Beltrami Flow of an Unsteady Dusty Fluid Between Parallel Plates in ...

EJTP 5, No. 17 (2008) 181–192

Electronic Journal of Theoretical Physics

Beltrami Flow of an Unsteady Dusty Fluid Between Parallel Plates in Anholonomic Co-ordinate System B. J. Gireesha, C. S. Bagewadi∗ and C. S. Vishalakshi Department of Mathematics, Kuvempu University, Shankaraghatta-577 451, Shimoga , Karnataka, India

Received 6 July 2007, Accepted 26 August 2007, Published 27 March 2008 Abstract: An analytical study of Beltrami flow of viscous dusty fluid between two parallel plates has been studied. The flow is due to influence of movement of plates. Flow analysis is carried out using differential geometry techniques and exact solutions of the problem are obtained using Laplace Transform technique also which are discussed with the help of graphs drawn for different values of Reynolds number. Further the expressions for skin-friction are obtained at the boundaries. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Frenet Frame Field System; Beltrami Flow, Dusty Fluid; Velocity of Dust Phase and Fluid Phase, Unsteady Flow, Vorticity, Reynolds Number PACS (2006): 47.10.g; 47.10.A-; 47.10.Fg; 47.11.j; 96.50.Dj AMS Subject Classification (2000): 76T10; 76T15

1.

Introduction

A dusty fluid is a mixture of fluid and fine dust particles. Its study is important in areas like environmental pollution, smoke emission from vehicles, emission of effluents from industries, cooling effects of air conditioners, flying ash produced from thermal reactors and formation of raindrops, etc. Also it is useful in the study of lunar ash flow which explains many features of lunar soil. P.G.Saffman [18] has discussed the stability of the laminar flow of a dusty gas in which the dust particles are uniformly distributed. Liu [11] has studied the Flow induced by an oscillating infinite plat plate in a dusty gas. Michael and Miller [12] investigated the motion of dusty gas with uniform distribution of the dust particles occupied in the semiinfinite space above a rigid plane boundary. T. M. Nabil [13] studied the Effect of couple ∗

prof [email protected] and bjgireesu@rediffmail.com

Electronic Journal of Theoretical Physics 5, No. 17 (2008) 181–192

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stresses on pulsatile hydromagnetic poiseuille flow, N.Datta [5] obtained the solutions for Pulsatile flow of heat transfer of a dusty fluid through an infinitly long annlur pipe. N. Rudraiah [17] has obtained the solutions of Magnetohydrodynamic Beltrami Flows, A.Eric [6] has studied the Quantitative Assessment of Steady and Pulsatile Flow Fields in a Parallel Plate Flow Chamber. Z. Yoshida and S.M. Mahajan [20] have discussed the Simultaneous Beltrami Conditions in Coupled Vortex Dynamics. P.Nemanyi and R.Prim [14] studied some properties of rotational flow of a perfect gas using Beltrami condition. Yu.A.Gostintsev, P.F.Pokhil and O.A.Uspenskii [15] studied Gromeka-beltrami flow in a semiinfinite cylindrical pipe. Some researchers like Kanwal [10], Truesdell [19], Indrasena [9], Purushotham [16], Bagewadi, Shantharajappa and Gireesha [1, 2, 3] have applied differential geometry techniques to investigate the kinematical properties of fluid flows in the field of fluid mechanics. Further, the authors [2, 3] have studied two-dimensional dusty fluid flow in Frenet frame field system. Recently the authors [7, 8] have studied the flow of unsteady dusty fluid under varying different pressure gradients like constant, periodic and exponential. In this paper we study the Beltrami flow of a dusty fluid between two infinite parallel plates in anholonomic co-ordinate system. Further by considering the fluid and dust particles are at rest initially, the analytical expressions are obtained for velocities of fluid and dust particles using Laplace transform technique in different three cases. The changes in the velocity profiles for different Reynold’s number are shown graphically.

2.

Equations of Motion

The governing equations of motion of unsteady viscous incompressible fluid with uniform distribution of dust particles are [18]: For fluid phase

For dust phase

→ ∇.− u =0 (Continuity) → kN − ∂− u − → → → → + (→ u .∇)− u = −ρ−1 ∇p + − (→ v −− u) u + g + ν∇2 − ∂t ρ (Linear Momentum)

→ ∇.− v =0 (Continuity) → − ∂v k → − → → − + (− v .∇)− v =→ g + (− u −→ v) ∂t m

(1) (2)

(3) (Linear Momentum)

(4)

We have following nomenclature: → − → u −velocity of the fluid phase, − v −velocity of dust phase, ρ−density of the gas, − → g = −∇ϕ, ϕ−gravitational potential, p−pressure of the fluid, N −number of density of dust particles, ν−kinematic viscosity, k = 6πaμ−Stoke’s resistance (drag coefficient), a−spherical radius of dust particle, m−mass of the dust particle, μ−the co-efficient of viscosity of fluid particles, t−time.


