On Global Solution of Incompressible Navier-Stokes Equations

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On Global Solution of Incompressible

Navier-Stokes Equations

Algirdas Antano Maknickas

Abstract—The fluid equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow. Due to specific of NS equations they could be transformed to full/partial inhomoge-neous parabolic differential equations: differential equations in respect of space variables and the full differential equation in respect of time variable and time dependent inhomogeneous part. Finally, orthogonal polynomials as the partial solutions of obtained Helmholtz equations were used for derivation of analytical solution of incompressible fluid equations in 1D, 2D and 3D space for rectangular boundary. Solution in 3D space for any shaped boundary is expressed in term of 3D global solution of 3D Helmholtz equation accordantly.

Index Terms—Analytical solution, Incompressible fluid, Helmholtz equation, Navier-Stokes equations.

I. INTRODUCTION

In physics, the fluid equations, named after Claude-Louis Navier and George Gabriel Stokes, describe fluid substances motion. These equations arise from applying Newton’s sec-ond law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow. Equations were introduced in 1822 by the French engineer Claude Louis Marie Henri Navier [1] and successively re-obtained, by different argu-ments, by a several authors including Augustin-Louis Cauchy in 1823 [2], Simeon Denis Poisson in 1829, Adhemar Jean Claude Barre de Saint-Venant in 1837, and, finally, George Gabriel Stokes in 1845 [3]. Detailed and thorough analysis of the history of the fluid equations could be found in by Olivier Darrigol [4]. The invention of the digital computer led to many changes. John von Neumann, one of the CFD founding fathers, predicted already in 1946 that automatic computing machines’ would replace the analytic solution of simplified flow equations by a numerical’ solution of the full nonlinear flow equations for arbitrary geometries. Von Neumann suggested that this numerical approach would even make experimental fluid dynamics obsolete. Von Neumann’s prediction did not fully come true, in the sense that both analytic theoretical and experimental research still coexist with CFD. Crucial properties of CFD methods such as consistency, stability and convergence need mathematical study [5].

Manuscript received December 01, 2013; revised February 14, 2014. This work was partly supported by the Project of Scientific Groups (Lithuanian Council of Science), Nr. MIP-13204.

A.A. Maknickas is with the Institute of Mechanical Sciences, Vilnius Ged-iminas Technical University, Saule˙_{tekio al. 11, LT-10223 Vilnius, Lithuania}

e-mail: (algirdas.maknickas@vgtu.lt).

Aim of this article is to propose new approach for solution of incompressible fluid equations. The article has three basic parts: first part explains how to solve NS in one dimension, second part extend solution into two-dimensional space and, finally, third part summarize with three-dimensional space.

II. PARABOLIC FORMULATION OF EQUATIONS

Incompressible fluid equations are expressed as follow

ρ

∂v

∂t + (v· ∇)v

−µ∆v+∇p = f, (1)

∂ρ

∂t +∇ ·(ρv) = 0, (2)

where equation (2) for incompressible flow reduces to dρ_dt = 0orρ=constdue to∇v= 0. Equations of fluid motion (1) could be expressed in full time derivative replacing covariant time derivative by

d dt =

∂

∂t+ (v· ∇). (3)

So, we obtain

dv

dt −a

2_∆_v₌ 1

ρ(−∇p+f). (4)

3 inhomogeneous parabolic like equation for full time deriva-tive, wherea=pµ/ρ.

III. ONE DIMENSIONAL INHOMOGENEOUS SOLUTION

Consider the initial-boundary value problem for v =

v(x, t)

dv dt −a

2_∆_v ₌ 1

ρ(−∇p+f)inΩ×(0,∞), (5)

v(x,0) = v0(x)x∈Ω, (6)

∂v

∂n = 0 on∂Ω×(0,∞), (7)

wherep=p(x, t)andf =f(x, t),Ω⊂Rn_{, n the exterior} unit normal at the smooth parts of∂Ω,a2_{a positive constant}

andv0(x)a given function.

So according to [6] equation (4), when x is normed to a= 1, could be rewritten as follow

dv

dt =vxx+Q(x, t), x∈Ω, t >0. (8)

We expand v and Q in the eigenfunctions sin (nπx) on space Ω ∈[0, L] where sin(nπx

L ) and sin( mπx

L ) functions orthogonality could be applied. So, we obtain

Q(x, t) =

∞

X

n=1

qn(t) sin (

nπx

L ), (9)

Proceedings of the World Congress on Engineering 2014 Vol II, WCE 2014, July 2 - 4, 2014, London, U.K.

