VETENSKAPOCH

KONST

KTH

TRITA-A SE0100210ReportISSN 1102-2051ISRN KTH/ALF/R--01/1--SE

On Analytical Solution of theNavier-Stokes Equations

J. Scheffel

Research and Training programme onCONTROLLED THERMONUCLEAR FUSION

AND PLASMA PHYSICS(Association EURATOM/NFR)

FUSION PLASMA PHYSICSALFVEN LABORATORY

ROYAL INSTITUTE OF TECHNOLOGYSE-100 44 STOCKHOLM SWEDEN

TRITA-ALF-2001-01ISRN KTH/ALF/R-01/l-SE

On Analytical Solution of theNavier-Stokes Equations

J. Scheffel

VETENSKAPOCH

KONST

Stockholm, April 2001

The Alfven Laboratory

Division of Fusion Plasma Physics

Royal Institute of Technology

SE-100 44 Stockholm, Sweden

(Association EURATOM/NFR)

ON ANALYTICAL SOLUTION OF THE NAVIER-STOKES EQUATIONS

J. Scheffel

Fusion Plasma Physics, Alfven Laboratory (Association EURATOM-NFR)

Royal Institute of Technology, SE-100 44 Stockholm, Sweden

http://www.fusion.kth.se

ABSTRACT

An analythical method for solving the dissipative, nonlinear and non-stationary Navier-Stokes equations is presented. Velocity and pressure is expanded in power series ofcartesian coordinates and time. The method is applied to 2-D incompressible gravitationalflow in a bounded, rectangular domain.

Keywords: Navier-Stokes equations, incompressible flow, viscous flow.

1. A LONG STANDING PROBLEM

The Navier-Stokes (NS) equations have since long been identified to model such diversephenomena as weather changes, ocean flow, airplane motion and thermonuclear plasmas. They arecentral in the study of naturally occuring turbulence and have thus been intensively studied.

So far only numerical solutions have been attainable [1]. Intense theoretical study of theequations has, however, been carried out since their early formulation by Louis Navier and GeorgeGabriel Stokes. The following theoretical results are presently established [2].

In two dimensions (plane-parallel flow) there exists a unique solution to Eqs.(l) at allinstants of time independently of the size of the gravitation g, the initial condition or the domainS. The proof holds for generalized (or weak) solutions, but it has also been shown that thegeneralized solution becomes a classical (smooth) solution if g (or more generally, the externalforce) is Holder continuous inside S.

Since this is the case here, we can assume the solution to be real and analytic. It is, in thepresent method, expanded as a power series in x, y and t within the domains x 6 [0,1], y e [0,1]and t e [0,°°[. Since the solutions to Eqs. (1) are unique, convergence can be determined from therequirement that the error should remain less than some number e when further expansion termsare added.

From a physical point of view, smooth solutions are both desired and expected. TheNavier-Stokes equations are indeed routinely solved numerically in applications ofhydrodynamics and magnetohydrodynamics (MHD), since they appear in the one-fluid equationsobtained by taking velocity moments of the more fundamental Boltzmann equation.

Dissipation in the form of viscosity is central in the study of the NS equations and isretained (in scalar form) throughout this study. We will assume that the fluid is incompressible.

The success of the present expansion method for a bounded domain depends on threefactors. First, the established existence and uniqueness of solutions to the initial-value problemwith the boundary condition v = 0 (no slip or normal flow at the boundary). This removes anyproblem associated with bifurcation points in the solution algorithm. Second, it was essential todetermine solutions of diw = 0 that satisfy the boundary conditions for all orders of theexpansion. Third, modern symbolic computer math programs (like MAPLE), running on fast,high-memory computers are required; the analytic method suggested here certainly is intractablefor "hand" calculations.

2. THE 2-D, VISCOUS INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

We wish to solve the equations

p ( + (vV)v) = V/> + vAvge>, „ .

V v = 0

in the region V of Fig. (1) with boundary S, with v denoting viscosity, g gravitational acceleration,A the Laplace operator and ey a unit vector in the positive y-direction. The boundary condition isthat v = 0 on S (no normal flow, no slip at boundary due to viscosity). The initial condition v = Vomust satisfy vo = 0 on S and V- vo = 0 in V. Possible forms for Vo are discussed in Section 4.

