The Motion Of A Micropolar Fluid In Inclined Open Channels

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The Motion Of A Micropolar Fluid In Inclined Open Channels

Transcript Of The Motion Of A Micropolar Fluid In Inclined Open Channels

PUBLISHING HOUSE OF THE ROMANIAN ACADEMY

PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, Volume 4 , Number 2/2003, pp.000-000

THE MOTION OF A MICROPOLAR FLUID IN INCLINED OPEN CHANNELS
Ligia MUNTEANU, Veturia Chiroiu 1 , Calin Chiroiu 2 , Stefania Donescu 3
1 Institute of Solid Mechanics, Romanian Academy 2 UniTeam, Torino
3 Technical University of Civil Engineering of Bucharest Corresponding author: [email protected]
The turbulent flow of a micropolar fluid downwards on an inclined open channel is studied. The wave profile moves downstream as a linear superposition of solitons at a constant speed and without distortion. The model parameters are determined by using a genetic algorithm. Key words: turbulent flow, micropolar fluid, solitons, roll-waves, Chezy resisting force, genetic algorithm.
1. INTRODUCTION
In a micropolar fluid the motion is described not only by a deformation but also by a micro-rotation giving six degree of freedom (Eringen, 1966, 1970 [1,2], Brulin, 1982 [3]). The interaction between parts of the fluid is transmitted not only by a force but also by a torque, resulting in asymmetric stresses and couple stresses. The theory of hydrodynamic turbulence is still lacking a fundamental theory from which the physical phenomena can be predicted and understood. The micropolar theory is employed here to obtain solutions which are periodic with respect to distance, describing the phenomenon called "roll-waves" for water flow along a wide inclined channel and to discuss the behavior of the solutions. In this work we study the turbulent flow of a micropolar fluid in a wide channel inclined at an angle θ > 0 below the horizontal.
The principal aim of this paper is to represent the periodic waves as a linear superposition of equally spaced solitons. A similar situation exists for Korteweg–de Vries, various modified Korteweg–de Vries, Boussinesq and Burger equations (Whitham, 1974 [4], Lamb, 1980 [6], Munteanu, Donescu [7]). In the light of inverse scattering theory, this representation may be viewed as a "clean interaction" of solitons, in that they are superimposed but retain their identity and do not destroy each other under the non-linear coupling (Whitham, 1984 [5]).
This representation requires determination of the model parameters: the wave numbers, the frequencies and the phases. For computing these unknown parameters we propose a new method based on a genetic algorithm. Details on the genetic algorithm are available in Goldberg, 1989 [8] and Tanaka, Nakamura, 1994 [9]).
The soliton representation is not so surprisingly in this case because Dressler [10] in 1949 studying the roll-waves motion of the shallow water in inclined open channels has found an equivalent form expressed as a cnoidal wave for a flow subject to the Chezy turbulent resisting force. Our theory generalizes the Dressler theory to the case of a micropolar fluid flow subject to a Chezy resisting force, and demonstrates that the presence of a resistance force, which varies with velocity, is sufficient to permit the construction of periodic solutions.
Recommended by Radu VOINEA Member of the Romanian Academy

Ligia MUNTEANU, Veturia CHIROIU, Calin CHIROIU, Stefania DONESCU

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2. EQUATIONS OF THE NONLINEAR SHALOW MICROPOLAR FLUID

In the shallow flow the vertical dimensions are small compared to the horizontal dimensions. The motion equations of a micropolar, viscous fluid are given by Eringen [1,2]:

ρv! + ρvgradv = X − grad π − (µ + α)curlcurl v + (2µ + λ)grad div v + 2αcurl w,

(2.1)

ρJw! + ρJvgrad w = Y − (γ + ε)curlcurl w + (2γ + ζ)grad div w − 4αw + 2αcurl v,

(2.2)

where X is the exterior body forces, Y is the exterior body couples, π is the thermodynamic pressure, ρJ is the inertia tensor density, v is the velocity vector v = ∂ u , u the displacement vector, φ the micro-
∂t rotation vector, w the micro-rotation velocity w = ∂ φ , ρ the fluid density.
∂t The superposed dot indicates the partial differentiation with respect to time a! = ∂ a . In (2.1), (2.2) λ
∂t and µ are the classical viscosities coefficients of the Navier-Stokes theory. The constants α,ζ, γ and ε are
the micropolar coefficients of viscosity. The elastic coefficients must fulfill the condition

µ ≥ 0, 2µ + 3λ ≥ 0, α ≥ 0, γ ≥ 0, 2γ + 3ζ ≥ 0, ε ≥ 0.

(2.3)

Equations (2.1) and (2.2) are six equations with unknown vector fields the velocity v and microrotation w . These equations must be supplemented by the equation of continuity for an incompressible fluid

div v = 0 .

