Transformation of the Navier-Stokes Equations in Curvilinear Coordinate Systems with Maple

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1 Global Jounal of Pue and Applied Mathematics. ISSN Volume 12, Numbe , pp Reseach India Publications Tansfomation of the Navie-Stokes Equations in Cuvilinea Coodinate Systems with Maple Suattana Sungnul Depatment of Mathematics, Faculty of Applied Science, King Mongkut s Univesity of Technology Noth Bangkok, Bangkok 10800, Thailand Abstact In this eseach, we study the geneal fom of the Navie-Stokes equations in abitay cuvilinea coodinate system which it is the govening equation of fluid flow. We often found that some engineeing o applied sciences poblems ae in complex geometies domain such as iegula o infinite which it is not easy to solve thei poblems. Most Reseaches will be tansfom into a domain that is easy to calculate, such as ectangle domain fo the flow in two dimensions o ectangula shape domain fo the flow in thee dimensions. This makes it easy to calculate the basic paametes of the poblem, such as speed, pessue and so on. Theefoe, the main pupose of this wok discusses the detailed tansfomation of the Navie-Stokes equations in cuvilinea coodinate systems. In addition, we wite the code in symbolic computation fo the tansfomation by Maple Softwae. The Maple softwae was used to pefom calculate all opeatos of scala, vecto and tenso quantities in the Navie-Stokes equations. The pogam fo tansfomation the Navie-Stokes equations in othogonal cuvilinea coodinate systems was obtained. AMS subject classification: Keywods: Tansfomation, Navie-Stokes equations, Cuvilinea coodinate systems. 1. Intoduction A geneal fom of the Navie-Stokes equations in a fixed cuvilinea coodinate system wee investigated long ago using coodinate tansfomation. We can find the standad

2 3316 Suattana Sungnul fom of these equations and thei deivation in tenso calculus textbooks [1] [3]. Howeve, its have not been used widely in numeical simulations, because of the calculation of the covaiant deivatives in cuvilinea coodinate systems is geneally vey complicate and spent time to much fo calculation. Most eseaches have utilized paticula fom of the Navie-Stokes equations when deal with flows in complex geometies via mapping into egula domain. Many eseaches compute on cylindical coodinate system fo solving the poblem of blood flow in the ateies o choose Spheical coodinate system fo calculate pimitive vaiables in the poblem of fluid flow though cicula object. Ogawa and Ishiguo [4] poposed a new method fo computing flow fields with abitaily moving boundaies. Unde the concept of Lie deivatives the field equations in geneal moving coodinates. They deived the tempoal deivative of tenso vectos by consideing the infinitesimal geometic motion of the cuvilinea coodinates. Haoxiang Luo and Thomas R. Bewley [5] pesented the contavaiant fom of the Navie-Stokes equations in time-dependent cuvilinea coodinate systems. They deived the tempoal deivative of tenso vectos by using an altenative appoach and the quotient ule of tenso analysis. Suattana Sungnul [6] pesented the Navie-Stokes equation in cylindical bipola coodinate system because of this coodinate can be tansfom infinite domain into finite domain in the poblem of incompessible fluid flow ove two otating cicula cylindes., etc. In this wok, we intoduced main definitions of tenso-vecto calculus elated with cuvilinea coodinate system. We then pesent detailed tansfomation of the Navie-Stokes equation into othogonal cuvilinea coodinate system. The complete fom of the Navie-Stokes equations with espect covaiant, contavaiant and physical components of velocity vecto ae pesented. The pogam in Maple softwae fo tansfomation the Navie-Stokes equations in cuvilinea coodinate systems ae obtained. 2. Definition of Opeations in the Cuvilinea Coodinate Systems Let R 3 be an open set. A one-to-one and ecipocal continuously diffeentiable mapping fom to subset of R 3 is called a coodinate system [1] [3]. This mapping is defined by the fomula x X 1 x,x 2 x,x 3 x 1 The value of functions X i x ae called coodinates cuvilinea coodinates of the point x Basis and Cobasis Let a point x be fixed. At this point thee ae vectos e i = x, i = 1, 2, 3 Xi e i = Xi x = Xi, i = 1, 2, 3 2

