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Divergence and Curl Operators in Skew Coordinates

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Divergence and Curl Operators in Skew Coordinates International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015 Divergence and Curl Operators in Skew Coordinates Rajai S. Alassar and Mohammed A. Abushoshah where r is the position vector in any three dimensional space, Abstract—The divergence and curl operators appear in i.e. r r(,,) u1 u 2 u 3 . numerous differential equations governing engineering and physics problems. These operators, whose forms are well r Note that is a tangent vector to the ui curve where known in general orthogonal coordinates systems, assume ui different casts in different systems. In certain instances, one the other two coordinates variables remain constant. A unit needs to custom-make a coordinates system that my turn out to be skew (i.e. not orthogonal). Of course, the known formulas for tangent vector in this direction, therefore, is the divergence and curl operators in orthogonal coordinates are not useful in such cases, and one needs to derive their rr counterparts in skew systems. In this note, we derive two uuii r formulas for the divergence and curl operators in a general ei or hiei (3) coordinates system, whether orthogonal or not. These formulas r hi ui generalize the well known and widely used relations for ui orthogonal coordinates systems. In the process, we define an orthogonality indicator whose value ranges between zero and unity. The Divergence of a vector field over a control volume V bounded by the surface S , denoted by F , is defined by Index Terms—Coordinates systems, curl, divergence, Laplace, skew systems. F n dS F lim S (4) I. INTRODUCTION V 0 V The curl and divergence operators play significant roles in physical relations. They arise in fluid mechanics, elasticity where n is an outward unit normal vector to the surface S . theory and are fundamental in the theory of electromagnetism, The divergence actually measures the net outflow of a vector [1], [2]. field from an infinitesimal volume around a given point (or The physical significance of the Curl of a vector field F , how much a vector field "converges to" or "diverges from" a denoted by F , is that it measures the amount of rotation given point). The general expression of the divergence for or angular momentum of the contents of a given region of arbitrary orthogonal curvilinear coordinates is given by space. If the value of the curl is zero then the field is said to be irrotational. The curl is defined in an arbitrary orthogonal 1 F () F1 h 2 h 3 curvilinear coordinates (,,)u u u as h1 h 2 h 3 u 1 1 2 3 (5) ()()F h h F h h he h e h e 2 1 3 3 1 2 11 2 2 3 3 uu23 1 F (1) h1 h 2 h 3 u 1 u 2 u 3 The Laplacian operator of a scalar (,,)u1 u 2 u 3 , 2 F1 h 1 F 2 h 2 F 3 h 3 denoted by is defined as the divergence of the gradient of ; that is where e is a unit vector in the direction of u , and i i FFFF1e13 2 e 2 3 e . The length of the tangent vector in 1 hh 2 () 23 the direction of u is known as scale factor, h , and is defined i i h1 h 2 h 3 u 1 h 1 u 1 by (6) hh13 hh12 u h u u h u r 2 2 2 3 3 3 h (2) i u where i 1 1 1 e e e (7) Manuscript received March 29, 2015; revised May 19, 2015. This work h u1 h u 2 h u 3 was supported by King Fahd University of Petroleum & Minerals. 