(2.4)

In this case, the thermodynamic pressure π must be replaced by an unknown pressure p to be

determined through the solution of each problem. The constitutive relations are

σij = (− p + λvk,k )δij + (µ + α)v j,i + (µ − α)vi, j − 2αεkij wk ,

(2.5)

µij = ζwk,k δij + (γ + ε)wj,i + (γ − ε)wi, j ,

(2.6)

where σij is the stress tensor and µij is the couple-stress tensor.
The field of equations (2.1), (2.2) and (2.4) are subject to certain boundary and initial conditions: - traction conditions on the surface S of the body B

σkl nk = tl , µkl nk = µl on S ,

(2.7)

where tl are the surface traction and µl the surface couple acting on S . - velocity conditions of adherence of the fluid to a solid boundary

vk (x,0) = vk0 (x), wk (x,0) = wk0 (x) in B ,

(2.8)

where vk0 , wk0 are the velocity and micro-rotation velocity of the solid boundary. For a rigid stationary boundary we have vk0 = wk0 = 0 .

3. TWO-DIMENSIONAL FLOW
Consider a two-dimensional flow of a micropolar, isotropic, incompressible, viscous fluid in a wide channel over a rigid bottom. We have chosen a wide channel to be sure that the motion will bee twodimensional only. The x-axis is horizontally and the bottom is given by h(x). The channel bed is linear and is

Motion of a micropolar fluid in inclined open channels

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inclined at an angle θ > 0 below the horizontal y = −mx with m = tan θ > 0 (fig.1). The vertical distance of

the surface above the x-axis is denoted by η (x).

The horizontal and vertical components of v and w are, respectively v , v and w , w . We write

12

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X1 = −r2ρv1 | v1 | / R = −r2ρv1 | v1 | / η, X 2 = −ρg, Y1 = Y2 = 0 in (2.1) and (2.2). Thus, equations (2.1), (2.2)

and (2.4) are then

ρv! + ρv v + ρv v = − p + (µ + α)∆v − r2ρv1 | v1 | ,

1

1 1,x

2 1, y

,x

1

η

(3.1)

ρv!2 + ρv1v2,x + ρv2v2,y = − p,y + (µ + α)∆v2 − ρg ,

(3.2)

ρJw!1 + ρJv1w1,x + ρJv2w1,y = (2γ + ζ)w1,xx + (γ + ζ − ε)w2,xy + (γ + ε)w1,yy − 4αw1 ,

(3.3)

ρJw! 2 + ρJv1w2,x + ρJv2w2,y = (2γ + ζ)w2,yy + (γ + ζ − ε)w1,xy + (γ + ε)w2,xx − 4αw2 ,

(3.4)

v2,x − v1,y = 0 , w2,x − w1,y = 0 , v1,x + v2, y = 0 .

(3.5)

The coma represents the differentiation with respect to the shown variable. In (3.2) g is the

acceleration of gravity. In the right side of (3.1) the term −r2v1 | v1 | / R represents the resisting body force

always acting opposite to that of the flow, where r2 is a constant depending upon the roughness of the channel walls and R is the hydraulic radius.

According to this formula, the turbulent fluctuations exert

on the main flow a resistive body force at every point of magnitude r2v12 / R . Since the most flows in practice are

highly turbulent we take account of the resistive force due to

the momentum transport of the secondary flow exerted on the

average flow at each point. The resistance effects due to the

dynamic viscosity of the water are neglected. The above

expression of the resisting force was given by Chezy

(Dressler [4], 1949). The hydraulic radius is defined as the

ratio of the area of a cross section of the water normal to the

channel to its "wetted perimeter". That means that part of the

Fig. 1 Geometry of the flow

perimeter excluding the free surface of the water. The Chezy

formula thus expresses the fact that the resistance will be

greater in shallow regions where all of the water is closer to

the rough boundary. The Chezy formula is valid only for uniform flows, and although it is used for nonuniform flows when the flow vary slowly with respect to x, y and t. In our case R = η . In equations (3.1)-

(3.5) the unknown functions are vi , wi , i = 1, 2 , p and η . The boundary conditions are

η! + v1η,x = v2 at y = η ,

(3.6)

p = 0 at y = η ,

(3.7)

v1m − v2 = 0 at y = h ,

(3.8)

The condition (3.6) says that a particle at the surface will remain at the surface and (3.8) - the velocity at the bottom is tangential to the bottom. We add the following condition

wi = 0, i = 1, 2 at y = h .

(3.9)

Ligia MUNTEANU, Veturia CHIROIU, Calin CHIROIU, Stefania DONESCU

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Let the constant H be a typical vertical dimension of the resulting flow, and L a typical horizontal dimension. We H 2 / L2 = δ the expansion parameter which we will use for a perturbation procedure. New dimensionless variables are defined by

α = x , β = y , τ = gH t, Υ = η , P = p , V = v1 ,

L

H

L

H

ρgH 1 gH

d = h , V = H v2 , W = w1 , W = Hw2 ,

H 2 L gH

1 g/H

2 L g/H

(3.10)

µ = µ , α = α , r 2 = r2H , ζ = ζ , γ = γ , ε = ε .

