Exact solution of Navier-Stokes equations

8/12/2019 Exact solution of Navier-Stokes equations

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PHYSICS OF FLUIDS 25 , 073102 (2013)

Exact solutions of the Navier-Stokes equations with spiralor elliptical oscillation between two innite planes

Ming-Jie Zhang and Wei-Dong Su a)State Key Laboratory for Turbulence and Complex Systems and Department of Mechanics

and Engineering Science, College of Engineering, Peking University, Beijing 100871, China(Received 9 September 2012; accepted 24 May 2013; published online 19 July 2013)

Exact solutions of the Navier-Stokes equations between two innite planes are con-sidered, where the velocity components parallel to the planes depend linearly on twospatial coordinates, and the third component depends only on the coordinate perpen-dicular to the planes. A class of unsteady exact solutions is found in this form withspiralor elliptical oscillation as an eigenmode of exponential time dependence, whichcan be arbitrarily superposed, while the pressure and boundary conditions remain un-changed. As a specic case, theow between two innite rotating disks is considered,and the corresponding eigenvalue problems are numerically investigated. Multiplesolutions have been taken into account under axisymmetric and non-axisymmetric

distributions of pressure. The eigenvalues, which are dependenton the Reynoldsnum-ber, the rotation ratio, and the pressure parameter ratio, are calculated, and the phasediagrams containing neutral curvesare presented. It is shown that some axisymmetricows between two parallel rotating disks can be associated with an added periodicoscillation at low frequency proportional to the rotation ratio and with arbitrarilylarge amplitude. C 2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4813629 ]

I. INTRODUCTION

Exactsolutions of the Navier-Stokes equationsare important, as they represent fundamental uidmotionand serveas standardsfor checking theaccuracy of numericaland asymptotic methods. In thiscontext, exact solutions are analytical solutions or a set of simpler ordinary or partial differentialequations reduced from the Navier-Stokes equations without any approximation. Extensive reviewson this subject were presented by Wang 13 and by Drazin and Riley. 4 In this paper, we introducea wide class of exact solutions of the incompressible Navier-Stokes equations which was rstdiscovered by Lin 5 and studied by Aristov et al. 6 in reduced forms. The velocity and pressure eldshave the form

u = f ( x , t ), (1)

v =g ( x , t ) +k ( x , t ) y +h ( x , t ) z, (2)

w = ( x , t ) +( x , t ) y +( x , t ) z, (3)

p = p0 12 C 2 y2 +2C 5 yz +C 4 z2 +2C 1 y +2C 3 z + f 2 2 f x , (4)where u, v, and w are the velocity components in Cartesian coordinates ( x , y, z); t is the time; p0 andC 15 are arbitrary functions of t ; and and are the uniform density and the kinematic viscosityof the uid, respectively. There are seven unknown functions f , g, k , h, , , and in the exact

a) Electronic mail: [email protected] .

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073102-2 M.-J. Zhang and W.-D. Su Phys. Fluids 25 , 073102 (2013)

solutions, which are governed by seven partial differential equations with x and t as variables. Forsteady solutions, the variable t disappears, and the partial differential equations turn into ordinarydifferential equations with respect to x .

This form of velocity and pressure leads to a wide class of exact solutions of the Navier-Stokes equations, describing the ow between two parallel innite planes with suction or injectionat each plane, such as plane ow, stretching ow, stagnation-point ow, and rotating disk ow.The points in the two planes can have velocities with a linear dependence on two spatial vari-ables y and z, i.e., allowing elliptical, hyperbolic, parabolic, and spiral motions. In the presentpaper, we construct a class of unsteady exact solutions and the corresponding eigenvalue problem,where g( x , t ) = n=1 gn ( x ) exp( n t ), ( x , t ) = n=1 n ( x ) exp( n t ) and the other functions areall time-independent. We note that there are many other exact solutions that contain exponentialtime dependence. For example, Merkin 7 studied mixed convection boundary layer ow, in whichexponential terms were introduced to test the linear temporal stability of the solutions. Weidmanet al. 810 extended Merkins method to study self-similar boundary layer ow driven by a movingsurface, uniform shear ow, and radial stagnation ow. Hui 11 and Shapiro 12 studied wave-like ow.Weidman and Mahalingam 13 considered axisymmetric stagnation-point ow impinging on an os-cillating plate with suction. Aristov and Gitman 14 considered the ow between two moving paralleldisks. Zaturska and Banks 15 studied the ow in a channel with porous walls and considered theeffect of three-dimensional disturbances on the temporal stability of the ow. Many solutions canalso be found in the book by Drazin and Riley. 4

