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УДК 517.958,532.516
Nonlinear Effects in Poiseuille Problem
Alexander V. Koptev*
Makarov State University of Maritime and Inland Shipping, Dvinskaya, 5/7, S-Petersburg, 198035,
Russia
Received 10.01.2013, received in revised form 25.02.2013, accepted 22.03.2013 Poiseuille problem is the first problem in theoretical hydromechanics for which the exact solution has been found. The solution is a steady state solution of Navier-Stokes equations and it gives the velocity profile known as "Poiseuille parabola". Experimental studies show that parabolic profile occurs very seldom in fluid flows. Usually more complex structures are observed. This fact makes us again focus attention on the problem to obtain other solutions. This paper presents an approach that takes onto consideration all nonlinear terms of Navier-Stokes equations. New solutions of the Poiseuille problem are obtained and their nonlinear properties are identified.
Keywords: partial differential equation, viscous incompressible fluid, nonlinearity, exact solution.
Introduction
Poiseuille problem is the classic problem of hydromechanics for which the exact solution has been found. The essence of the problem is to define fluid motion when pressure drop is specified. At first that problem was considered for one-dimensional incompressible fluid motion in channel of circular cross-section [1-2]. Later some other variants of the problem were considered. Thus the Poiseuille problem for channels of elliptical and rectangular cross-sections were also studied.
Two-dimensional Poiseuille problem for fluid flow between two parallel plates is considered in this paper. Let us suppose that two parallel plates are boundary surfaces of a channel. The spacing between plates is 2H and they are parallel to the OX axis. Let us consider the plane problem only. So that fluid motion is identical in all planes perpendicular to the boundary surfaces. Suppose that pressure drop between two fixed cross-sections is equal to AP. Pressure drop is the main cause of fluid motion and it sets overall direction of fluid motion.
In the Poiseuille problem one needs to determine the basic characteristics of motion — velocities and pressure. Let us consider the problem based on 2D steady state Navier-Stokes equations for viscous incompressible fluid flow when mass forces are absent. In the non-dimensional form these equations are
du + du dp +1 / d2u + d 2u\ dx dy dx Re \ dx2 dy2 J '
dv dv dp 1 f d2v d2 v \ dx dy dy Re \ dx2 dy2) '
du dv o (o\
dx dy
* [email protected] © Siberian Federal University. All rights reserved
The velocities and pressure are main unknowns. They are designated as u, v and p. Re is positive parameter. It is the Reynolds number
n HUo
Re =-,
H is the half of the channel width and it is the scale of length. The scale of velocity is U0 = \JAP/p, where AP is pressure drop. The scale of pressure is pressure drop AP, p is fluid density and v is kinematic viscosity.
Because we consider the problem for incompressible fluid only, p and v are constant.
Formulation of the Poiseuille problem requires the assignment of certain boundary conditions. The adhesion conditions at the boundary y = ±1 should be held [1,2]. These conditions are
u(x, 1)=0, v(x, 1)=0, (4)
u(x,-1)=0, v(x,-1)=0. (5)
Another boundary condition should be set for pressure drop between two cross-sections of the channel. Let us chose two cross-sections at x = ±L/2, where L is the positive parameter. Then boundary condition for non-dimensional pressure is
-L.») -»(L,0) = 1. (6)
Thus the Poiseuille problem for the channel is described by equations (1)-(3) with boundary conditions (4)-(6), where Re and L should be specified as parameters.
1. Trivial solution
The trivial solution of the Poiseuille problem is well known [1,2]. The solution is derived in the assumption that fluid flow is one-dimensional and stream lines are parallel to the OX axis. In this case the transversal velocity is equal to zero. The axial velocity depends only on y and pressure is the linear function of x
v = 0, u =RL • (1 - y2) , (7)
P - Po = -L, Po = const. (8)
The velocity profile defined by these formulas is named as "Poiseuille parabola". This solution of the problem is an exact one but it occurs rarely in practice. Experimental data leads to conclusion that other more complex velocity profiles occur more frequently. Thus there is a dilemma [3]. Either the Navier-Stokes equations (1)-(3) inadequately describe fluid flows or there are solutions that describe fluid flows more accurately. Assuming that such solutions exist and they can be found.