ρL gH

ρL gH

L

ρLJ gH

ρLJ gH

ρLJ gH

In terms of the dimensionless variables (3.1)-(3.9) become δY[V1,τ + V1V1,α − (µ + α)V1,αα + P,α ] + V2V1,βY − (µ + α)V1,ββY + r 2V12 = 0 ,

δ[V2,τ + V1V2,α − (µ + α)V2,αα + P,β + 1] + V2V2,β − (µ + α)V2,ββ = 0 ,

(3.11) (3.12)

δ[W1,τ + V1W1,α − (2γ + ζ)W1,αα + 4αW1] + V2W1,β − ( γ + ε)W1,ββ − ( γ + ζ − ε)W2,αβ = 0 ,

(3.13)

δ[W2,τ + V1W2,α − ( γ + ε)W2,αα + 4αW2 ] + V2W2,β − (2γ + ζ)W2,ββ − ( γ + ζ − ε)W1,αβ = 0 ,

(3.14)

V2,α − V1,β = 0 , W2,α − W1,β = 0 , δV1,α + V2,β = 0 , δ(Y,τ + V1Y,α ) = V2 at β = Y ,

(3.15)

P = 0 at β = Y , δV1m / H = V2 at β = d , W1 = W2 = 0 at β = d .

(3.16)

We assume that the unknowns can be expressed as power series in terms of δ

∑ ∑ ∞



Vi = Vi(k) (α,β, τ)δk , i = 1, 2 , Wi = Wi(k) (α,β, τ)δk , i = 1, 2 ,

k =0

k =0

∑ ∑ ∞
P = P(k) (α,β, τ)δk ,
k =0


Y = Y (k) (α, τ)δk .
k =0

(3.17)

The series (3.17) are inserted in (3.11)-(3.16) and the resulting coefficients of like powers of δ are equated. Consider that

where

Mi(k) = ∂∂αkk log fi(k) (α,β, τ) , M = (V1,V2 ,W1,W2 , P ,Y ) and

k = 1, 2,...N ,

i = 1, 2...6 ,

(3.18)

f (1) i

(α,β,

τ)

=

1

+

exp

θ1i

,

fi(2) (α,β, τ) = 1 + exp θ1i + exp θ2i + exp(θ1i + θ2i ) ,

……….

∑ ∑ ∑ N

N

N

fi(N ) (α,β, τ) = 1 + exp θ ji + exp(θ ji + θli ) +

exp(θ ji + θli + θri ) +.... ,

j =1

j≠l =1

j≠l ≠r =1

(3.19)

θki = akiα + bkiβ − ωki τ + ςki , k = 1, 2,...N , i = 1, 2...6 ,

Motion of a micropolar fluid in inclined open channels

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and aki , bki the nondimensional wave numbers, ωki the nondimensional frequencies and ςki the nondimensional phases. The parameters in this formulation aki , bki , ωki and ςki , k = 1, 2,...N , i = 1, 2...6 are computable from (3.18),(3.19) and the like powers of δ equations. The numerical determination of these parameters are discussed in the next section. We find that asymptotically the solutions become

M (k) i

=

Aik

sec h2 (akiα

+

bkiβ −

ωkit



2∆ki )

,

k = 1, 2,...N , i = 1, 2...6 at t → ±∞ ,

(3.20)

where the constants A, ∆ can be easily calculated with respect to aki , bki , ωki and ςki . The functions

M (k) i

are periodic with the period

2∆ki . These solutions represent a linear superposition of solitons, a row

of solitons, spaced 2∆ki apart.

4. APPLICATION OF GENETIC ALGORITHM TO PARAMETERS DETERMINATION

Next step is to use (3.18),(3.19) and the like powers of δ equations to determine 23 × N parameters

p = {aki ,bki ,ωki ,ςki} , k = 1, 2,...N , i = 1, 2...6 ,

(4.1)

The wave numbers, frequencies and constant phases are also vectors

aki = (a11, a12 , a13 ,.....aN 6 ), bki = (b11,b12 ,b13 ,....bN5 ), ωki = (ω11, ω12 , ω13 ,....ωN 6 ), ςki = (ς11, ς12 , ς13 ,....ςN 6 ).