1 1 2 2 3 3 The authors are with the Department of Math and Stat, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia (e-mail: Equations (1) and (5) are valid for orthogonal systems [email protected], [email protected]). DOI: 10.7763/IJMO.2015.V5.461 198 International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015 only (i.e. e1 , e2 , and e3 are orthogonal), [3]-[5]. ee31e2 w 3 h 1 h 2 du 1 du 2 du 3 (11) u3 II. DIVERGENCE FOR GENERAL COORDINATES SYSTEMS Using the same argument, the net flux through the The Divergence Theorem relates the flow, or the flux, of a remaining two pairs of surfaces are: vector field through a surface to the behavior of the vector field inside the surface. More precisely, it states that the outward flux of a vector field through a closed surface is ee31e2 w 2 h 1 h 3 du 1 du 2 du 3 equal to the volume integral of the divergence over the region u2 inside the surface, i.e. ee31e2 w 1 h 2 h 3 du 1 du 2 du 3 u1 w n dS wdv (8) SD w n dS on the left hand side of equation (8) becomes where D is a closed bounded region with piecewise smooth boundary S, n is an outer unit vector normal to the surface S , w we w e w e , wn represents the component ewhh3 1 2 ewhh 2 1 3 ewhh 1 2 3 11 2 2 3 3 u3 u 2 u 1 (12) of w in the direction of n , and dv is the volume bounded du1 du 2 du 3 by the region D. where eeee31 2 is an orthognality indicator of the system, 01e . The volume element, dv, is given by dv h3e3 du 3 h 1 e 1 du 1 h 2 e 2 du 2 (13) The right hand side of equation (8), then, is Fig. 1. An infinitesimal control volume bounded by the surface. wdv w he du h e du h e du Considering the bottom shaded surface dS of the given 33 3 1 1 1 2 2 2 control volume, Fig. 1, one can find that wee31 e2 hhhdududu 1 2 3 1 2 3 (14) we hhh1 2 3 dududu 1 2 3 rr uu12 rr In the limit, by equating (12) and (14), an expression for n , and dS du12 du (9) rr uu12 the divergence of a vector field w in a general coordinates uu12 system, whether orthogonal or not, is given by Hence 1 w () e w1 h 2 h 3 eh1 h 2 h 3 u 1 rr (15) uu12rr ()()e w2 h 1 h 3 e w 3 h 1 h 2 w n dS w du12 du uu23 rruu 12 uu 12 It is easy to see that when e 1 (orthogonal system), ()w1e1 w 2 e 2 w 3 e 3 h 1 e 1 h 2 e 2 du 1 du 2 (10) equation (15) reduces to (5). w3e3 e 1× e 2 h 1 h 2 du 1 du 2 III. CURL FOR GENERAL COORDINATES SYSTEMS On the upper surface, using Taylor series, the outward flux is w3e 3 e12 e h 1 h 2 du 1 du 2 we e e h h du du du ... u 3 312 1 2 1 2 3 3 The net flux through the upper and lower surfaces, then, is Fig. 2. An infinitesimal surface enclosed by the closed path c. 199 International Journal of Modeling and Optimization, Vol. 5, No. 3, June 2015 Stokes' Theorem relates the surface integral of the curl of a 1 vector field over a surface to the line integral of the vector ()[][]w1 h 3 wee32 h 2 w (21) e h h u u field over the surface boundary, or 3 2 2 3 Equations (19), (20) and (21) can be written in the concise w n dS w dr (16) form Sc he h e h e Considering the bottom shaded surface of the given 11 2 2 3 3 1 control volume as shown in Fig. 2, w (22) This surface is bounded by the curve c which is traced eh1 h 2 h 3 u 1 u 2 u 3 counterclockwise. This curve is comprised of the four h1 we1 h 2 w e 2 h 3 w e 3 segments c , c , c , and c . We evaluate the right hand side 1 2 3 4 and the left hand side of (16) over this surface and two other Obviously, the difference between orthogonal and surfaces not opposite to each other. On the shown shaded nonorthogonal coordinates systems lies in the introduction of surface, one can write the scalar triple product eeee into the expressions of 31 2 w n dS F , and F . This product returns a value of unity if the coordinates system is orthogonal. ()()()we w e w e 11 2 2 3 3 hhee 1212hee h du du (17) 112 2 1 2 IV. EXAMPLE hh12ee12 Consider the simple right-handed coordinates system () we e e h h du du 33 1 2 1 2 1 2 (,,)u v z which is related to the Cartesian system by the ( w )3 e h 1 h 2 du 1 du 2 relations x u v where ()w is the component of w in the direction i yu (23) of e . i zz The term w dr in equation (16), over the four segments comprising the curve c, is This system is composed of the planes: yu which are planes parallel to the xz plane, w dr w dr w dr w dr w dr y x v which are planes that are extrusions of the straight c1 c 2 c 3 c 4 w dr w dr w dr w dr lines y x u into the z -direction, and planes that are c1 c 3 c 4 c 2 parallel to the xy plane.
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