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− → → − Let − s ,→ n , b be triply orthogonal unit vectors tangent, principal normal, binormal respectively to the spatial curves of congruences formed by fluid phase velocity and dusty phase velocity lines respectively as shown in the figure-1.

Fig. 1 Frenet Frame Field System

Geometrical relations are given by Frenet formulae [4] − → → → → − ∂− s ∂− n ∂b − → → − → i) n = ks n , = τs b − ks s , = −τs − ∂s ∂s ∂s − → → → → − ∂b ∂− s ∂− n → − − − = kn s , = −σn → = σn b − kn → s, n ii) ∂n ∂n ∂n − → → → − → ∂− n ∂− s ∂b → → → = kb − = −σb − = σb − s, s, n − kb b iii) ∂b ∂b ∂b − → − → − → iv) ∇. s = θns + θbs ; ∇. n = θbn − ks ; ∇. b = θnb

(5)

where ∂/∂s, ∂/∂n and ∂/∂b are the intrinsic differential operators along fluid phase velocity (or dust phase velocity ) lines, principal normal and binormal. The functions (ks , kn , kb ) and (τs , σn , σb ) are the curvatures and torsion of the above curves and θns and θbs are normal deformations of these spatial curves along their principal normal and binormal respectively.

3.

Formulation and Solution of the Problem

Let us consider an unsteady flow of an incompressible viscous fluid with uniform distribution of dust particles between two infinite parallel plates separated by a distance h under conservative body forces as shown in the figure-2. The flow is due to the influence of movement of the plates. Both the fluid and the dust particle clouds are supposed to be static at the beginning. The dust particles are assumed to be spherical in shape and uniform in size. The number density of the dust particles is taken as a constant throughout the flow. Under these assumptions the flow will be a parallel flow in which the streamlines are along the tangential direction and the velocities are varies along binormal direction and with time t, since we extended the fluid to infinity in the principal normal direction.

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Fig. 2 Geometry of the flow.

For the above described flow the velocities of fluid and dust are of the form → − → s, u = us −

− → − v = vs → s

where (us , un , ub ) and (vs , vn , vb ) are velocity components of fluid and dust particles respectively. By virtue of system of equations (5) the intrinsic decomposition of equations (2) and (4) give the following forms; From the vector identities we know that → → → → u ×− w) (− u .∇)− u = ∇ u2 /2 − (−

(6)

− → where → w = ∇×− u i.e., the vorticity vector for fluid phase, similarly the identity (6) is → →=∇×− v. true for dust phase also with vorticity vector − w p − → Since the forces are conservative in nature, we have g = −∇ϕ, here ϕ is the gravitational potential. Hence using equation (6) in (2) and (4) one can obtain → ∂− u kN − → → → → + ∇ u2 /2 − (− (→ v −− u) u ×− w ) = −ρ−1 ∇p − ∇ϕ + ν∇2 − u + ∂t ρ → ∂− v → → → →) = −∇ϕ + k (− + ∇ v 2 /2 − (− u −− v) v ×− w p ∂t m

(7) (8)

→ → → → = ∇×→ − → For a Beltrami flow, we know that − w = ∇×− u = α− u and − w v = α− v for p fluid and dust phase respectively, where α is a scalar quantity. So we get − → → → → = 0. u ×− w = 0 and − v ×− w p

(9)

By taking curl on both sides of equation (7) and (8) and using (9), we see that → kN − ∂− w → → →−− w+ w) = ν∇2 − (w p ∂t ρ → k → − ∂− w p →) = (− w −w p ∂t m

(10) (11)