The gravitational term is of potential form, enabling the substitution p - h-gy. Afternormalization we obtain the following scalar equations, implicitly containing gravitation;

dh . ,d\ (?V

dy

dv,, dv,, dv,, dh 3Y, 92v

V —— + V —— = 1-Vl — H —)

3. EXPANSION METHOD

The velocity v = (vx,vy) and potential h are expanded as power series in cartesian coordinates x, yand time t:

k,m,n

E K^y"? (3)k,m,n

X dkmnxkymt»

The desired order of expansion must be given. The expansion order (K,M,N) is here definedas the order in x, y, and t up to which Eqs. (2) hold identically. From Eqs. (2) it is apparent thatlow order coefficients couple to higher order coefficients because of space and time derivatives.Thus the expansions (3) must be carried out to higher order than (K,M,N), this order being givenby the degree of the derivatives operating on vx, vy and h, respectively.

When a solution is obtained, there will thus remain free higher order coefficients in (3). Thismakes the solution multi-valued, although exact to the given order. If the method converges, theremaing coefficients will have diminishing influence at higher orders.

4. NO SIMPLE SOLUTIONS

We will here show that there are no simple solutions to Eq.(l). This will become clear as wediscuss the possible forms for the initial velocity field.

Let us, for clarity, introduce the time series

(4)

Using Eq. (2), there results

S , f "JO , „ "JO _ " " 0 , , , ( " JO , " JO \

dx dy dx dx dy

,~.

,x dy

We now wish to determine possible initial conditions. We immediately exclude the cases/0 = 0, go* 0 and/) * 0, g0 = 0, since they do not satisfy both the zero order incompressibility condition in(5) and the boundary conditions.

A. Zero initial flowNow assume that there is no initial flow; \(t=0) = 0. Thus/) = go = 0. We obtain

dy dx (6)

dx dy

and, using the Laplace operator,

A/; =o(7)

Ag,=0.

Since v = 0 on S, we must have/i = g\ = 0 on S. A well known theorem of potential theory thenstates that Eq.(7) implies that/j = g\ = 0 in V. Thus, if the initial flow is zero, the flow must bezero everywhere for all times.

This result may seem surprising, since one can imagine the physical situation that thesystem is started from a static state with a spatially varying pressure gradient, which thus woulddrive a flow in the fluid.

B. Finite initial flowWhat classes of initial flow satisfy Eq.(5) and the boundary conditions? In order that theboundary conditions should be satisfied, we must require that

/ 0 ° c ^ ( i - x ) V 3 ( l - ; 0 i 4 (8)

where kh i = 1..4, are integers > 1. The ansatz (8) is, however too simple to satisfy theincompressibility condition dfo/dx + dgo/dy = 0. The reason is that

which expression does not satisfy the boundary conditions at y = 1. However, on differentiating(8) w.r.t. x, there results

x \ \ x ) y ( l y ) ( k , ( l x ) - k 2 x ) . (10)dx

This expression can be used for g0, since then fo = - j(dg(Jdy)dx is exactly integrable. Hence, thesimplest possible initial conditions that satisfy both the incompressibility and the boundaryconditions, are

where Kh i = 1..4, are integers > 2 in order to satisfy the boundary conditions. Any linearcombination of these initial conditions is also a possible initial condition.

C. Higher ordersUsing the result (11) in the first two of Eqs. (5), we observe that it is required that Kt > 4 for theviscous terms to satisfy the boundary conditions. The simplest relevant initial condition for firstorder (in time) accuracy, omitting multiplying constants, is thus

= -x\l-xfy\l-y)\\-2x).

This in turn forces the nonlinear convective terms to include terms of order (14,15,N) and(15,14JV) in Eqs. (5). We thus see, that already f\ and g\ must feature a considerable complexity.By similar reasoning, higher orders of ft and gt feature even higher degrees of complexity anddetail. This sheds some light on the difficulty to obtain closed analytical solutions to the NSequations.