In this paper, we concentrate on theowsproduced by therotationof thetwo planesto investigatethe behavior of the unsteady solutions. The ow between two innite parallel rotating disks withexact solutions has been the subject of numerous studies. 16 Most works were focused on solutionswith axisymmetric velocity ( k , h ), which were initially considered by Batchelor 17 andStewartson. 18 The existence and non-uniqueness of these solutions were investigated. Keller andSzeto 19 obtained solutions for s

[ 1, 1] (angular velocity ratio of the top disk to the bottom disk)with Re up to 1000, where Re is the Reynolds number based on the angular velocity of the bottom

disk and the distance between the two disks. The authors concluded that the solution is unique for Re 55. As Re increases, the solution branches become increasingly complex. Holodniok et al. 20

studied the ow at Re = 625 for s [ 1, 1] and identied 19 solution branches. Using ananalytical method, Kreiss and Parter 21 established the existence and non-uniqueness of solutions fors [1, 1] when Re is sufciently large. Furthermore, Rajagopal et al. 22, 23 considered the owbetween two rotating disks about a common axis or distinct axes. A class of asymmetric solutions,which is the superposition of the K arm an ow and a rigid-body translation in each y z plane,was studied. Recently, solutions with non-axisymmetric velocity bifurcating from axisymmetricsolutions were obtained for the ow between two rotating disks 2426 and the ow between twoporous planes with suction/injection. 27, 28 Overall, all of these solutions belong to Lins class of exact solutions, as formulated in Eqs. (1 )(4) , wherein g and either vanish or do not depend ontime. Our unsteady exact solutions also belong to Lins class of solutions. However, to the best of our knowledge, the solutions have never been postulated and thus justify further investigation.

In Sec. II of this paper, a mathematical formulation of the unsteady exact solutions is described,and Sec. III focuses on the ow between two innite parallel rotating disks under an axisymmetricpressure eld. Both the axisymmetric and non-axisymmetric velocity solutions are considered.

Section IV considers suction and injection. Section V considers ow between two rotating disksunder non-axisymmetric pressure elds ( C 2 = C 4 , C 5 = 0). This is a natural extension to theexisting studies covering only axisymmetric pressure elds ( C 2 =C 4 , C 5 =0). The discussion andconclusions are presented in Sec. VI .

II. FORMULATION

Substituting Eqs. (1)(4) into the Navier-Stokes equations for an incompressible viscousuid and introducing velocity scale U , length scale L, time scale L / U , and the Reynolds number



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Re =UL / , we derive the following nondimensional governing equations: f x +k + =0, (5)

ht + f

h x h

f x

1 Re

2h x 2 =C 5(t ), (6)

t + f

x

f x

1 Re

2 x 2 =C 5(t ), (7)

k t + f

k x +k

2

+h 1

Re 2k x 2 =C 2(t ), (8)

t + f

x +

2 +h 1

Re 2 x 2 =C 4(t ), (9)

gt + f

g x +kg +h

1 Re

2 g x 2 =C 1(t ), (10)

t + f

x +g +

1 Re

2 x 2 =C 3(t ). (11)

Substituting the solutions of Eqs. (5)(9) into Eqs. (10) and (11) , we obtain the solutions for gand . We restrict x to the unit interval [0, 1], and thus the ow is bounded by two innite parallelplanes at x =0 and x =1.In this paper, we consider a class of unsteady ow with

g( x , t ) = g( x ) exp( t ), ( x , t ) = ( x ) exp( t ), (12)and we suppose that f , k , , h , and are independent of t ; that C 2 , C 4 , and C 5 are constants; andthat C 1 =C 3 =0. Furthermore, by a proper coordinate transform about ( y, z), the parameter C 5 canvanish in this case. Thus, without loss of generality, we set C 5 =0 hereinafter.Substituting the above equations into Eqs. (10) and (11) yields

g + f g +kg +h 1

Reg =0, (13)