3 The Navie-Stokes Equations in Cuvilinea Coodinate Systems 3317 which fom a basis, e i, and cobasis, e i,inr 3. These basis ae called a coodinate basis and cobasis of the coodinate system X at the point x. A vecto v can be decomposed along eithe basis o cobasis vectos v = v i e i = v i e i. The components v i and v i ae called covaiant and contavaiant components of vecto v espectively. A coodinate system is called othogonal if its basis is othogonal, 2.2. Fundamental Tenso e i e j = 0, e i e j = 0, i = j. 3 The fundamental tenso and its invese ae defined by g ij = e i e j, g ij = e i e j. 4 Coodinates of the fundamental tenso g with espect to an othogonal coodinate system ae g g g ij = 0 g 22 0, g ij = 0 g g g 33 whee g ij = e i 2 = x 2 X, g ij = e i 2 = X i 2, i = 1, 2, 3 and g = [ e 1 e 2 e 3 ] 2. Deivatives of the basis and cobasis vectos with espect to cuvilinea coodinates can be epesented in tems of the basis { e i } and cobasis { e i } espectively, e i X j = Ɣs ij e s, e i X j = Ɣi js es, 5 whee the coefficients Ɣij s ae called Chistoffel s symbols of second ode. The Chistoffel symbols ae elated to the deivatives of the fundamental tenso Ɣij l = 1 gis 2 X j + g js X i g ij X s g ls. 6 Covaiant deivatives ae expessed in tems of patial deivatives with espect to coesponding coodinates, Chistoffel s symbols and components of a tenso. The simplest ae covaiant deivatives of scala field, F, which coincide with usual patial deivative F,i = F X i. The covaiant deivatives of the covaiant and contavaiant components of a second ode tenso ae ij,l = ij X l Ɣ s li sj Ɣ s lj is, 7

4 3318 Suattana Sungnul ij,l = ij X l + Ɣ i ls sj + Ɣ j ls is. 8 Similaly, the covaiant deivatives of the mixed components ae.j i.,l =.j i. X l Ɣ s li.j s. + Ɣj ls.s i., 9 j..i,l = j..i X l Ɣli s j..s + Ɣj ls s..i. 10 In the above equations and eveywhee below, a comma with an index in a subscipt denotes covaiant diffeentiation. A deivative of the vecto field, v, is the second ode tenso which is denoted by the symbol, v ae x v x ij v i x,j v x. Covaiant and mixed coodinates of the = v i X j Ɣs ij v s = v i,j = vi X j Ɣi js v s = v i,j Divegence of a vecto field The divegence of a vecto field, v, is a scala. Divegence can be expessed in tems of the covaiant deivatives of the contavaiant components of vecto field, v, div v = v,i i = vi X i + Ɣi is vs = vi X i + vi g g X i 11 = 1 g v i g X i Gadient of the scala function The gadient of the scala function F is defined by F = F x = F X i ei. 12 Covaiant components of the gadient vecto ae F i = F,i = F X i, and contavaiant components of the gadient vecto ae F i = g is F s.

5 The Navie-Stokes Equations in Cuvilinea Coodinate Systems Laplace opeato of the scala function The Laplace opeato of the scala function F is defined by F = div F = F i,i We can ewite this fomula in the following fom 2.6. Cul of a vecto field F = div F = 1 g X i = g is F s,i = g is 2 F X s X i Ɣα is F X α. g g is F X s. 13 A cul of a vecto field, v, is the vecto field. In a ight-handed basis ε 123 = e 1 e 2 e 3 = g, the contavaiant components of cul v ae 2.7. Divegence of a tenso field cul v 1 = 1 v3 g X 2 v 2 X 3, cul v 2 = 1 v1 g X 3 v 3 X 1, 14 cul v 3 = 1 v2 g X 1 v 1 X 2, Divegence of a tenso field is a vecto. The s-th contavaiant component of this vecto is divp s = P sj,j = div P s + Ɣjα s P jα, 15 whee P s = P s1,p s2,p s3 is s-th ow of a matix which epesents second ode tenso P and div P s = Psj X j + Ɣj jα P sα Laplace opeato of vecto field The Laplace opeato of a vecto field is a vecto field and the contavaiant components of this vecto ae v l = v l + 2g ij Ɣis l v s Ɣ l X j + is gij X j Ɣα ji Ɣl αs + Ɣl jα Ɣα is v s, 16

6 3320 Suattana Sungnul whee [ v l = g ij 2 v l X j X i v l ] Ɣs ji X s is the Laplace opeato of scala function v l Covaiant and contavaiant components of the acceleation of a vecto field The covaiant and contavaiant components of the acceleation of a vecto field, v, ae d v = v i + v s v i dt X s Ɣj is vs v j, 17 i d v i = vi dt + v s vi X s Ɣi js vj v s. 18 If vectos of coodinate bases and cobases ae not nomed, then components of tensos have diffeent numeical values in diffeent baseds even if diections of basis vectos coincides. numeical values of tenso components divided by the length of coesponding basis o cobasis vectos, which define these components ae called physical components of the tenso. Fo example, if u i = u e i ae covaiant coodinates of a vecto u, then the physical components ae ũ i = u i e i, the covaiant components of a tenso, T ij, elated with the physical components as T ij = 3. The Navie-Stokes Equations T ij e i e j. Fluid obey the geneal laws of continuum mechanics: consevation of mass, linea momentum and enegy. In the Euleian epesentation of the flow, we epesent the density ρ x,t as a function of the position x and time t. The consevation of mass is expessed by the continuity equation ρ + divρ v = If we assume that the fluid is incompessible and homogeneous, then the density is constant in space and time: ρ x,t = ρ 0. Then the continuity 19 becomes div v = The momentum equation, which is base on Newton s second law, epesents the balance between vaious foces acting on a fluid element. The foces ae

