bzÞ þ m5 ; other λ2

ðA1Þ

ðA2Þ

ðA3Þ Q76

ðA4Þ

● Eqs. (23)–(26) satisfy the following boundary conditions uðx; 0; t Þ ¼ m1 þ m2 eλ1 ðaxþct Þ þ

abg   ðax þ ct Þ 2 ðc þ m0 Þ a2 þ b

ðA5Þ

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

uðx; H; t Þ ¼ m1 þ m2 eλ1 bH eλ1 ðaxþctÞ þ

592 593 595 Q77 596

598

ðA7Þ

2 3 m0 a 4 abg λ1 bH λ1 ðaxþct Þ   ðax þ ct þ bHÞ5 wðx; H; t Þ ¼ e þ − m1 þ m2 e 2 b b ðc þ m0 Þ a2 þ b

pðx; 0; t Þ ¼ −

bρg 2

ðax þ ct Þ þ m3

2

ðax þ ct þ bH Þ þ m3

a2 þ b bρg

pðx; H; t Þ ¼ −

a2 þ b

ðA8Þ

ðA9Þ

8 2λ1 ðaxþct Þ 2λ1 ðaxþct Þ > eλ1 ðaxþct Þ þ M 4 ðax þ ct Þ þ m5 ; υ ¼ 2α > < M 1 ðax þ ct Þeλ1 ðaxþctÞ þ M2 eλ1 ðaxþct Þ þ M 32λ þ M6 e þ M 7 e 1 ðaxþct Þ þ M4 ðax þ ct Þ þ m5 ; υ ¼ α θðx; 0; t Þ ¼ M 5 ðax þ ct Þe m4 λ2 ðaxþct Þ > 2λ1 ðaxþct Þ λ1 ðaxþct Þ > þ M9 e þ e þ M4 ðax þ ct Þ þ m5 ; other : M8 e λ2

8 2λ1 bH 2λ1 ðaxþct Þ 2λ1 bH 2λ1 ðaxþct Þ > þ M 3 eλ1 bH eλ1 ðaxþct Þ þ M4 ðax þ ct þ bH Þ þ m5 ; υ ¼ 2α > < M 1 ðax þ ct þ bLÞe λ bH eλ ðaxþct Þ þ M 2λe bH λ eðaxþct Þ 2λ1 bH 2λ1 ðaxþct Þ 1 1 1 1 M ð ax þ ct þ bH Þe e þ M e e þ M e þ M 4 ðax þ ct þ bH Þ þ m5 ; υ ¼ α 5 7e 6 : θðx; H; t Þ ¼ m > 4 2λ bH 2λ ð axþct Þ λ bH λ ð axþct Þ λ bH λ ð axþct Þ > M8 e 1 e 1 : þ M9 e 1 e 1 þ e 2 e 2 þ M 4 ðax þ ct þ bHÞ þ m5 ; other λ2

P

599 Q78

2 3 m0 a 4 abg λ1 ðaxþct Þ   ðax þ ct Þ5 wðx; 0; t Þ ¼ þ m1 þ m2 e − 2 b b ðc þ m0 Þ a2 þ b

F

589 590

ðA6Þ

● Eq. (25), Eqs. (34)–(36) satisfy the following initial conditions

607 609 610

bρg

pðx; z; 0Þ ¼ −

2

a2 þ b

E

C

2 3 m0 a 4 abg 2  ðax þ bzÞ þ m6 ðax þ bzÞ þ m7 5 −  − wðx; z; 0Þ ¼ 2 b b 2υ a2 þ b ðax þ bzÞ þ m3 4

3

R

606

T

603

E

abg   ðax þ bzÞ2 þ m6 ðax þ bzÞ þ m7 2 2υ a2 þ b

uðx; z; 0Þ ¼ −

2

θðx; z; 0Þ ¼ M 10 ðax þ bzÞ þ M 11 ðax þ bzÞ þ M12 ðax þ bzÞ þ m8 ðax þ bzÞ þ m9 :