The existence of more complex solutions is supported by the following reasoning. Equations (1)-(3) contain twelve differential terms (five plus five plus two). Solution (7)-(8) assumes that only two out of twelve terms are nonzero. These terms are -dp/dx and Re-1d2u/dy2. The other ten terms of equations (1)-(3) are set equal to zero.
We face the following question. Can the solution obtained on the basis of only two terms out of twelve represent a real fluid flow? The most likely answer is no. Solution (7)-(8) is only
v
special solution. It does not describe the features of two-dimensional nonlinear problem. This solution can be called as trivial solution. It is impossible to understand the problem in full if we limit ourselves to the consideration of the trivial solution only. Hence for today the Poiseuille problem is still largely unexplored.
Thus we need to focus on the search for nontrivial solutions. These solutions should take into account two-dimensional character of the motion and the presence of nonlinear terms.
2. Nontrivial solution
Derivation of nontrivial solutions is based on the first integral of equations (1)-(3) considered in [4, 5]. In the case of 2D steady state incompressible viscous fluid flow the first integral is reduced to three equations:
U2
p + d = a + P, (9)
2 2 2 ( du dv\ d2^2 d2^2 ,
u2 -v2 + R (-dX + dy) = "dx2 + V + 2(a-P)' (10)
1 (dv du N = d2 Re \ dx dy J dxdy
The new unknown term together with main terms u,v and p is present in relations (9)-(11). This variable is not included in the original equations (1)-(3) but it is the result of first integration. The meaning of other symbols is the following: U = %/u2 + v2 is absolute value of velocity; d is the dissipative term defined as
d = -1 (+ ^); (12)
a and p are arbitrarily selected functions of one variable, a depends only on y and p depends only on x.
Relations (9)-(11) have advantages in comparison to original equations (1)-(3). The main advantage is that there are only first order derivatives for the main unknown in equations (9)-(11). So it is better to use these equations to derive nontrivial solutions.
Together with equations (9)-(11) we should also consider the continuity equation (3). Then we obtain the set of four equations for four unknowns u, v, p and . Three equations form the defining set. These equations are (10), (11) and (3). The unknowns u, v and are defined from these equations. The last unknown p can be easily obtained from equation (9). For simplicity we assume that a = 0 and p = 0. The form of trivial solution (7)-(8) suggests that nontrivial solution should be represented in the form of a power series in x and y. Let us suppose that is the well-known stream function for a plane fluid motion [1, 2]:
= d = d^1 dy ' dx
A nontrivial solution is represented in the form of polynomials of degree N:
N N-n N N-n
anmXnym, = £ bnmXnym,
n=0 m=0 n=0 m=0
where anm and bnm are some coefficients, N is the approximation degree and N > 3.
To begin with we impose restrictions on the coefficients anm to satisfy boundary conditions (4)-(5) and continuity equation (3).
Preliminary analysis indicates that if these restrictions are satisfied then for 3 < N < 5 we arrive only to trivial solution (7). For N > 6 another possibility arises and nontrivial solution may be obtained. In this case velocities should be represented by fifth degree polynomial of the special form:
u = (1 - y2) • [aoi + anx + 2a02y + (aoi + 3-o3)y2 - 4awxy - Aa20x2y-
-5anxy2 + (2ao2 + 4ao4)y3], (13)
v = -(1 - y2)2 • (aio + 2a2ox + any) . (14)
Expressions (13), (14) contain seven unknown coefficients. These coefficients are aoi, aio, ao2, aii, a2o, ao3 and ao4. Let us take them as the basic unknown coefficients. They should be determined in such a way as to satisfy equations (10)-(11) with an accuracy of up to fourth degree.