7 The Navie-Stokes Equations in Cuvilinea Coodinate Systems The foce due to ate of change of momentum, geneally efeed to as the inetia foce 2 Body foces, f : such as buoyancy foce, magnetic foce, and electostatic foce 3 Pessue foce 4 Viscous foces causing shea stess. Fo a fluid element unde equilibium, o we can wite Inetia foce + body foce + pessue foce + viscous foce = 0. ρ v + v v + f = p + µ v, 21 whee µ is called the coefficient of dynamic viscosity. Equations 20 and 21 ae govening equation the motion of a viscous incompessible fluid, they ae called the Navie-Stokes equations fo viscous incompessible flow. In case of the motion of viscous incompessible fluid flow which has no body foce can be descibed by the Navie-Stokes equations in vecto fom 22 v + v v = 1 p + ν v, ρ div v = 0, 22 whee v is the velocity vecto of fluid, p is the pessue of fluid, ρ is the density of fluid and ν = µ is the kinematics viscosity. ρ 4. Main Results We emind main definitions of vecto-tenso opeation in the cuvilinea coodinate system and the Navie-Stokes equations fo incompessible fluid flow in pevious section. Equation 22 epesents invaiant fom of the Navie-Stokes equations in a cuvilinea coodinate system. In index fom equation 22 have the following contavaiant fom 23 v i + v s vi X s + Ɣi js vj v s = 1 ρ gis p s + ν v i, v i,i = 0, 23 and the covaiant fom 24 v i + v s v i X s Ɣj is vs v j = 1 ρ p i + ν v i, v i,i = 0. 24

8 3322 Suattana Sungnul We can wite out the Navie-Stokes equations with espect to the contavaiant and covaiant components by using definitions and fomulae fom pevious section. We wite the code in Maple softwae to pefom calculation fo the tansfomation of the Navie-Stokes equations into the cuvilinea coodinate systemse with espect to physical component. Let e i be a fixed ight-handed othonomal Catesian basis in R 3. A vecto x R 3 has components xx,y,z; x = x e 1,y = x e 2,z= x e 3 in this basis. Cuvilinea coodinates ae tansfomation fom R 3 to subset of R 3 X x = X 1 x, y, z, X 2 x, y, z, X 3 x,y,z The Navie-Stokes equations in cylindical coodinate system In the cylindical coodinate, we use notation = X 1 x, y, z, θ = X 2 x, y, z, z = X 3 x,y,z. 26 The cylindical coodinate system can be defined by invese tansfomation X 1 : x = cos θ, y = sin θ, z = z, 27 whee >0, 0 θ 2π, z,. The Navie-Stokes equations in cylindical coodinate system with espect to physical component ae following v v + v + v θ = 1 ρ p + ν v θ v2 θ 1 { + v v z z v v 2 θ θ + 2 v z 2 }, 28 + v + v θ = 1 ρ p θ + ν θ + v v θ + v z z { 1 v θ v θ 2 θ } v 2 θ + 2 v θ z 2, 29 v z v z + v + v θ = 1 ρ v z θ + v v z z z { 1 p z + ν v z } v z 2 θ v z z 2, 30 1 v + v θ + v z = θ z Equations epesent the consevation of linea momentum equations in, θ andz diections, espectively. Equation 31 epesents the continuity equation.