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By virtue of system of equations (5) the intrinsic decomposition of equations (10) and → → → → = α− v one can obtain the following forms: (11) and using the fact that − w = α− u and − w p 2 ∂ us ∂us kN =ν (vs − us ) − Cr u s + (12) 2 ∂t ∂b ρ ∂us − us ks2 (13) 2u2s ks = ν 2σb ∂b ∂us 0 = ν us ks τs − 2kb (14) ∂b k ∂vs = (us − vs ) (15) ∂t m (16) vs2 ks = 0. where Cr = (σn2 + kn2 + kb2 + σb2 ) is called curvature number [3]. From equation (16) we see that vs2 ks = 0 which implies either vs = 0 or ks = 0. The choice vs = 0 is impossible, since if it happens then us = 0, which shows that the flow doesn’t exist. Hence ks = 0, it suggests that the curvature of the streamline along tangential direction is zero. Thus no radial flow exists. CASE-1: In this case, the equations (12) and (15) are to be solved when subjected to the following initial and boundary conditions; Initial condition; at t = 0; us = 0, vs = 0 Boundary condition; for t > 0; us = 0, at b = 0 and us = u0 , at b = h where u0 is a constant. Let us consider the following non-dimensional quantities, u∗s =

us h vs h tU , vs∗ = , b∗ = b/h, t∗ = 2 ; U U h

where U is the characteristic velocity. Using the above non-dimensional quantities we get the non-dimensionalized form of the equations (12), (15) and the boundary conditions as follows; ∂us h ∂ 2 us h3 Cr kN h2 = u (vs − us ) − + s ∂t Re ∂b2 Re ρU kh2 ∂vs = (us − vs ) ∂t mU hu0 at b = 1. us = 0 at b = 0 and us = U

(17) (18) (19)

where Re = U h/ν is the Reynold’s number. We define Laplace transformations of us and vs as ∞ Us =

e 0

−xt

∞ us dt and Vs = 0

e−xt vs dt

(20)


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Applying the Laplace transform to equations (17), (18) and to (19), then by using initial conditions one obtains h d2 Us h3 Cr h2 l − + U (Vs − Us ) s Re db2 Re Uτ h2 (Us − Vs ) xVs = Uτ hu0 Us = 0 at b = 0 and Us = at b = 1 Ux

xUs =

where l =

mN ρ

and τ =

m . k

(21) (22) (23)

Equation (22) implies Vs =

h2 Us (h2 + xU τ )

(24)

Eliminating Vs from (21) and (24) we obtain the following equation d2 Us − Q2 Us = 0 (25) db2 lh2 1 + . whereQ2 = h2 Cr + xRe h (xU τ +h2 ) The velocities of fluid and dust particle are obtained by solving the equation (25) subjected to the boundary conditions (23) as follows hu0 Us = U

sinh(Qb) x sinh(Q)

Using Us in (24) we obtain Vs as u0 h3 Vs = U (xU τ + h2 )

sinh(Qb) x sinh(Q)

By taking inverse Laplace transform to Us and Vs , one can obtain us and vs , as; ∞ hu0 sinh(Xb) 2πh2 u0 (−1)r r sin(rπb) us = − U sinh(X) U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 × + x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] ∞ hu0 sinh(Xb) 2πh4 u0 (−1)r r sin(rπb) − vs = U sinh(X) U Re r=0 ex1 t (h2 + x1 U τ ) ex2 t (h2 + x2 U τ ) + × x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] Shearing Stress (Skin Friction): The Shear stress at the boundaries b = 0 and b = 1


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are given by ∞ 2π 2 μh2 u0 u0 μhX D0 = − (−1)r r2 U sinh(X) U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 × + x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ]

2π 2 μh2 u0 2 u0 hμ X coth(X) − r D1 = U U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 + × x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] ∞

CASE-2: In this case, solve the equations (12) and (15) when subjected to the following initial and boundary conditions i.e., both lower and upper plates are moving with uniform velocity as; Initial condition; at t = 0; us = 0, vs = 0 Boundary condition; for t > 0; us = u0 , at b = 0 and us = u1 , at b = h where u0 and u1 are constants. By applying the same procedure as in case-1, one can obtain us and vs as follows; ∞ h u1 sinh(Xb) − u0 sinh(X(b − 1)) 2πh2 r[u1 (−1)r − uo ] sin(rπb) us = − U sinh(X) U Re r=0 x1 t 2 2 x2 t 2 e (h + x2 U τ )2 e (h + x1 U τ ) + × x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] ∞ h u1 sinh(Xb) − u0 sinh(X(b − 1)) 2h4 π r[u1 (−1)r − u0 ] sin(rπb) − vs = U sinh(X) U Re r=0 x1 t 2 x2 t 2 e (h + x1 U τ ) e (h + x2 U τ ) + × x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] Shearing Stress (Skin Friction): The Shear stress at the boundaries b = 0 and b = 1 are given by ∞ μhX u1 − u0 cosh(X) 2π 2 h2 μ 2 r [u1 (−1)r − u0 ] D0 = − U sinh(X) U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 + × x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] ∞ μhX u1 cosh(X) − u0 2π 2 h2 μ 2 r [u1 − u0 (−1)r ] − D1 = U sinh(X) U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 × + x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ]