5. ANALYTICAL METHOD

In essence, the analytical method consists in substituting the expansions (3) in Eqs.(2), whereafterthe coefficients are determined to desired order.

A problem of the present complexity (three variables, nonlinear, high orders) requires acomputer math program. Large resources of primary memory and high performance processorsare also essential. In this work MAPLE is used on Mac G3 and Sun Blade 1000 platforms.

First, boundary and initial conditions are imposed. These are v(0,y,0 = \(\,y,f) = \(x,0,i) =v(x,l/) = 0, and \(x,y,0) = vo = (fo,go), respectively. A method for successive solution of theassociated linear equations for the coefficients in (3) is used.

An algorithm has been written to determine the coefficients of (3) from each power of theNS equations. The nonlinearity does not generate spurious solutions due to the provenuniqueness of the problem and the design of the algorithm. The solution contains viscosity v andcoefficients in v0 as free parameters.

The combined Eqs.(2) and (3) can also be solved as a system of recurrence relations for thecoefficients. Since this system may be nonlinear and implicit, it was found to lead to tedious andoperations requiring high memory-consumption.

6. RESULTS

An analytical solution to Eqs. (2), being correct to order (8,8,1), has been determined using thesimple initial condition (12) multiplied with an arbitrary magnitude C. The resulting orders of vx,vy and h are thus (10,10,2), (10,10,2) and (9,9,1), respectively. We have further imposed thecondition h(l,y,f) = 0. A solution, where the remaining coefficients have been set to zero, is givenin Appendix. It is clearly seen that the boundary conditions are satisfied for all times. Viscositybecomes included up to second order. It should be noted that the present "solution" is onlycorrect upp to order (8,8,1) and is probably not representative for an exact solution of the NSequations with initial condition (12). It serves to show, however, that the present method worksalgorithmically and it also gives some insight into the structure of solutions to the NS equations.

In Figs. 2-4 the solution with C = 0.001 and v = 0.1 is displayed for various times. It maybe noted that the changes in velocity appear on a faster time scale than changes in pressure. Thismay also be seen by comparing the global potential energy of the system W'= JJ hdxdy with itsglobal kinetic energy Q = l/2\\ (vx

2 + vx2)dxdy. We find

W = const + 0.001351vC - 0.02846C2 - Cv(0.8700C + 0.2232v)f (13)

2 n (2.549 -10"7C-l. 185 -10"*v)f

]

The potential energy is continuously decreasing in time, as expected. The kinetic energy may belinearly decreasing in time but nonlinearly increasing in time. For C = 0.001 and v = 0.1, Qincreases for all times except very early.

One should note that energy is not conserved in the system because of the dissipationbrought about by viscosity. The sum of W and Q is thus not a constant.

7. DISCUSSION

A method for obtaining analytical solutions to the viscous, nonlinear and non-stationary Navier-Stokes equations has been demonstrated. Using power expansions in x, y, and t, the two-dimensional motion of a fluid under the influence of gravity is computed analytically, satisfyingthe Navier-Stokes equations up to eighth order in space and first order in time.

The algorithm is fairly efficient, but need more development to avoid computer memoryconsumption much above 100 Mb. It may then be used for solving the three-dimensional Navier-Stokes equations, for which less is known analytically.

8. REFERENCES

The literature in this field is enormous. Every week 20-30 articles, related to the NS equations, arepublished. These two references summarises numerical and theoretical aspects of the problem:

[1] C. A. J. Fletcher, Computational Techniques for Fluid Dynamics,Vols 1 &2, Springer, 2000.

[2] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,Gordon and Breach, 1969.