ISBN: 978-988-19253-5-0

ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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with

qn(t) =

1

I1 Z

Ω

Q(x, t) sin (nπx

L )dx, (10)

I1 =

Z

Ω

sin2(nπx

L )dx (11)

and

v(x, t) =

∞

X

n=1

un(t) sin (

nπx

L ). (12)

Thus we get the inhomogeneous ODE

˙

un(t) +

nπ

L 2

un(t) =qn(t), (13)

whose solution is

un(t) =un(0) exp (−(nπ/L)2t)

+

t

R

0

qn(τ) exp (−(nπ/L)2(t−τ))dτ, (14)

where

un(0) =

1

I1 Z

Ω

v0(x) sin ( nπx

L )dx. (15)

Again, we substitute all obtained equations into (12), denote

sin2(x, y) =sin(πx)sin(πy)and have

v(x, t) =

R

Ω

ds(v0(s)(

∞

P

n=1 1

I1sin

2₍ns L,

nx

L) exp (−(nπ/L)

2_t_{)) +}

t

R

0

dτ Q(s, τ)

× ∞

P

n=1 1

I1sin

2₍ns L,

nx

L) exp (−(nπ/L)

2₍_t

−τ))). (16)

Now we must apply continuity condition

∇ ·v=∂v(x, t)

∂x = 0. (17)

This is equation of extreme for coordinate x. Solving this equation gives extreme point xex. Finally, solution of 1D incompressible Navier-Stokes equation is

v(x, t) =

R

Ω

ds(v0(s)

∞

P

n=1 1

I1sin

2₍ns L,

nxex

L ) exp (−(nπ/L)

2_t_{) +}

t

R

0

dτ Q(s, τ)

× ∞

P

n=1 1

I1sin

2₍ns L,

nxex

L ) exp (−(nπ/L)

2₍_t

−τ))). (18)

If we investigate each point of fluid in moving coordinate system of this point, Galilean transform must by applied v(r₀−vt, t).

IV. TWO DIMENSIONAL INHOMOGENEOUS SOLUTION

v(x, y, t)

dvi

dt −a

2_∆_vi ₌ 1

ρ(−∇ip+fi)inΩ×(0,∞), (19) vi(x, y,0) = v0i(x, y)x, y∈Ω, (20)

∂vi

∂n = 0on∂Ω×(0,∞), (21)

where p = p(x, y, t) and f = f(x, y, t), Ω ⊂ R2n_{, n the} exterior unit normal at the smooth parts of∂Ω,a2_{a positive}

constant andvx

0(x, y), v0y(x, y)a given function.

So, whenxandy are normed toa= 1, equation (4) could be rewritten as follow

dvi

dt =v

i

xx+viyy+Qi(x, y, t), x, y∈Ω, t >0. (22)

A. Rectangular boundary

We expand v and Q in the eigenfunctions exp (jnπx_L x ) exp (jmπy

Ly )on spaceΩ∈[0, Lx]×[0, Ly]whereexp (

jnπx Lx ) exp (jmπy_L

y ) and exp (

jn′_πx Lx ) exp (

jm′_πy

Ly ) functions

orthog-onality could be applied, where j = √−1. Let’s denote dΩ = dxdy and exp2₍_{x, y}_{) = exp (}_jπx₊_jπy₎_{. So, we}

obtain

Qi₍_{x, y, t}_{) =} ∞

X

m,n=1 qi

mn(t) exp2(

nx Lx

,my Ly

), (23)

with

qmni (t) =

1

I2 Z Z

Ω

Qi(x, y, t) exp2(nx

Lx

,my Ly

)dΩ,(24)

I2 = Z Z

Ω

(exp2(nx

Lx

,my Ly

))2dΩ (25)

and

vi₍_{x, y, t}_{) =} ∞

X

m,n=1 ui

mn(t) exp

2₍nx Lx

,my Ly

). (26)

˙

ui

mn(t) +k2m,nuimn(t) =qmni (t), (27)

k2

m,n=

nπ Lx

2

+mπ Ly

2

. (28)

whose solution is

ui

mn(t) =uimn(0) exp (−k2m,nt)