After substitution of and into equations (10)-(11) we obtain the relationship between coefficients anm and bkl. These equations are satisfied with the specified accuracy only if all coefficients bkl for k + l < 6 are determined from the basic coefficients. Analysis shows that it is possible only if the basic coefficients satisfy some additional conditions. These conditions result in the following system of nonlinear equations:
4
— (3ao4 - 2a2o) + aio(3ao3 + 2aoi) = 0, (15)
Re
—^ - 2aioaii + 2aoia2o + 3a2oao3 = 0, (16)
Re
— (aoi + 3ao3) + (a2o - ao2 - 3ao4)--j1(2aoi + ao3) = 0, (17)
Re 3 4
i - 2k) =°, (18) —(3a2o - 5ao2 - 1°ao4) - -io ^-oi +--+ -ii (-t + -o*4) = °, (19)
5aii , -2o -2o , f-o2 \ „ fon,
2ke + T" - t+-2n T +H = °. (20)
Then we have six equations for seven unknowns anm and Re is the parameter. The simplest equation in the system is equation (18). It presents two special solutions: a2o = 0 or aii -3/(2Re) = 0. Calculations show that the first solution leads to -io = 0, ao2 = 0, aii = 0, ao4 = 0 and ao3 = -aoi/3. This corresponds to trivial solution (7). The second special solution gives more interesting results. In this case coefficients are
-ii = —, (21) ii 2Re V '
4 2 3 8 2 3
-io + 27Re -io _ 2aio + gjRe aw -oi = -/ ==, ao3 =--/ ==, (22)
R^ i0-2o + R5 R^y i0-2o + Rf-
55 2 I 375
o,2Q = 7\/10aio + , «02 = - 6 ±U 4Re , (23)
4 V 10 Re
Vio«2o + RQ5
3 2 , 105 55aïo +
" —2, a02 —--
2
16 a2 + 105
ao4 — 3 a10 + 2Re2 . (24)
10a2 + i05 10a10 + Re2
Let us consider some features of the obtained relations. Coefficient a11 is uniquely expressed in terms of Re and coefficients a01, a03, a20, a02 and a04 are expressed in terms of a10 and Re. Coefficient a10 must be defined from boundary condition (6). To use this condition we need to express bkl in terms of basic coefficients and obtain relation for p according to equation (9). As a result condition (6) leads to the following equation in one unknown a10
£3 - 3x3 £ + 2X4 — 0, (25)
where £ is defined as
2 , 2X1
e = «lo + ^Xx, (26)
parameters Xj are defined by the following formulas
L ( t2 315\ (517 o 15\ , ,
X1 — R? (201 + L23rj , X2 — ^517 + L2y) , (27)
-if Xi -3 { 1 3 10 105 >
X3 = Xi (y + y) ' X4 = Xi (-27Xi + yxi - 2Reixi ). (28)
Equation (25) is a cubic equation and it has at least one real root. Thus it is possible to determine all coefficients and obtain nontrivial solution of the problem.
3. Results
The obtained above expressions allow us to calculate basic coefficients anm at given values of Re and L. To do this we use (27)-(28) and solve cubic equation (25). Then one should chose appropriate roots according to condition
£ - > 0, (29)
3X2
which follows from (26). When the value of aio is known one can find the remaining basic coefficients by the formulas (21)-(24).
The results of calculations for five sets of parameters are presented in Tab. 1. The results presented in Tab. 1 can be used to determine velocities u and v by the formulas (13)-(14). The results of calculations are presented in Tab. 2. In this table velocities are functions of y for —1 ^ y ^ 1 at x = 0 and x =1. Calculations have been performed for Re = 10 and L = 2. For comparison the last column of Tab. 2 presents the velocity profile of trivial solution calculated with the same values of parameters Re and L. Each vertical column of table 1 gives seven basic coefficients for the specified values of parameters Re = 0.5; 1; 2; 10 and L = 2; 5. As can be seen from the table nontrivial solutions not always exist. For example, there is no solution for Re = 0.5 and L = 2 because inequality (29) is violated. When nontrivial solutions exist the values of coefficients are in a range which depends on parameters Re and L.
Table 1. Values of basic coefficients
№ 1 2 3 4 5
parameters / coefficients Re=10 L=5 Re=2 L=2 Re=1 L=5 Re=10 L=2 Re=0.5 L=2
1 a10 9,487•10-3 5.700 • 10-3 1.523 • 10-1 3.050 • 10-2 -
2 «11 1.500 • 10-1 7.500 • 10-1 1.500 1.500 • 10-1 3.000
3 aoi 9.270 • 10-4 5.560 • 10-4 1.489 • 10-2 2.977 • 10-3 -
4 «03 -1.850 • 10-3 -1.110 • 10-3 -2.973 • 10-2 -5.950 • 10-3 -
5 «20 7.689 • 10-1 3.843 7.694 7.686 -
6 «02 -9.153 • 10-1 -4.575 -9.160 -9.150 -
7 «04 5.126 • 10-1 5.562 5.129 5.124 -
Table 2. Velocity profiles
№ y u(0,y) v(0,y) u(1,y) v(1,y) utriv
1 1.0 0.0 0.0 0.0 0.0 0.0
2 0.8 -4.866 -0.020 -13.664 -2.018 0.9
3 0.6 -6.724 -0.049 -18.431 -6.346 1.6
4 0.4 -6.028 -0.064 -16.232 -10.910 2.1
5 0.2 -3.494 -0.056 -9.253 -14.223 2.4
6 0.0 0.003 -0.031 0.153 -15.403 2.5
7 -0.2 3.500 -0.001 9.546 14.167 2.4
8 -0.4 6.033 0.021 16.489 10.826 2.1
9 -0.6 6.736 0.025 18.637 6.272 1.6
10 -0.8 4.867 0.012 13.775 1.981 0.9
11 -1.0 0.0 0.0 0.0 0.0 0.0
One can see the following feature. The axial and transverse velocities are comparable in magnitude. Hence we have a real two-dimensional flow.