9. APPENDIX

-xy ( - l + y ) ( x - l ) C ( - 4 7 0 4 0 x V +211680 vx5? + 70560x3 n11760'

+ 1270080 xt2 v2- 70560x6tv-6511680x6 ? v2 + 30764160y612 v2 - 24978240x417 v2

+ 30764160 y5 ? v2 - 11760 x6 / - 176400 x4 y4 - 4480 x8 tv + 40320 x1t2 v7

- 7922880 y7 f v2 + 47040 x6 / - 7197120 x2 i7 v2 - 2634240 x3 / tv + 1599360 x2 y5 n<

+ 53760x2 y6 ? v + 32034240x/ t2 v2 + 388080 x*/tC + 635040x6y tv

- 3405696 x4 y4 rv - 14978880 x6 y2 *2 v2 - 235536 x7 y5 f v - 868864 x6 y8 t v

+ 1905120 vx4> f+ 1962128xV f v - 3 5 2 8 0 x 4 / +625737x8 / f C - 2040384x 6 / f v

- 236928 x 6 / t >'+ 298224612x s / t7 v C + 6437235x7 / / C - 211680 vx41

- 1 5 6 4 0 8 x 5 / t C + 298224612x8ys t2vC+ 167233248x7y417 v2 + 23567040x2y612 v2

~ 32901120 x4 y1t7 v7 + 40320 xs f v2 + 48507498 x8 / t7 v2 + 1270080 xyt2 v2

- 196224 x4 / t v - 67200 x3 / t v + 4445280 x5 y2 t v + 2798880 rx3 y2 v

- 33445440 x4y2 f v2 - 1607920 x8 y1 tv + 1715168 x5 / t v - 29211840x 5 / t7 v7

+ 48507498 x8 / t2 v2 - 14434560 x6 / f v7 + 61824 x5 y6 tv + 35138880 xy312 v2

-498960 x6 y7 tC+ 89376x7y5 t C-498960x6ys t C- 15120000x2ys f v2

- 564480x8y f2 v 2 + 1 1 7 1 5 2 0 x / f v + 1 2 0 3 4 4 x s / / C - 1 5 6 8 0 x 8 y 3 t C - 3684303 x^y1 tC

- 3147564x8 / f C - 6511680x6 y t2 v7 + 1219008 x 6 / t v+ 1938048x 4 / t v

- 26880 x7 / f v - 6720 x7 y2tv + 58800 x7 y3 ?v - 208320 x2y1tv- 806064 x4 y7 i v

+ 260400 x3 y7 tv + 543648 Xs y7 tv + 336000 x6 y4 tv - 6348416 x8 ys tv

+ 5785920 x4y612 v2 + 29967840 x7 y2 ? v2 - 6652800 xy7 t7 v7 + 19051200 x5 y t2 v2

-54774720x6 y4 f2 v2 + 356028x7y7 tv + 52920000 x5y'1 f v2 + 52073280 x3y312 v2

- 132160 xsy2tv+ 393344 xs y41 v + 1431920 x3 y8 tv + 48968640 x3 y6 ;2 v2

+ 32034240 xy612 v2 - 587440 xs / t v - 125340768 x7 / t2 v2 - 564480 x7 y t2 v2

+ 167233248 xs y4 f v2 - 14434560 x6 y8 f v2 + 18204480 x3 y t2 v7 - 24978240 x4 y t7 v2

+ 23567040 x2/fv7 + 33868800 y3 f v7 - 46992960 x y4 f v2 - 73241280 x4 y412 v7

- 114095520 x8 y312 v7 + 705600 v x y41 + 6585600 / / t v - 6256320 x3 y3 tv

- 564480 vxy3 t + 25872 x6/ tC+ 2127195x7y7 f C+ 11128320x5 y7 f2 v2

- 156408 x V t C+ 26671680x2y3 r2 v2 + 19051200x517 v2 - 3998400x2y41 v

- 7197120 x/f v2 + 57696 x7 / tv + 1737540 x7 /12 v C+ 1737540 x8 /fvC

+ 92117844x8 / t7 v C+ 298224612x7y7 t2 vC+ 3198720 f x2y3 v+ 141120 vxy2 t

- 282240 xy5 t v -7922880 ys ? v7 + 48968640 x3 / f v2 + 9737280 x3 y2 t2 v2

- 114095520 x7 y312 v2 - 15120000 x7 / f v7 + 48507498 x7 y8 r2 v2 + 29967840 x8 y2 t2 v7