+ f +g + 1

Re =0. (14)

Thus, an eigenvalue problem is derived for g, , and under homogeneous boundary condi-tions of g(0) = g(1) = (0) = (1) = 0. When the eigenvalues are discrete, more generalunsteady velocity components, expressed as the arbitrarily superposition of eigenmodes, viz.,g( x , t ) n=1 gn ( x ) exp( n t ) and ( x , t ) n=1 n ( x ) exp( n t ), can be added to the steady baseow described by the time-independent forms of Eqs. (5)(11) , where gn , n , and n are the eigen-functions and eigenvalues of Eqs. (13) and (14) . In general, we have gn = gnr + igni , n = nr +i ni , and n = nr +i ni in complex forms. The oscillation nature of the unsteady part depends onthe characteristics of the eigenvalues, especially the sign of the largest real parts of the eigenvalues;hence, we can interpret the unsteady components as a type of horizontal disturbance and therebyexamine the stability of the base ow. However, it should be emphasized that such disturbancescan have arbitrary amplitudes and are not restricted to innitesimal linear disturbances.

We dene the differential operator as

L =1

ReE

dd x

(d

d x ) A , = exp Ref dx , A = k h , (15)

and the inner product of ( x ) and ( x ) as

, = 1

0[1 1 +2 2]dx , (16)



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where E is the unit matrix, =( 1 , 2)T and =( 1 , 2)T are vectors satisfying the homogeneousboundary conditions (0) = (1) = (0) = (1) = 0, and the bar denotes a complex conjugate.Equations (13) and (14) are transformed into a vectorial Sturm-Liouville problemL (g, )T = (g, )T . (17)

For the case of a symmetric matrix A (h ), ( x ), and ( x ) satisfy, L L , =

1 Re

[ ]|10 =0. (18)

Thus, the eigenvalues are real, and the eigenfunctions are orthogonal. Conversely, for the case of an asymmetric matrix A , such as in rotating disk ow, the eigenvalues and eigenfunctions may becomplex, with nonvanishing imaginary parts.

The asymptotic forms of the eigenvalues and eigenfunctions for large eigenvalues can beobtained from a similar treatment to that used for the scalar Sturm-Liouville problem. Introduc-ing the new vectorial function q( x ) 1/2 ( x ), Eq. (17) can be transformed to regular Liouvilleform

1

Re

E

d2q

d x 2

I

+A

+( Re f 2

4

f

2) I

q

=0. (19)

For large , from the asymptotic expansionq ( x ) = e

i x Re q0( x ) +( )1/ 2q1( x ) + (20)

the estimate for the nth eigenvalue n and eigenfunction n can be derived as

n (n )2/ Re, n ( x ) 1/ sin(n x )0 , n , (21)

where 0 is a constant vector. These estimates are useful in determining the convergence rate of series expansion in g( x , t ) and ( x , t ).

In the case of k h = 0, for each x , there exists a stagnation point ( x , y0 , z0) that satisesv, w =0, where

y0 =1

k h

n=1e

nr t (h nr gnr ) cos ni t (h ni gni )sin ni t , (22)

z0 =1

k h

n=1e nr t (gnr k nr ) cos ni t (gni k ni ) sin ni t . (23)

For simplicity, we only add one eigenmode, say the one with the largest real part, to the base ow;hence, at each xed x or in each plane parallel to the boundary planes, the locus of ( x , y0 , z0) is anellipse for r =0 and i =0, and a spiral for r =0 and i =0. When is real, at each x , the locusof ( x , y0 , z0) will degenerate into a line. Furthermore, when k = h + 0 and i = 0, thelocus of ( x , y0 , z0) will be a circle ( r =0) or a logarithmic spiral ( r = 0). The elliptical or spiraloscillation of the locus of ( x , y0 , z0) corresponds to the same oscillatory behavior of the ow.