+

t

R

0 qi

jmn(τ) exp (−km,n2 (t−τ))dτ, (29)

where

uimn(0) =

1

I2 Z Z

Ω

vi0(x, y) exp2( nx Lx

,my Ly

)dΩ. (30)

Again, we substitute all obtained equations into (26) and have

vi₍_{x, y, t}_{) =}

RR

Ω ds′

ds(vi

0(s

′

, s)S(s, s′

, x, y) exp (−k2

m,nt) +

t

R

0 Qi₍_s′

, s, τ)S(s, s′

, x, y) exp (−k2

m,n(t−τ)))dτ),

S(s, s′

, x, y) =

∞

P

m=1

n=1 1

I2exp

2₍ns Lx,

ms′ Ly ) exp

2₍nx Lx,

my

Ly). (31)

∇ ·v= ∂v

x₍_{x, y, t}₎

∂x +

∂vy₍_{x, y, t}₎

∂y = 0. (32)

ISBN: 978-988-19253-5-0

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So, we obtain relation conditions betweenvx

0mnf andv y

0mnf,

Qx

mnf andQ y mnf

(vx

0nmf+Qxnmf)

nπ Lx

+ (vy₀_nmf+Qy_nmf)mπ

Ly

= 0, (33)

wherevi

0f, Qif are

vi

0nmf =

1

I2

RR

Ω vi

0(s

′

, s) exp2₍ns Lx,

ms′ Ly )ds

′

ds, (34)

Qi nmf =

1

I2

RR

Ω ds′

ds

× t

R

0 Qi₍_s′

, s, τ) exp2₍ns Lx,

ms′ Ly ) exp (k

2

mnτ)dτ. (35)

Finally, solutions of 2D incompressible Navier-Stokes equa-tions are

vx₍_{x, y, t}_{) =} ∞

P

m,n=1

(vx

0mnf+Qxmnf)

×exp (−jπmx(v

y

0mnf+Q y mnf)

Ly(v0xmnf+Qxmnf) +

jπmy Ly −k

2

mnt), (36)

vy₍_{x, y, t}_{) =} ∞

P

m,n=1

(v₀y_mnf+Qy_mnf)

×exp (jπnx_L

x −

jπny(vx

0mnf+Qxmnf)

Lx(vy0mnf+Q y

mnf) −k

2

mnt). (37)

B. Any shaped boundary

For any shaped boundary ∂Ω, equation (23) could be replaced by

Qi(x, y, t) =

∞

X

m,n=1

qmni (t)H∂mnΩ(x)H∂mnΩ(y) (38)

and equation (26) by

vi(x, y, t) =

∞

X

m,n=1

uimn(t)H∂mnΩ(x)H∂mnΩ(y). (39)

where Hmn

∂Ω(x)H∂mnΩ(y) are partial solutions of Helmholtz

2D equation for given boundary∂Ωand could be taken for example from [7]. So equation (31) transforms to

vi₍_{x, y, t}_{) =} ∞

P

m,n=1

(vi

0mnf+Qimnf)

×Hmn

∂Ω(x)H∂mnΩ(y) exp (−kmn2 t), (40)

vi mnf =

1

I2mn

RR

Ω vi

0(s

′

, s)Hmn

∂Ω(s)H∂mnΩ(s

′

)ds′

ds, (41)

Qi mnf =

1

I2mn

RR

Ω ds′

dsRt 0dτ

×Qi₍_s′

, s, τ)Hmn

∂Ω(s)H∂mnΩ(s

′

) exp (k2

mnτ)), (42)

whereI2mn is expressed as follow

I2mn=

Z Z

∂Ω

(H∂mnΩ(x)H∂mnΩ(y))2dxdy. (43)

Applying of continuity condition∇ ·v= 0 gives similar to equations (33)(36)(37) relations between n and m. Finally, we obtain

vi₍_{x, y, t}_{) =} ∞

P

n=1

(vi

0m(n)nf +Qim(n)nf)

×H_∂m_Ω(n)n(x)H_∂m_Ω(n)n(y) exp (−k2

m(n)nt), (44)

wherem(n)notes dependence ofmandn. If we investigate each point of fluid in moving coordinate system of this point, Galilean transform must by appliedv(r₀−vt, t).