- 18878328 x8 y4 t7 v C+ 298224612 x7 y8 t2vC+ 92117844xsy6 i2 v C

+ 92117844 x7 y6 t7vC- 125340768 x7 /12 v7 + 18204480x3 f2 v2 + 10584000 x5 y2 t7 v2

+ 517440 x6/tv+ 1171520 ys tv + 27357120 x6 y3 f v2 - 125340768 x8 y6 f v2

+ 11128320 x5 y817 v7 + 49815360 x*y6 ? v2+ 92117844x7y5 t7 vC~ 1428000x6y2 t v

+ 11760 x3 y2 - 634116 x7y6 t C- 113232 x6 y 6 1 C - 8467200 y2 t7 v7 - 310464 x5y5 t v

- 32901120 x4 y812 v2 + 10281600 x3 y8 ? v2 - 1905120 x5 y tv + 56448 x5 y4 f v

- 7197120 x2 y ?2 v2 + 8890560 x4 y3 f v2 + 49815360 x5 / f v2 - 125340768 x8 / f v2

- 48263040 y4 f v2 - 328672 x8 / t v - 30058560 x3 y4 t2 v2 - 4384468 x7 y 8 1 v

- 15664320 x2 y2 t7 v2 + 5785920 x4 / f v2 - 6652800 x y 8 f v2 + 82880 x 8 y ( v

- 799680 t x2 / v + 963200 x2 y*tv + 365456 x4 / t v + 24252480 x6 y6 I2 v2

+ 48507498 x7 / t2 \>2 + 10281600 x 3 / f v2 - 10584x 4 / t C- 1 0 5 8 4 x 4 / tC

- 55460160 x2 / f v2 - 635040 x3 y tv + 5597760 x4 / fv - 18878328 x7 / 1 2 v C

- 2963520 x5 / t v- 5103840 x 4 / f v + 35280 x 3 / + 141120 x 4 / + 58800 x3 /

+ 23520 x6 / - 23520 x3 / - 70560 x5 / - 58800 x6 / + 176400 x5 / + 70560 x4 /

- 141120x5/ + 24252480x6/ t2 v2)

= T^7^xy (-1 + y) ( x - 1) C( 7197120/ ?2 v2 + 211680/1 v + 70560 / tv11760

+ 8467200 x2fv2+ 181440 / f2 v2 - 211680 / t v - 1270080 y t2 v2 - 18204480 / t2 v2

- 1176000 x81 v + 3366195 x7 / t C + 18404064 x5 / f v2 - 70560 / tv + 24978240 / f2 v2

- 30764160 x6 r2 v2 + 181440 / f v2 - 4445280 x2 / t v - 25595136 x4 / ? v2

- 134769600 x4 / f v2 + 67200 xy* t v - 383705280 x2 / t2 v2 - 14055552 x5 / t2 v2

+ 7599543 xs / t C + 47370960 x8 y412 v2 + 167308596 x7 / t2 v C - 1270080 xyt2 v2

+ 11340000 x3 / f v2 + 26880 x6 / t v - 420672 x4 / t v - 2782080 x2 / r2 v2

- 25492320 x7yr2 v 2-75264 x 4 / f v - 310464 xs/tv + 910968 x 8 / f v

- 14597226 x7 / t2 v2 - 134769600 x6 / f2 v2 - 25492320 x8 y t2 v2 + 117321876 x6 / f2 v C

- 2798880 t x2 / v + 46569600 x5 y ?2 v* - 38976 x4 / t v - 2247264 x7 y61 v

+ 21611520 x3 / f2 v2 + 46569600 x6 y t2 v2 + 12700800 x5 / t2 v2 - 677376 x5 / ?v

-503580 x7 / t C + 7 5 6 0 0 x s / t v - 9 8 4 4 8 x 6 / ? C + 1481760x 2 / t v- 1417920x 3 / f v

+ 253680 x1 / t v - 25872 C / f x5 + 282240 x5 y tv + 28224 f Cx7 / - 20160 x / tv

- 635040 x / t v + 981456 x7 / t v - 11760 x2 y3 + 388080 x7 / ? v C + 70560 x5 y5