III. FLOW BETWEEN TWO ROTATING DISKS

The governing equations (5)(9) in Cartesian coordinates can be recast as

f f +12

[ f 2 +(k )2

+(h +)2

(h )2] +

1 Re

f =C 2 +C 4 , (24)

f (k ) f (k ) 1

Re(k ) =C 2 C 4 , (25)

f (h ) f (h ) 1

Re(h ) =0, (26)



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f (h +) f (h +) 1

Re(h +) =0, (27)

with the pressure eld

p p0 =

1

2

2 L2(C 2 y2

+C 4 z2)

1

2 ( Re f 2

2 f ), (28)

where f , k , , h, and are functions of x and C 2 and C 4 are constants.For ow between two rotating disks, the boundary conditions are

f (0) = f (0) = k (0) =(0) = 0, (0) = 1, h(0) = 1, (29)

f (1) = f (1) = k (1) =(1) = 0, (1) = s, h(1) = s, (30)where s [ 1, 1] is the rotation ratio of the two disks; is the angular velocity of the bottomdisk ( x = 0); s is the angular velocity of the top disk ( x = 1); L is the distance between the twodisks; U = L is the velocity scale; and Re = 2 L / is the Reynolds number based on the motionof the bottom disk. There are ten boundary conditions for the ninth-order system with two pressureparameters of C 2 and C 4 . We can prescribe the value of one parameter and leave the other to be

determined by the boundary conditions.In this section, we consider the ow under an axisymmetric pressure eld ( C 2 = C 4), wherek and h + satisfy the same homogeneous equations and boundary conditions. In this eld,C 2 =C 4 is the parameter to be determined. When k = h + 0, Eqs. (24)(27) are reducedto the equations of the well-known von K arm an ow for two rotating disks with an axisymmetric

velocity. For counter-rotating disks ( s < 0), k and h + may have nonvanishing solutions withk = cos and h + = sin , corresponding to the non-axisymmetric solutions obtainedby Hewitt and Al-Azhari, 26 where is a function of x and is an arbitrary constant. In theirstudy, the bifurcation points at which the non-axisymmetric solutions appear in the ( s, Re) spacewere numerically calculated, and the solution structures upon approaching the critical boundaries of Re =0 and s =0 were asymptotically analyzed. The existence domain of this solution is shown inFig. 1(b) (see also below).

We investigate two types of base ow: the axisymmetric ow (type I) and the non-axisymmetricow (type II). The fundamental parameters governing this problem are the Reynolds number Re

FIG. 1. Phase diagrams of the ow between two rotating disks under an axisymmetric pressure eld ( C 2 = C 4). (a) Thesolid circles are the points when 1r =0 for type II. A is the domain for which 1r > 0 for type II. (b) The solid circles arethe points when type II vanishes. The hollow circles are the points when 1r =0 for type I. B is the existence domain of typeII; C is the domain for which 1r > 0 for type I. The solid curve near s =0 is the lower boundary of B. It does not intersectwith the axis s =0.



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TABLE I. Values of Re1 , Re2 , and Re3 at various s.

s 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Re1 47.29 49.82 52.72 56.12 60.16 65.07 71.22 79.21 89.96 Re2 48.55 51.15 54.18 57.75 62.10 67.51 74.54 84.30 99.52

Re3 88.74 93.57 99.39 106.57 115.80 128.36 147.34 183.20 None

and the rotation ratio s. Type I considered here is the only branch of the von K arm an ow for tworotating disks for which the existence interval of Re is (0, ). Here, we rst calculate the solutionat small values of Re and subsequently obtain the solution continuously at large values of Re. In thecase of type II, for 1 s 0.088, the solution exists when Re is smaller than a critical value Re2 and turns into type I when Re > Re2 ; for 0.088 s 0.055, the solution exists at all Re; for0.055 s < 0, there is an s-dependent interval of Re wherein the solution vanishes. The pressureparameters C 2 of types I and II are different. For type I, C 2 < 0 when Re is small and increasesgradually with increasing Re. For type II, C 2

as Re

0 and decreases with increasing Re

until it coincides with that of type I; whereafter, type II vanishes.Many studies of unsteady rotating disk ow have focused on the initial-value problem. 29, 30

Here, we consider the eigenvalue problem, as formulated in Sec. II. Let 1 = 1r + i 1i bethe eigenvalue with the largest real part. There are three groups of Reynolds numbers to be dis-cussed in this section: Re1 is the Reynolds number when 1r = 0 for type II, Re2 is the criticalReynolds number beyond which type II vanishes, and Re3 is the Reynolds number when 1r =0 fortype I.

