Indian Streams Research Journal ISSN: 2230-7850 Impact Factor : 4.1625 (UIF) Volume - 6 | Issue - 11 | December – 2016 ______LEMMA AND THEOREM WHICH HELP US TO FIND A REPRESENTATION FOR A SOLENOIDAL VECTOR FIELD IN CARTESIAN CO-ORDINATES IN TERMS OF TWO SCALAR FUNCTIONS. Katta Koteswari Research Scholar , Department of Mathematics , Krishna University. Abstract: boundaries. By In many physical making use of this situations, the representation, we governing equations of establish the motion are usually completeness of some given as a vector general solutions of equation where the Stokes, Brinkman quantity of physical and Oseen equations interest may be a proposed for plane vector, like the fluid boundaries in a later velocity in fluid flows. chapter. Despite the However, it has been similarity in the observed that when very simple form. structure of the these quantities are relations in terms of these Chadwick and solution in both the expressed in terms of a scalars. Trowbridge [6] spherical [7] and scalar field, the scalar In 1967, Chadwick and observed that the plane geometries, the satisfies a much Trowbridge [6] showed that any results can be proof of simpler partial divergence free vector field V extended to an completeness of the differential equation can be expressed as infinitedomain (r2oo) solution in the plane than the given vector. with boundedness boundaries case is not conditions On V. obtained by merely where A and B are scalar KEYWORDS : This extension was mimicking the proof functions on any bounded Lemma and theorem , used in [7] given in [7] and annular domain quantity of physical to prove the requires completely a interest , velocity completeness of a different approach potential. certain solution of altogether. where are spherical Stokes equations. The INTRODUCTION : polar co-ordinates. proof of this 2.2Solenoidal Vector A simple example is This result was found to be extension was given Fields in Spherical the role played by a extremely useful in the context by Padmavathi and Polar Co-ordinates velocity potential in of Stokes equations where the Amaranath [9]. This We now discuss irrotaticnal flows or a velocity vector is solenoidal as it representation may some results given in stream function in two gives rise to a complete general not be convenient to [9] which establish dimensional, solution of Stokes use for plane the completeness of incompressible, equations [7], which in turn boundaries. In this certain solutions of inviscid flows. Such a proves to be very convenient in chapter, a new Stokes and Brinkman representation of problems involving spherical representation is equations in infinite vector fields using boundaries, since the scalar given for divergence domains involving scalar functions is, in functions occurring in it satisfy free (solenoidal) spherical bound-aries. particular, useful in simple partial differential vector fields which is The following lemma boundary value equations which can be solved useful in problems and theorem for problems when the easily. Moreover, the boundary dealing with plane infinite domains are boundary conditions conditionrexpressed in terms of due to Padmavathi reduce to simple these scalar functions are of a and Amaranath [9].
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LEMMA AND THEOREM WHICH HELP US TO FIND A REPRESENTATION … Volume - 6 | Issue - 11 | December – 2016
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2 Lemma 1: Let Z ∈ C on S 1, where S 1 is given by
If Z satisfies r • Z = 0, (2.2.1) r•CurlZ = 0, (2.2.2) ∆ • Z = 0, (2.2.3) on S 1, then Z = 0, (2.2.4) ons l.
2 Theorem 1 : If V ∈ C on S 1 and satisfies ∆ •V = 0, then we can find scalar functions A and B such that on S 1, V can he represented as
V = Curl Curl(rA) +Curl(rB), (2.2.5)
where A and B are solutions of the following equations
LA = —r • V, (2.2.6) LB = —r • Cur1V, (2.2.7)
where L is the transverse part of the Laplace operator except for the factor 1/r 2 in spherical polar co- ordinates . From the theorem discussed above, it is possible to express the velocity vector which is solenoidal in terms of two scalars A and B. This representation is used in Chapter-3 to prove the completeness of the solutions of homo¬geneous and non-homogeneous unsteady Stokes equations. In the next section, we shall discuss the representation of solenoidal vector fields in cartesian co-ordinates.
2.3 Solenoidal Vector Fields in Cartesian Co-ordinates We state a lemma and theorem which enable us to find a representation of a solenoidal vector field in cartesian co-ordinates in terms -of certain scalar functions and also determine the partial differential equations satisfied by the scalars themselves. The proofs of the lemma and theorem given here cannot be obtained by merely emulating the proof given in the case of spherical boundaries stated in the previous section.
Lemma 2 : If Z E C 2(IR 3) and satisfies