V. THREE DIMENSIONAL INHOMOGENEOUS SOLUTION

v(x, y, z, t)

dvi

dt −a

2_∆_vi ₌ 1

ρ(−∇ip+fi)inΩ×(0,∞),(45) vi(x, y, z,0) = v0i(x, y, z)x, y, z∈Ω, (46)

∂vi

∂n = 0 on∂Ω×(0,∞), (47)

where p = p(x, y, z, t) and f = f(x, y, z, t), Ω ⊂ R3n_, n the exterior unit normal at the smooth parts of ∂Ω, a2

a positive constant and vx

0(x, y, z), vy0(x, y, z), v0z(x, y, z) a

given function.

So, whenx, y and z are normed to a= 1, equation (4) could be rewritten as follow

dvi

dt =v

i

xx+viyy+vzzi +Qi(x, y, z, t), (48)

where x, y, z∈Ω, t >0.

A. Rectangular boundary

Let’s denote

exp3(x, y, z) = exp(jπx) exp(jπy) exp(jπz). (49)

We expandvandQin the eigenfunctionsexp3₍nx Lx,

my Ly,

pz Lz)

on space Ω ∈ [0, Lx] × [0, Ly] × [0, Lz] where

exp (jnπx Lx ) exp (

jmπy Ly ) exp (

jpπz Lz ),exp (

jn′_πx Lx ) exp (

jm′_πy Ly ) exp (jp_L′πz_z ) functions orthogonality could be applied. Let’s denotedΩ =dxdydz. So, we obtain

Qi_(Ω_{, t}_{) =} ∞

X

m=1

n=1

p=1 qi

mn(t) exp3(

nx Lx

,my Ly

,pz Lz

), (50)

with

qi

mnp(t) = I13

RRR

Ω

Qi_(Ω_{, t}_{) exp}3₍nx Lx,

my Ly,

pz

Lz)dΩ, (51)

I3=RRR Ω

(exp3₍nx Lx,

my Ly,

pz Lz))

2_d_Ω ₍₅₂₎

and

vi(x, y, z, t) =

∞

X

m,n,p=1

uimn(t) exp3(

nx Lx

,my Ly

, pz Lz

). (53)

˙

ui

mnp(t) +k2mnpuimnp(t) =qimnp(t), (54)

k2

mnp =

nπ Lx

2

+mπ Ly

2

+pπ_L y

2

, (55)

whose solution is

ui

mnp(t) =uimnp(0) exp (−kmnp2 t)

+

t

R

0 qi

mnp(τ) exp (−kmnp2 (t−τ))dτ, (56) Proceedings of the World Congress on Engineering 2014 Vol II,

WCE 2014, July 2 - 4, 2014, London, U.K.

ISBN: 978-988-19253-5-0

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where

ui

mnp(0) =

1

I3

RRR

Ω vi

0(x, y, z) exp3(nπxLx ,

mπy Ly ,

pπz

Lz)dΩ. (57)

Again, we substitute all obtained equations into (53) and have

vi₍_{x, y, z, t}_{) =}

RRR

Ω vi

0(s

′′

, s′

, s)(S(Ω′

,Ω) exp (−k2

mnpt))dΩ ′

+

RRR

Ω dΩ′

t

R

0 dτ

×Qi₍_s′′

, s′

, s, τ)(S(Ω′

,Ω) exp (−k2

mnp(t−τ))), (58)

S(Ω′

,Ω) =

∞

P

m,n,p=1 1

I3Smnp(s, s ′

, s′′

)Smnp(x, y, z),(59)

Smnp(x, y, z) = exp3(nπx_L_x ,mπy_L_y ,pπz_L_z). (60)

∇ ·v= ∂vx₍_x,y,z,t₎

∂x +

∂vy₍_x,y,z,t₎

∂y +

∂vz₍_x,y,z,t₎

∂z = 0. (61)

So, we obtain relation conditions between n,mandp

(vx

0nmpf+Qxnmpf)nπLx + (v

y

0nmpf+Q y nmpf)

mπ Ly +(vz

0nmpf+Qznmpf) pπ

Lz = 0, (62)

wherevi

0f, Qif are

vi

0mnpf =

1

I3

RRR

Ω vi

0(s

′′

, s′

, s)Smnp(s, s′, s′′)dΩ′, (63)