In the case of type I, numerical results show that the eigenvalues are real when s = 1 andcomplex when s = 1. For 0.206 s 1, 1r < 0; and for 1 s 0.206, 1r > 0 when Reis larger than a critical value Re3 . In the case of type II, for 0.0195 s < 0, 1r < 0; for 1 s0.0195, 1r > 0 when Re is smaller than a critical value Re1 . The phase diagrams consisting of separate domains divided by the three Reynolds numbers are shown in Fig. 1. The curves for Re1

and Re3 are neutral curves. An interesting observation shows Re1 < Re2 < Re3 for xed s. It seemsthat the curves for Re2 and Re3 have asymptotes of s 0.088 and s 0.206, respectively, when Re . Tables I and II present values of Re1 , Re2 , and Re3 at various s.Then, we consider the case of 1r =0. For Re = Re3 , Figs. 2 and 3 show steady velocity proles f , k , h, and and periodical velocity proles in the y-direction gr cos 1it gisin 1it at various s. Notethat =k , =h, r =g i , and i =gr . Figure 4 shows values of | 1i| for type I for various s. Theow is periodically oscillating in the same whirling direction as the bottom disk. It is observed thatthe circular oscillation frequency | 1i| approximately linearly increases with s, which can be tted as| 1i| =0.1576( s +1). The locus of the stagnation points at each plane parallel to the disks is a circle.Figure 5 gives the radius of the circle R( x ) = y20 + z20 varying with s, where the R( x ) has beennormalized by its maximum within x

[0, 1]. It can be seen that the plane with maximum oscillation

amplitude R gradually approaches the bottom disk (the disk with a larger rotation velocity) when sincreases. For type II, in contrast, numerical results show that 1i

=0 when 1r

=0. For Re

= Re1 ,

Figs. 6 and 7 show steady velocity proles f , k , h, , gr , and r at various s. Here, we set k 0and h + = .

TABLE II. Values of Re1 at various s.

s 0.1 0.06 0.03 0.025 0.021 0.02 0.0196 0.019526 0.019525 Re1 103.10 103.46 78.05 61.54 34.43 19.92 8.00 1.13 0.65



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FIG. 2. Type I, velocity proles f ( x ), k ( x ), h( x ), and ( x ) for various values of s at Re3 . =k , = h.

FIG. 3. Type I, normalized instantaneous velocity proles g r ( x )cos 1i t g i( x )sin 1i t at different moments for s = 0.8and 0.21 at Re3 . 1i t =n /4, n =0, 1, 2, . . . , 7, r =gi , i = gr .

FIG. 4. Type I, variation of | 1i| with s at Re3 .



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FIG. 5. Type I, normalized R( x ) = y20 + z20 by its maximum for various s at Re3 .

IV. SUCTION AND INJECTION

In this section, for the type I solution, we consider the inuence of suction ( f (0) < 0, f (1) > 0)and injection ( f (0) > 0, f (1) < 0) of the disks on the eigenvalues. Let us integrate Eqs. (13) and (14) ,yielding

r 1

0(|g|

2

+ | |2)dx =

1

0k |g|

2

12

f (|g|2

+ | |2)

+(h

+)(gr r

+g i i )

+

|

|

2

+

1 Re

(

|g

|

2

+ |

|

2) dx . (31)

For axisymmetric ows, including type I, k =h + 0. Equation (31) is then reduced to

r 1

0(|g|2 + | |2)dx =

1

0 f (|g|2 + | |2)

1 Re

(|g |2 + | |2) dx . (32)

FIG. 6. Type II, velocity proles f ( x ), k ( x ), h( x ), and ( x ) for various values of s at Re1 . k 0, h + = .