It is also interesting to compare the velocity distribution given in equation (7) with the velocity distribution that follows from nontrivial solution. This comparison shows that velocity magnitude in nontrivial solution is greater than velocity magnitude in trivial solution. To see this one can compare u(0, y), u(1, y) and utriv. In trivial solution the maximum value of velocity is achieved at y = 0 and it is 2.5 but in nontrivial solution the maximum value of velocity is more than 18.
Another interesting feature is observed if we compare the velocities sign. For trivial solution the sign of axial velocity is not changed: utriv > 0 for all —1 < y < 1. Then direction of motion is from left to right. For nontrivial solution the velocities u(0, y) and u(1,y) change their signs and thus the reverse flow is observed. The fluid flow from left to right corresponds to positive sign of axial velocity and reverse flow corresponds to negative sign of axial velocity. As can be seen from Tab. 2 zones of reverse flow take considerable space.
It is interesting to note that sign changes are also observed for velocities v(0, y) and v(1,y). Such changes correspond to vortex flow.
Conclusion
Poiseuille problem have various solutions. The trivial solution is well known. In some cases nontrivial solutions can exist. If nontrivial solution exists it may be not unique. The number of possible solutions equals to the number of real roots of cubic equation (25). Thus the existence of various types of flow in Poiseuille problem is theoretically substantiated. They were observed earlier in experimental studies.
All discussed above features of nontrivial solutions are due to nonlinear effects. These effects are taken into account as nonlinear terms in equations (1)-(2), (10)-(11), (15)-(20) are considered. The relative contribution of nonlinear terms is increased with increasing Reynolds number Re. Then nonlinear effects are amplified. The nonlinear effects are also amplified as characteristic length L is reduced.
References
[1] L.G.Loitsanskiy, Fluid and gas mechanics, Nauka, Moscow, 1987 (in Russian).
[2] N.E.Kochin, I.A.Kibel, N.V.Rose, Theoretical hydromechanics, part 2, Nauka, Moscow, 1967 (in Russian).
[3] O.A.Ladizhenskaya, The Mathematical Theory of Viscous Incompressible Fluid, Gordon and Breach, New York, 1969.
[4] A.V.Koptev, Integrals of Navier-Stokes equations, Trudy sredne-volzhskogo matematiches-kogo obshchestva, Saransk, 6(2004), no. 1, 215-225 (in Russian).
[5] A.V.Koptev, First integral and ways of further integration of Navier-Stokes equations, Izves-tia universiteta im. Gertsena, Zhurnal gumanitarnyh nauk, 147(2012), 7-17 (in Russian).
Нелинейные эффекты в задаче Пуазейля
Александр В. Коптев
Задача Пуазейля представляет одну из первых задач теоретической гидромеханики, для которой было найдено точное решение. Процедура построения решения основывается на уравнениях Навье-Стокса и дает профиль скорости в виде "параболы Пуазейля". Однако проблема состоит в том, что данное решение очень редко реализуется на практике. Гораздо чаще наблюдаются другие законы, существенно более сложные. Это обстоятельство заставляет вновь обращаться к этой известной задаче и предпринимать поиск других решений, отличных от классической "параболы Пуазейля". В данной работе предлагается исследование задачи с учетом полного рассмотрения нелинейных членов. В результате построены новые решения, изучены их свойства и выявлены нелинейные эффекты.
Ключевые слова: дифференциальные уравнения, частная производная, нелинейность, вязкая несжимаемая жидкость, точное решение.