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CASE-3: For this case, consider an initial and boundary conditions as; Initial condition; at t = 0; us = 0, vs = 0 Boundary condition; for t > 0; us = u0 + u1 t, at b = 0 and us = u2 + u3 t, at b = h. where u0 , u1 , u2 and u3 are constants. Now we obtain us and vs as follows; ∞ h u2 sinh(Xb) − u0 sinh(X(b − 1)) 2πh2 us = r[u2 (−1)r − uo ] sin(rπb) − U sinh(X) U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 + × x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] 3 th h2 Re(1 + l) + − coth(X) U2 2XU 2 u3 sinh(Xb) − u1 sinh(X(b − 1)) × sinh(X) 2 h Re u3 b cosh(Xb) − u1 (b − 1) cosh(X(b − 1)) + (1 + l)) 2XU 2 sinh(X) ∞ 4 2h π r[u3 (−1)r − u1 ] sin(rπb) − 2 U Re r=0 ex1 t (h2 + x1 U τ )2 ex2 t (h2 + x2 U τ )2 × 2 2 + x1 [(h + x1 U τ )2 + lh4 ] x22 [(h2 + x2 U τ )2 + lh4 ] h u2 sinh(Xb) − u0 sinh(X(b − 1)) vs = U sinh(X) ∞ 2h4 π r[u2 (−1)r − u0 ] sin(rπb) − U Re r=0 ex2 t (h2 + x2 U τ ) ex1 t (h2 + x1 U τ ) + × x [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] 13 h2 Re(1 + l) hτ th − − coth(X) + U2 U 2XU 2 u3 sinh(Xb) − u1 sinh(X(b − 1)) × sinh(X) 2 h Re(1 + l) u3 b cosh(Xb) − u1 (b − 1) cosh(X(b − 1)) + 2XU 2 sinh(X) ∞

2hπ r[u3 (−1)r − u1 ] sin(rπb) − Re r=0 ex1 t (h2 + x1 U τ ) ex2 t (h2 + x2 U τ ) + × 2 2 x1 [(h + x1 U τ )2 + lh4 ] x22 [(h2 + x2 U τ )2 + lh4 ] Shearing Stress (Skin Friction): The Shear stress at the boundaries b = 0 and b = 1 are given by


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∞ μhX u2 − u0 cosh(X) 2π 2 h2 μ 2 D0 = r [u2 (−1)r − u0 ] − U sinh(X) U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 + × x1 [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] 3 th X h2 Re(1 + l) u3 − u1 cosh(X) +μ − coth(X) U2 2U 2 sinh(X) ∞ μh2 Re(1 + l)u1 2h4 π 2 μ 2 − 2 r [u3 (−1)r − u1 ] + 2U 2 U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 + × 2 2 x1 [(h + x1 U τ )2 + lh4 ] x22 [(h2 + x2 U τ )2 + lh4 ] ∞ μhX u2 cosh(X) − u0 2π 2 h2 μ 2 r [u2 − u0 (−1)r ] − D1 = U sinh(X) U Re r=0 ex2 t (h2 + x2 U τ )2 ex1 t (h2 + x1 U τ )2 + × x [(h2 + x1 U τ )2 + lh4 ] x2 [(h2 + x2 U τ )2 + lh4 ] 1 3 h2 Re(1 + l) u3 cosh(X) − u1 th X − coth(X) +μ U2 2U 2 sinh(X) ∞ 2 4 2 μh Re(1 + l)u3 2h π μ − 2 r2 [u3 − u1 (−1)r ] + 2U 2 U Re r=0 ex1 t (h2 + x1 U τ )2 ex2 t (h2 + x2 U τ )2 + × 2 2 x1 [(h + x1 U τ )2 + lh4 ] x22 [(h2 + x2 U τ )2 + lh4 ] where 2 1 x1 = − (h Cr + r2 π 2 )hU τ + Reh2 (1 + l) 2ReU τ 1 + ((h2 Cr + r2 π 2 )hU τ + Reh2 (1 + l))2 − 4ReU τ (h2 Cr + r2 π 2 ) h3 2ReU τ 2 1 x2 = − (h Cr + r2 π 2 )hU τ + Reh2 (1 + l) 2ReU τ 1 − ((h2 Cr + r2 π 2 )hU τ + Reh2 (1 + l))2 − 4ReU τ (h2 Cr + r2 π 2 ) h3 2ReU τ X = h2 Cr