(4.2)

The resulting system is a system of 36 equations to determine a number of 23 × N unknowns. In this paper a new method is proposed to determine the model parameters (Goldberg 1989 [8]). It is assumed the parameters p are discretized into discrete values with the step width ∆p = {∆aki , ∆bki , ∆ωki , ∆ςki} . The set of parameters for arbitrary values p = {aki,m ,bki,n , ωki,q , ςki,s} can be expressed as 6N numbers

Nikmnqs = (m −1)NikQik Sik + (n −1)Qik Sik + (q −1)Sik + s ,

where M ki , Nki ,Qki and Ski denote the total number of discretized values for each parameter p . These

numbers represent an individual in a population and for the discretized parameters indicate a specific

solution (Tanaka, Nakamura, 1994 [9], Chiroiu et al., 1999 [13]). An individual is expressed as a row of the integer number with Ngen = 6N genes. To compute the fitness F we write (3.18)-(3.19) in the form

L(m) k

=

π(m) k

,

m = 0,1, 2 ,

k

= 1, 2,...12 , and note the square sum of differences

L(m) k

− π(km)

by



∑ ∑ 2 12

ℑ=

(L(kj) − π(k j) )2 .

j =0 k =1

(4.3)

∑ ∑ 2 12

We define fitness as follows F = ℑ0 / ℑ , with ℑ0 =

(

π

( k

j

)

)

2

.

As the convergence criterion of

j =0 k =1

iterative computations we use the expression Z to be maximum Z = 12 log10 ℑℑ0 → max. Numerical

simulation is carried out for λ =1.055 × 10 −3 Kg/ms, and µ =1.205 × 10 −3 Kg/ms. The micropolar coefficients

of viscosity have values α = ζ = ε =1.035 × 10 −3 mKg/s. We consider m = tan θ with m ∈ [0.2, 0.8]. The value m = 0.8 represents an upper limit on the slopes for which the shallow fluid theory furnish a good

approximation. The number r2 must satisfy the condition 4r2 ≤ 0.7m, , which is important for existence of

waves. If the resistance is too large, the waves cannot form. This condition is obtained numerically. We take r2 ∈ [0.035, 0.14]. The value r2 = 0.14 was chosen as the greatest value for the resistance since it satisfies

Ligia MUNTEANU, Veturia CHIROIU, Calin CHIROIU, Stefania DONESCU

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the condition (4.8) for m = 0.8. The intervals for the model parameters are evaluated from the condition that the total mass of fluid per wavelength is constant and the same in all approximations. In order to illustrate the results three cases are considered (N = 4):
- case 1 θ = 45 0 ( m = 1) , r2 = 0.17 ,
- case 2 θ = 31 0 ( m = 0.6) , r2 = 0.1,
- case 3 θ = 22 0 ( m = 0.4) , r2 = 0.06 .
In all cases we have assumed that the number of populations is 25, ratio of reproduction is 1, number of multi-point crossovers is 1, probability of mutation is 0.2 and maximum number of generations is 250.
The linear summation of the solution Y (α, τ) for τ → ∞ is given in fig. 2-4 ( τ → ∞ means in the numerical simulation the time interval after that the solutions have a permanent profile in time). In all cases the fluid velocity is greater in the region of the crests than in the shallower regions, but nowhere will the fluid velocity be as great as the wave speed. For example, in the case 1 the average fluid velocity is about 3.05 m/s while the wave velocity is about 4.1 m/s. From numerical simulations we conclude that the remaining solutions have a similar evolution with respect to α : they increase and decrease in the same manner as Y . The micro-rotation components and the vertical component of the fluid velocity are greater in the crest regions than in the shallower regions. The model parameters were obtained after 149 iterations in the case 1, 167 in the case 2 and 187 in the case 3.
In conclusion, the solutions we have obtained in this paper describe the phenomenon called "rollwaves" for fluid flow along a wide inclined channel. This phenomenon appears in hydraulic applications like run-off channels and open aqueducts. When a liquid flows turbulently downwards on an inclined open channel, the wave profile represented as sums of solitons moves downstream as a progressing wave at a constant speed and without distortion, and such that the velocities of the fluid particles are everywhere less than the wave velocity. Comparing only the performance, the genetic algorithm is superior to other conjugate gradient methods because it is simple to be applied, is stable and the correct solutions are detected through a relative small number of iterations, without requiring the stopping criterion for them. We need more computer memory in order to store the data, but in view of today's computer capabilities we do not consider this as a real disadvantage.

Fig. 2 The profile of the wave Y (α, τ) in the case 1

Motion of a micropolar fluid in inclined open channels

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Fig. 3 The profile of the wave Y (α, τ) in the case 2

Fig. 4 The profile of the wave Y (α, τ) in the case 3

Ligia MUNTEANU, Veturia CHIROIU, Calin CHIROIU, Stefania DONESCU

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ACKNOWLEDGMENTS
The authors acknowledge the financial support of Ministry of Education and Research (MEC)National University Research Council (NURC-CNCSIS) Romania, Grant nr.33517/2002.

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Received January 17, 2003
FlowSolutionsAlgorithmSolitonsMicropolar Fluid