+ 117321876 x 5 / t2 v C-70560x5/+ 117321876x6 / r2 v C - 176400x 4 /

+ 5880 x3 / t C + 388080 x s / t2 v C+ 23520 x5 / + 3292800 x4 / f v + 11760 x2 y6

+ 6560316 x4 / f v C - 47040 x3 / + 58800 x4 / - 23520 x5 / - 182784 x6 / f v

- 141120 x7ytv + 20160 x7 y tv + 6039936 x 4 / t2 v C+ 6560316 x 4 / f v C+ 176400 x 4 /

+ 1640520 x5 / f2 v C- 58800 x4y3 + 1640520 x3 y8 ( 2 v C + 47040 x3 y3

+ 7472304 x5 / t2 v C - 141120 x3y4 + 4148333 x7 ys t C- 19051200 y5 r2 v2

+ 98189280 x8 y2 i2 v2 - 147364560 x7 y3 ? v2 + 89959968 Xs y6 /2 v2 + 26931744 x7 / t2 v2

+ 1646400 x8 y tv + 6350400 x / t2 v2 + 18404064 x6 y8 f2 v2 - 14597226 x7 y7 f2 t-2

+ 220147200 x4 y2 f v2 + 24978240 xy4 f v2 + 1471344Xs y8 f v - 19051200 xy5 ?2 v2

+ 12700800 x6 y2 f2 v2 + 48263040 x4 ? v2 + 18404064 x6 y7 t2 v2 - 14597226 xs / t2 v2

+ 107956800 x5 y4 I2 v2 -5597760 x3 y4 tv + 635040 v x y 3 1 - 1905120 vxy 4 f

- 1646400 x8 /tv+ 1360464 xs y5tv+ 1300560 xs y6 rv - 1089564 x8 y5 t C

+ 31187520 x3 / t2 v2 + 26931744 xa/?v2 + 95844 j ^ / f C - 2782080 x2 ys f2 v2

- 1908256 x6 / t C + 5103840 x2 y4 fv + 103096 x4/tC + 377300 x5 ys t C

+ 607656 x6 y7 t C- 18204480 xy312 v2 - 1141056 x5 y6 tv - 175812 x7y5 f C

- 705600 v x4 y f + 344064 x4 / tv + 3057600 x3 ystv + 6256320 x3 y3 t v - 241920 x7 y2 t v

+ 40320 x2 y7 f v + 78960 x3 y7 tv + 3844764 x8 y6 f C - 33868800 x3 f2 v2 + 56448 x6 / f v

+ 6350400 y6 i2 v2 + 89959968 x7 / f2 v2 + 6552000 x7 r2 v2 - 379008 x7 / t v - 35280 x2 /

- 160876800 x4 y t2 v2 - 25595136 x4 y8 t2 v2 - 14055552 x6 y5 t2 v2 + 35280 x2 /

+ 181440 xy112 v2 + 98189280 x1 y212 v2 +7197120 x y212 v2 - 134769600 x5 y312 v2

+ 127260x1 y1tv+ 107956800x6y412 v2-59040 x6y11 v + 4181184x4y6 f v2

+ 47370960 x1 y4 f v2 - 531552 x6 / tv + 18404064 x3 ys f v2 - 147364560 Xs / t2 v2

+ 107956800 x4 y4 t2 v2 - 46992960 x2 y5 ? v2 - 1599360 x5 y2 t v + 564480 x3 y t v

+ 205851744 / / t2 v C+ 167308596 x* / 1 2 v C- 4260060 x* y312 v C

+ 167308596 x* y1 t2vC- 5794740 x1 / t2vC- 3198720 t r5 / v - 6585600 x4y3 t v

+ 211891680 x7 y6 t2vC- 5794740 x*>412 v C+ 167308596 x1 ys t2 v C

- 4260060 x1 y f v C+ 211891680 x8 y6 fvC- 14597226 jr8 y1 f v2 + 2540160 x5 y3 tv