Conclusion The figures 3, 4 and 5 represents the velocity profiles for the fluid and dust particles respectively, which are parabolic in nature. It is observed that the path of fluid particles much steeper than that of dust particles. Further one can observe that if the dust is very fine i.e., mass of the dust particles is negligibly small then the relaxation time of dust particle decreases and ultimately as τ → 0 the velocities of fluid and dust particles will be the same. Also we see that the fluid particles will reach the steady state earlier than

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the dust particles. The Reynolds number (Re) which means the inertial motion over the viscous resistance. One can observed that the impressive effect of Reynolds number on the velocity field. It is seen that the Reynolds number is favorable to the velocity fields i.e., for a constant value t, the velocity profiles for both fluid and dust particles increases as Reynolds number increases. The graphs are drawn for the following values h = 1, U = 0.8, r = 2, τ = 0.5, Cr = 1, u0 = 0.5, u1 = 1, u2 = 1.5, u3 = 2, l = 1, t = 0.2.

Fig. 3 Variation of fluid and dust phase velocity with b



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References [1] C.S.Bagewadi and A.N.Shantharajappa, A study of unsteady dusty gas flow in Frenet Frame Field, Indian Journal Pure Appl. Math., 31, 1405-1420, (2000). [2] C.S.Bagewadi and B.J.Gireesha, A study of two dimensional steady dusty fluid flow under varying temperature, Int. Journal of Appl. Mech. & Eng., 09, 647-653, (2004). [3] C.S.Bagewadi and B.J.Gireesha, A study of two dimensional unsteady dusty fluid flow under varying pressure gradient, Tensor, N.S., 64, 232-240, (2003). [4] Barret O’ Nell, Elementary Differential Geometry, Academic Press, New York & London, (1966). [5] N.Datta & D.C.Dalal, Pulsatile flow of heat transfer of a dusty fluid through an infinitly long annlur pipe, Int. J. Multiphase flow, 21(3), 515-528, (1995). [6] A.Eric, Nauman, J.Kurtis, Risic, M.Tony, Keaveny, & L.Robert Satcher, Quantitative Assessment of Steady and Pulsatile Flow Fields in a Parallel Plate Flow Chamber , Annals of Biomedical Engineering, 27, 194-199, (1999). [7] B.J.Gireesha , C. S. Bagewadi & B.C.Prasannakumara, Flow of unsteady dusty fluid under varying periodic pressure gradient, ’Journal of Analysis and Computation’, 2(2), 183-189, (2006). [8] B.J.Gireesha , C. S. Bagewadi & B.C.Prasannakumara, Flow of unsteady dusty fluid between two parallel plates under constant pressure gradient,, Tensor.N.S. 68 (2007). [9] Indrasena, Steady rotating hydrodynamic-flows,, Tensor, N.S., 350-354, (1978). [10] R.P.Kanwal, Variation of flow quantities along streamlines, principal normals and bi-normals in three-dimensional gas flow, J.Math., 6, 621-628, (1957). [11] J.T.C.Liu, Flow induced by an oscillating infinite plat plate in a dusty gas, Phys. Fluids, 9, 1716-1720, (1966). [12] D.H.Michael and D.A.Miller, Plane parallel flow of a dusty gas,, Mathematika, 13, 97-109, (1966).

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[13] T.M.Nabil, EL-Dabe, M.G.Salwa and EL-Mohandis, Effect of couple stresses on pulsatile hydromagnetic poiseuille flow, Fluid Dynamic Research, 15, 313-324, (1995). [14] P.Nemanyi and R.Prim, Some properties of rotational flow of a perfect gas, PROC. N. A. S., 34, 119-124, (2007). [15] P.F.Pokhil and O.A.Uspenskii, Gromeka-beltrami flow in a semiinfinite cylindrical pipe., Jou. Fluid Dyn., 6(2), 281-284, (1971). [16] G.Purushotham and Indrasena, On intrinsic properties of steady gas flows, Appl.Sci. Res., A 15, 196-202,(1965). [17] N. Rudraiah, Magnetohydrodynamic Beltrami Flows, Journal of the Physical Society of Japan, 29(1), 205-209,(1970). [18] P.G.Saffman, On the stability of laminar flow of a dusty gas, Journal of Fluid Mechanics, 13, 120-128,(1962). [19] C.Truesdell, Intrinsic equations of spatial gas flows, Z.Angew.Math.Mech, 40, 9-14, (1960). [20] Z.Yoshida and S.M.Mahajan, Simultaneous Beltrami Conditions in Coupled Vortex Dynamics, Phy. Rev. Letters, 81(22) , 4863-4866,(1998).