ðA11Þ

ðA12Þ

ðA13Þ

ðA14Þ

ðA15Þ ðA16Þ

R

612

ðA10Þ

D

601

604

O

586 587

abg   ðax þ ct þ bHÞ 2 ðc þ m0 Þ a2 þ b

R O

583 584

11

abg   ðax þ ct Þ2 þ m6 ðax þ ct Þ þ m7 2 2υ a2 þ b

ðA17Þ

abg   ðax þ ct þ bH Þ2 þ m6 ðax þ ct þ bH Þ þ m7 2 2υ a2 þ b

ðA18Þ

2 3 m0 a 4 abg 2  ðax þ ct Þ þ m6 ðax þ ct Þ þ m7 5 −  − wðx; 0; t Þ ¼ 2 b b 2υ a2 þ b

ðA19Þ

2 3 m0 a 4 abg 2  ðax þ ct þ bH Þ þ m6 ðax þ ct þ bHÞ þ m7 5 −  − wðx; H; t Þ ¼ 2 b b 2υ a2 þ b

ðA20Þ

uðx; 0; t Þ ¼ − 614 615

620 621

623 624 626 627 629 630 632

U

uðx; H; t Þ ¼ − 617 618

N C O

● Eq. (25), Eqs. (34)–(36) satisfy the following boundary conditions

pðx; 0; t Þ ¼ −

bρg 2

ðax þ ct Þ þ m3

ðA21Þ

2

ðax þ ct þ bH Þ þ m3

ðA22Þ

a2 þ b

pðx; H; t Þ ¼ −

bρg a2 þ b

θðx; 0; t Þ ¼ M 10 ðax þ ct Þ4 þ M11 ðax þ ct Þ3 þ M12 ðax þ ct Þ2 þ m8 ðax þ ct Þ þ m9

ðA23Þ

Please cite this article as: Y. Zhao, et al., Traveling wave solutions to incompressible unsteady 2-D laminar flows with heat transfer boundary, Int. Commun. Heat Mass Transf. (2015), http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.05.006

12

633

Y. Zhao et al. / International Communications in Heat and Mass Transfer xxx (2015) xxx–xxx

4

3

2

θðx; H; t Þ ¼ M10 ðax þ ct þ bH Þ þ M 11 ðax þ ct þ bH Þ þ M12 ðax þ ct þ bHÞ þ m8 ðax þ ct þ bHÞ þ m9 :

ðA24Þ

635

● Eqs. (37)–(40) satisfy the following initial conditions uðx; 0Þ ¼ u0

638

648 649 651 652 654 655

ðA26Þ

pðx; 0Þ ¼ p0

ðA27Þ

8 < M13 axe2λ3 ax þ M 14 e2λ3 ax þ M15 eλ3 ax þ M16 ax þ m13 ; υ ¼ 2α θðx; 0Þ ¼ M17 axeλ3 ax þ M18 e2λ3 ax þ M 19 eλ3 ax þ M 16 ax þ m13 ; υ ¼ α : : M20 e2λ3 ax þ M21 eλ3 ax þ M22 eλ4 ax þ M16 ax þ m13 ; other

ðA28Þ

● Eqs. (37), Eqs. (39), and Eqs. (41)–(42) satisfy the following initial conditions uðx; 0Þ ¼ u0 wðx; 0Þ ¼

g 2 x þ m14 ax þ m15 2υ

pðx; 0Þ ¼ p0 

υ g 2 4 m14 ga 3 m214 a2 2 x x θðx; 0Þ ¼ − x þ þ þ m ax þ m17 : 16 C p α 12υ2 3υ 2

ðA29Þ ðA30Þ ðA31Þ ðA32Þ

D

657

658

F

646 Q80

m10 λ3 ax ga e − x þ m11 c þ au0 λ3

O

644 Q79

wðx; 0Þ ¼

R O

640 641 643

ðA25Þ

P

637

Similarly, other initial and boundary conditions can be obtained when the governing Eqs. (1)–(4) are considered in slanted plates, plates that are aligned with gravity's direction and other cases.

660

References

661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 Q81 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699

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