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FIG. 7. Type II, normalized velocity proles gr ( x ) and r ( x ) for various values of s at Re1 .

Therefore, we observe that r < 0 when f is always negative. It seems that suction is more likelyto lead to r > 0 than injection, i.e., Re3 may decrease with the increasing suction velocity orincrease with the increasing injection velocity. Two instances are given by numerical calculation,respectively, where the suction and injection on the disks are assumed to be symmetrical ( f (1)

= f (0)) for convenience of discussion. For exact counter-rotating disks( s = 1), the results inTable III indicate that Re3 increases when the boundary conditions change from suction to injection.Moreover, 1r canbe positive andzero forco-rotating disks ( s > 0) with suction on thedisks, whereas 1r is negative without suction. The data in Table IV provide values of Re3 at various suction andinjection velocities in the case of s =0.5.

V. DISK FLOW UNDER NON-AXISYMMETRIC PRESSURE FIELDS

In this section, we consider the ow between two rotating disks under a non-axisymmetricpressure eld ( C C 4 / C 2 =1) and concentrate on C [0, 1) in most cases. The pressure contoursare a series of ellipses ( C

(0, 1)) or degenerate cases parallel lines ( C = 0). C 2 and C 4 are tobe determined by a given ratio of the pressure parameter C , the ratio of rotation velocity s, and the

Reynolds number. It was found that h + 0 for all of the solutions we obtained.

A. 1 s < 0

For C

(0, 1), we found three coexisting solutions (I, II, III) when Re is smaller than a criticalReynolds number Re c , beyond which two solutions (I and II) vanish. Solution III exists for Re (0,

) and passes through a point in the ( Re, C 2) plane where C 2

= C 4

= 0, corresponding to a

zero pressure gradient in the ( y, z) plane. It should be noted that these three branches of solutionsbifurcate out of the two type of solutions at C =1.For the eigenvalue problem, here, we introduce Re1 , Re2 , and Re3 as the Reynolds numbers forwhich 1r =0 for solutions I, II, and III, respectively. It was found that C 2 and 1r of solution II are

TABLE III. Values of Re3 at various suction/injection velocities ( s = 1). f (0) 0.4 0.3 0.2 0.1 0.05 0 0.05 0.1 0.2 Re3 10.61 14.12 20.92 37.52 55.63 88.74 149.74 253.33 941.39



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FIG. 9. Variations of Rec , Re1 , Re2 , and Re3 with C for s = 1. Rec is the critical Reynolds number beyond which solutionsI and II vanish, and Re1 , Re2 , and Re3 are the Reynolds numbers for which 1r =0 for solutions I, II, and III, respectively. Ais the domain for which 1r < 0 for solution I, and B is the domain for which 1r < 0 for solutions III.

opposite C 2 . Figure 8(b) shows ReC 2 curves for C = 1, 0.8, 0.4, 0.2, and 0, and Fig. 8(d)shows Rec for C [ 1, 1].Then, we consider the eigenvalue problem. Here, ( C 0 , Rec0) = (0.9580, 45.3965), as shown inFig. 9. For solution III, if C 0.852, there is a domain in the ( C , Re) plane where 1r < 0; if C 0.852, 1r is always positive. Figure 9 shows variations of Rec , Re1 , Re2 , and Re3 with C . It was

found that Re1 and Re2 decrease as C decreases.

2. 1 < s 0.088

For C

[0, 1), Fig. 10(a) shows C Rec curves for s = 0.8, 0.6, 0.5, 0.4, and 0.2. Recis nite when C =0. Here, we consider the case of s = 0.5 as an example. When C 1, Rec 67.51, at which point, the type II solution vanishes. Solutions III passes through the point (58.13,0). Figure 10(b) displays ReC 2 curves for C = 0.8, 0.4, and 0. Figure 10(c) shows variations of Rec , Re1 , and Re3 with C . Here ( C 0 , Rec0) = (0.9453, 61.9529), as shown in Fig. 10(d) . For solutionIII, if C 0.460, there is a domain in the ( C , Re) plane where 1r < 0; if C 0.460, 1r is alwayspositive.