+ 3998400 x4y2tv+ 205851744x1y5 i2vC + 24978240x3 / f v2 + 232424640 x2 y4 f v2

+ 48972672 x5 y612 v2 - 33868800 x3 / ? v2 - 58847040 / y512 v2 + 48972672 x6 y612 v2

+ 799680 t x2 / v + 1905120 x/tv- 147329280 x3 y2 t2 v2 + 8487360 x2 y6 t2 v2

+ 11340000 Jy1 f v2 -69007680 fyt2 v2 +473472 x6 y6 tv + 123621120 x3 y t2 v2

+ 252745920 x2 y2 t2 v2 + 425040 x3 y8 t v + 1232328 x7 ystv + 2016036 x8 y8 t v

- 30764160 x5 r2 v2 - 134400 x2 y8 tv + 181440 x y8 ? v2 + 6552000 x812 v2

+ 1432368 x6 y5 i2 v C+ 117321876 x5 y7 t2 v C+ 1432368 x5 y5 t2 v C + 7472304x6 y6 f v C

+ 141120 x3 y5 + 8381681x8ys f C + 23520x8 y4 ? C)

| x 8 y 4 C2 + 54x8y5 C 2 + | y » x 4 C 2 - ^ p x 5 y 8 C 2 + ^ y ? x 5 C2 - ^ - ^ x 7 y 7 C2

f 111 4v+ 480x 2 y 3 C+168x 6 y C + 4 8 2 4 x 2 y 5 C - 5 0 4 0 x 2 y 6 C + — y ? C + — C + 2 4 y 5 C

10

+ 5040 x6 / C + 11760 x6 / C- 168 xy6 C - 10920 x4 / C - 4824 x5 y2 C + 9072 x5 y5 C9792

- 1 2 0 x / C + 4 8 x / C + 2 1 6 x / C + — - ^ / C - 4 8 0 x 3 / C - 5 0 4 0 A-3/C

-2400 / / C + 3120x2/ C+ 24Cxy3 +7152x3y5 C+ 1260x8/ C + 4320x1 y* C1404 24

- 5280x3 / C + - y - J t 3 / C - 24x3)- C-3120x7/C- 2 7 6 x * / C - y

24- 1260x2/ C+ 2400x*y2C+ 12600x4/C-8040x4/C + —x8yC+ 5880x4

+ 2160x3 / C+120x4> C-48x7yC-16800x'/C-3360 x 7 /C-2352x^/40

- 11200 x6y5 C-216x5yC+ 13632x5/ C - 8 0 / C- 6 0 / C + 60x8C-—

^ i / x 4 C-536/x» C+6800/x 3C-840x s /C + ̂ / x » C

+ 80x6c|v2 If

11

10. FIGURE CAPTIONS

Fig. 1 Fluid domain V with boundary S.

Fig. 2 Pressure, shown as h(x,y,t) with h(x,Q,0) = 0. Here C = 1.0-10"3 and v = 0.1.a) t= 0, b) t = 0.01, c) t = 0.1, d) t = 0.5.

Fig. 3 Velocity v^xj/,0 for C = 1.0-10'3 and v = 0.1.a)t=O, b) /= 0.0001, c) / = 0.001, d) t= 0.01.

Fig. 4 Velocity Vjfoy,/) for C = 1.0-10-3 and v = 0.1.a) t = 0, b) f = 0.0001, c) / = 0.001, d) t = 0.01.

12

Fig. 1

x ^

' v

1

gr

13

Fig. 2

a) b)

c) d)

14

Fig. 3

a) b)

c) d)

15

Fig. 4

a) b)

c)

16

TRITA-ALF-2001-01ReportISSN 1102-2051ISRN KTH/ALF/R 01/1-SE

ON ANALYTICAL SOLUTION OF THE NAVIER-STOKES EQUATIONS

J. Scheffel

ABSTRACT

An analythical method for solving the dissipative, nonlinear and non-stationary Navier-Stokes equations is presented. Velocity and pressure is expanded in power series ofcartesian coordinates and time. The method is applied to 2-D incompressible gravitationalflow in a bounded, rectangular domain.

Keywords: Navier-Stokes equations, incompressible flow, viscous flow.

17