Qi mnpf =

1

I3

RRR

Ω dΩ′

× t

R

0

dτ Qi₍_s′′

, s′

, s, τ)Smnp(s, s′, s′′). (64)

Finally, solutions of 3D incompressible Navier-Stokes are

vi₍_{x, y, z, t}_{) =} ∞

P

m,n=1

(vi

0mnpf+Qimnpf)

×Smnp(m,n)(x, y, z) exp (−kmnp2 (m,n)t), (65) Smnp(m,n)(x, y, z) = exp3(nx_L_x,my_L_y,p(m,n_L_z)z), (66)

where integersp(m, n)must satisfy (62) relations for eachm andnand could be expressed by substituting ofp/Lz corre-sponding expression as the sum of two othersn/Lx, m/Ly. If we investigate each point of fluid in moving coordinate system of this point, Galilean transform must by applied v(r₀−vt, t).

B. Any shaped boundary

For any shaped boundary ∂Ω, equation (50) could be replaced by

Qi(x, y, z, t) =

∞

X

m,n,p=1

qi(t)H_∂mnp_Ω_,k(x, y, z) (67)

and

vi(x, y, z, t) =

∞

X

m,n,p=1

ui(t)H_∂mnp_Ω_,k(x, y, z), (68)

whereH_∂mnp_Ω_,k(x, y, z)are partial solutions of Helmholtz 3D equation for given boundary ∂Ω. and could be taken for example from [8].So equation (58) transforms to

vi₍_{x, y, z, t}_{) =} ∞

P

m,n=1

(vi

0mnpf+Qimnp)

×H_∂mnp_Ω_,k(x, y, z) exp (−k2

mnpt), (69)

vi

0mnpf =

RRR

Ω vi

0(s

′′

, s′

, s)H_∂mnp_Ω_,k(s, s′

, s′′

)dΩ′

, (70)

Qi mnp=

RRR

Ω dΩ′

t

R

0 dτ

×Qi₍_s′′

, s′

, s, τ)H_∂mnp_Ω_,k(s, s′

, s′′

) exp (k2

mnpτ). (71)

Applying of continuity condition∇ ·v = 0gives relations betweenn, mandpor p=p(m, n). Finally, we obtain

vi₍_{x, y, z, t}_{) =} ∞

P

m,n=1

(vi

0mnp(m,n)f+Q i

mnp(m,n))

×H_∂mnpΩ,k(m,n)(x, y, z) exp (−k

2

mnp(m,n)t). (72)

VI. CONCLUSIONS

Due to the form of fluid equations they could be trans-formed into the full/partial inhomogeneous parabolic differ-ential equations: differdiffer-ential equations in respect to space variables and full differential equations in respect to the time variable and inhomogeneous time dependent part. Finally, orthogonal polynomials as the partial solutions of obtained Helmholtz equations were used for derivation of analytical solution of velocities for incompressible fluid in 1D, 2D and 3D space for rectangular boundary. Solution in 3D space for any shaped boundary is expressed in term of 3D global solution of 3D Helmholtz equation accordantly.

REFERENCES

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[2] A. L. Cauchy, “Recherches sur l’eqnilibre et le mouvement interieur des corps solides et fluides, elastiques ou non elastiques,”Bull. Philomat., vol. 1, pp. 9–13, 1823.

[3] G. Stokes, “On the theories of the internal friction of fluids in motion,” Transactions of the Cambridge Philosophial Society, vol. 8, pp. 287– 305, 1845.

[4] O. Darrigol, “Between hydrodynamics and elasticity theory: The first five births of the navier-stokes equation.”Arch. Hist. Exact Sci., vol. 56, pp. 95–150, 2002.

[5] Centrum Wiskunde & Informatica, “History of theoretical fluid dynam-ics,” retrieved 11/29/2013.

[6] B. Neta,Partial Differential Equations, MA 3132 Lecture Notes, Depart-ment of Mathematics Naval Postgraduate School, Monterey, California,. Online, 2012.

[7] W. Read, “Analytical solutions for a helmholtz equation with dirichlet boundary conditions and arbitrary boundaries,” , Mathematical and Computer Modelling, vol. 24, no. 2, pp. 23–34, July 1996.

[8] R. H. Hoppe,Computational Electromagnetics, Handout of the Course Held at the AIMS Muizenberg. Online, December 17–21 2007.

ISBN: 978-988-19253-5-0