3. 0.088 s 0.055

Consider the case of s = 0.08 as an example. For C =1, types I and II solutions exist for Re

(0, ), as discussed in Sec. III . It should be noted that the C Rec curve is discontinuous at C 0.9977, from Rec =

144.12 at C

=0.99765 to Rec

=578.94 at C

=0.9977, as shown in Fig. 11 .

Rec when C 1. Here, ( C 0 , Rec0) = (0.9140, 97.2676). For solution III, 1r > 0 when Re < Re3 and 1r < 0 when Re > Re3 . Figures 11(a) and 11(b) present ReC 2 curves for solutions I, II,and III with C = 0.998, 0.995, and 0.99. Variations of Rec , Re1 , Re2 , and Re3 with C are shown inFigs. 11(c) and 11(d) .4. 0.055 s < 0

Consider the case of s = 0.04 as an example. For C = 1, the type I solution exists for Re(0, ); the type II solution exists for Re (0, 103.96) and (152.92, ). We notice that for C 0.936, one pair of solutions (IV, V) appears when Re ( Rec1 , Re c2). Rec1 152.92 when C 1, and Rec2 when C 1. We have ( C 0 , Rec0) = (0.8128, 88.0608), as shown in Fig. 12(c) .



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FIG. 10. (a) Variation of Rec with C for s = 0.8, 0.6, 0.5, 0.4, and 0.2. (b) s = 0.5, ReC 2 curves for solutionsI, II, and III with C = 0.8, 0.4, and 0. The bifurcation points at which solutions I and II vanish are (58.2670, 0.0106),(70.3130, 0.1114), and (103.2659, 0.0540). (c) and (d) s = 0.5, variations of Rec , Re1 , Re2 , and Re3 with C . A is the domainfor which 1r < 0 for solution I, and B is the domain for which 1r < 0 for solution III.

For solution III, 1r > 0 when Re < Re3 , and 1r < 0 when Re > Re3 . For solutions IV and V, 1r is always negative. Figure 12(a) shows ReC 2 curves for solutions I, II, III, IV, and V with C =0.995 and 0.99. Variations of Rec , Re1 , Re2 , and Re3 with C are shown in Fig. 12(c) . Figure 12(b)presents ReC 2 curves for solutions IV and V with C =0.95, 0.94, and 0.936, and Fig. 12(d) givesthe existence interval of solutions IV and V.

As discussed above, for each investigated s [1, 0), there exists a point ( C 0 , Rec0) on theC Rec curve where 1r =0. We plot the velocity proles f ( x ), k ( x ), ( x ), h( x ), ( x ), g( x ), and ( x ) atthese points for selected values of the rotation ratio s = 1, 0.5, 0.08, and 0.04, as shown inFigs. 13 and 14. A numerical calculation shows that 1i =0, meaning there are no oscillatory owsunder these marginal parameters.

B. 0 s 1

For C

[0, 1), we found one solution (I) that exists when Re is smaller than a critical Reynoldsnumber Re1c . Re1c when C 1. This solution should be the natural continuation for type I atC =1. Meanwhile, a new solution (II) appears when Re is smaller than a critical Reynolds number Re2c . Figure 15 shows the case of s = 0 and s = 0.5 as examples. A numerical calculation showsthat 1r < 0 for solution I and 1r > 0 for solution II.

VI. DISCUSSION AND CONCLUSIONS

In this paper, we considered a class of unsteady exact solutions of the Navier-Stokes equationsbetween two innite parallel planes based on the general form of exact solutions that Lin initially



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FIG. 11. s = 0.08. (a) ReC 2 curves for solutions I and II with C =0.998, 0.995, and 0.99. The bifurcation points at whichsolutions I and II vanish are (632.38, 0.0200), (124.75, 0.0148), and (116.51, 0.0132). (b) ReC 2 curves for solutionsI, II, and III with C =0.998, 0.995, and 0.99. This is locally enlarged for (a). (c) Variations of Rec , Re1 , Re2 , and Re3 with C

[0, 0.9977). (d) Variation of Rec with C [0.9977, 1).

found. As the superposition of (0, gn( x )exp ( n t ), n( x )exp ( n t )), each eigenmode of which givesrise to spiral or elliptical oscillation, the unsteady velocity components can be added to the steadybase ow.