9 The Navie-Stokes Equations in Cuvilinea Coodinate Systems The Navie-Stokes equations in spheical coodinate system In the spheical coodinate, we use notation = X 1 x, y, z, φ = X 2 x, y, z, θ = X 3 x,y,z. 32 The spheical coodinate system can be defined by invese tansfomation X 1 : x = sin φ cos θ, y = sin φ sin θ, z = cos φ, 33 whee >0, 0 φ π, 0 θ 2π. The Navie-Stokes equations in spheical coodinate system with espect to physical component ae following v v + v + v φ v φ + = 1 ρ p + ν 2 { v sin 2 φ θ 2 2 v θ v sin φ θ 1 v 2 φ + vθ 2 2 v + 2 v φ 2 + cot φ v φ + 34 vφ φ + 1 sin φ θ + v + cot φv θ }, v φ v φ + v + v φ + ν 2 { 2 v φ φ + v θ v φ sin φ θ + 1 vφ v cot φv 2 1 p θ = ρ φ + vφ + 1 sin φ v φ 1 2 v φ + sin φ φ φ sin 2 φ θ vφ v φ cos φ sin 2 φ θ 1 2 sin 2 φ v φ }, + v + v φ φ + sin φ + ν 2 { 2 vθ +2 2 vθ v θ θ p vθ v + cot φv θ v φ = ρ sin φ θ + + sin φ vθ v θ φ φ sin φ sin 2 φ θ cos φ vθ + 1 v φ sin φ sin φ θ + csc φ cot φ v φ }, θ 1 2 v + 1 sin φ φ sin φv φ + 1 sin φ θ = Equations epesent the consevation of linea momentum equations in, φ and θ diections, espectively. Equation 37 epesents the continuity equation.

10 3324 Suattana Sungnul 4.3. The Navie-Stokes equations in cylindical bipola coodinate system In the cylindical bipola coodinate, we use notation ξ = X 1 x, y, z, η = X 2 x, y, z, z = X 3 x,y,z. 38 The cylindical bipola coodinate system can be defined by invese tansfomation X 1 : x = a sinh η cosh η cos ξ, y = a sin ξ, z = z, 39 cosh η cos ξ whee 0 ξ 2π, η,, z, and a is chaacteistic length in the cylindical bipola coodinate system which is positive. The Navie-Stokes equations in cylindical bipola system with espect to physical component ae following v ξ + ν h v η + ν h + 1 h { 1 h + 1 h { 1 h v ξ v ξ 2 v ξ ξ 2 v η v ξ 2 v η ξ 2 ξ + v v ξ v ξ η + v z η z 1 sinh ηvξ v η sin ξv η 2 = 1 1 p a h ρ ξ + 2 v ξ η 2 2 sinh η v η a ξ sin ξ v η cosh η + cos ξ η a v ξ },40 ξ + v v η v η η + v z η z + 1 sinh ηvξ 2 sin ξv ξ v η = 1 1 p a h ρ η + 2 v η η sinh η v ξ a ξ sin ξ v ξ cosh η + cos ξ η a v η },41 v z + 1 h = 1 ρ v ξ v z ξ + v v z η η { 1 h 2 p z + ν + v z v z z 2 v z ξ v z η v ξ z 2 }, 42 { 1 hvξ h 2 + hv } η + v z = 0, 43 ξ η z a whee h =. Equations epesent the consevation of linea cosh η cos ξ momentum equation in ξ,η and z diections, espectively. Equation 43 epesents the continuity equation. Similaly, we can wite out the Navie-Stokes Equations in abitay cuvilinea coodinates systems.

11 The Navie-Stokes Equations in Cuvilinea Coodinate Systems Conclusion We have pesented the complete fom of the Navie-Stokes equations with espect covaiant, contavaiant and physical components of velocity vecto. In addition, we obtained the code Maple pogam fo tansfomation of the Navie-Stokes equations in cuvilinea coodinate systems. In genealized, it is not difficult to tansfom the Navie-Stokes Equations to anothe cuvilinea coodinate system. Acknowledgement This eseach was funded by King Mongkut s Univesity of Technology Noth Bangkok Contact no. KMUTNB-GEN Refeences [1] Chen To Tai, Genealized Vecto and Dyadic Analysis, IEEE Pess, New Yok, 134, [2] D.A. Danielson, Vectos and Tensos in Engineeing and Physics, Addison-Wesley Publishing Company, 280, [3] R.Ais, Vectos, Tensos, and the Basic Equations of Fluid Mechanics, Pentice-Hall, Englewood Clifffs, NJ., 890, [4] S. Ogawa and T. Ishiguo,A method fo computing flow fields aound moving bodies, J. Comput. Phys., vol. 69, 49 68, [5] Haoxiang Luo and Thomas R. Bewley, On the contavaiant fom of the Navie- Stokes equations in time-dependent cuvilinea coodinate systems, J. Comput. Phys. vol. 199, , [6] S. Sungnul, On the Repesentation of the Navie-Stokes Equations in Cylindical Bipola Coodinate System, Poceedings, ANSCSE9, The 9 th Annual National Symposium on Computational Science and Engineeing, Mahidol Univesity Thailand, Mach, , [7] T.C. Papanastasiou, G.C. Geogiou and A.N. Alexandou, Viscous Fluid Flow, CRC Pess LLC, Floida, 418, 1999.

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