Under homogeneous boundary conditions, the solutions of gn( x ), n ( x ), and n correspond toa vectorial Sturm-Liouville eigenvalue problem. We focused particularly on the properties of theeigenvalues 1 = 1r + i 1i with the largest real part. The unsteady velocity components grow toinnity when 1r > 0 or decay to zero when 1r < 0. A periodic or steady solution exists when 1r =0.The most intriguing feature of the ow is that the pressure and boundary conditions can besteady, whereas the velocity is unsteady. In this ow, the unsteadiness is completely caused by theimbalance between the convective acceleration, the viscous term, and the steady pressure gradient,and we wonder whether other exact solutions have the same property. It should be emphasized that

although the unsteady components can be regarded as a type of linear disturbance, the strength of theunsteady components is unlimited. This property comes from the fact that the unsteady disturbanceitself is free of acceleration and hence does not produce a high-order nonlinear term.

The ow between two innite rotating disks was investigated. For the ow under an axisym-metric pressure eld ( C 2 =C 4), the inuence of the rotation ratio s, the Reynolds number Re (basedon the angular velocity of the bottom disk), suction and injection on the disks, and the pressureparameter C 2 were numerically studied for two types of base ow: axisymmetric ow (type I) andnon-axisymmetric ow (type II). In the cases without suction and injection, phase diagrams contain-ing the neutral curves were presented. It was found that the periodic oscillation can occur in the baseow of type I when 1 s 0.206. The circular oscillation frequency | 1i|, which depends ap-proximately linearlyon s as | 1i|0.1576( s +1), is substantially smaller than the frequency scale .



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FIG. 13. Velocity proles f ( x ), k ( x ), ( x ), h( x ), and ( x ) for s

= 1,

0.5,

0.08, and

0.04 at ( C 0 , Rec0), where 1r

=0.

(C 0 , Re c0) is (0.9580, 45.3965), (0.9453, 61.9529), (0.9140, 97.2676), and (0.8128, 88.0608), respectively. The pressureparameter C 2 is 0.08206, 0.04140, 0.00684, and 0.00390, respectively.

FIG. 14. Normalized velocity proles gr ( x ) and r ( x ) for s = 1, 0.5, 0.08, and 0.04 at ( C 0 , Rec0 ), where 1r = 1i=0.



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FIG. 15. (a) s =0. (b) s =0.5. Variations of Re1c and Re2c with C . Re1c and Re2c are the Reynolds numbers beyond whichsolutions I and II vanish, respectively.

It should be noted that in this paper, we only considered the branches of the solutions at small ormoderate values of Re because of the tremendous difculties with exhausting all branches at largevalues of Re . Meanwhile, much attention was paid to the eigenvalue properties rather than to theow structures. On the other hand, more axisymmetric and non-axisymmetric base ows may beobtained by solving Eqs. (5)(11) or Eqs. (24) (27) under injection or suction boundary conditionsand possible dynamic pressure parameters to study the possible additional periodic oscillating owsand/or other more complicated unsteady ows under a more extensive background.

The solutions given by Lin can be regarded as the local Taylor expansion for the generalincompressible viscous ow; thus, many ows in the vicinity of some crucial locations of particularinterest can be modeled approximately using this solution. The oscillation ows are of specialsignicance in nature and industrial applications. The unsteady exact solutions that we proposed

may correspond to the real oscillating ow or describe the local oscillation of certain complicatedows when theboundaryconditions andpressureparameters areappropriatelygiven. A fewexamplesinclude the spiral vortex breakdown, the low-frequency pressure uctuation in certain giant Francishydraulic turbines, and certain oscillations in the atmosphere and the oceans. Further work onbridging real oscillating ows and the unsteady exact solutions proposed in the present paper isrequired.

ACKNOWLEDGMENTS

The authors acknowledge the discussions with Professor Jian-Jun Tao and Professor Qing-DongCai. We are indebted to two anonymous referees for valuable comments. This work is supported byMOST of China (Grant No. 2009CB724100) and NSFC (Grant No. 10921202).

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Exact solution of Navier-Stokes equations

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