GALERKIN FINITE ELEMENT MODELING OF THERMOHALINE FLUID IN AN INCLINED CAVITY

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The knowledge of the behavior of thermo-solute in a confined space shall open up many application areas in control process. Flow analysis of a Thermohaline fluid inside a square cavity inclined at angle ‘φ0’ with X-axis under steady state conditions has been investigated has been investigated.. The equations are modified with respect to rotational symmetry of groups. The resulting partial differential equations are modelled using Finite Element Technique. Also we have established a set of shape functions applicable to fourth order partial differential equations. Shape functions and the corresponding surfaces are characterized with respect to its properties. Algorithm has been developed for the computation. This model can be extended to analysis of largescale double diffusive convection subjected to complex boundaries.


INTRODUCTION
Mathematical models for Benard convection with respect to boundary layer, temporal and steady state behavior has been investigated and analysed since 1975. Study of Cavity flow plays a vital role in several engineering applications extending to automobiles, aircrafts to riverbed channels and large scale flows. Gill [1] studied the cavity flow in a two dimensional rectangular cavity with vertical walls maintained at different temperature with huge difference in temperature. Garandet [2] proposed an analytic solution to the problem of buoyancy driven flow via two-dimensional shallow cavity with a transverse magnetic field. The resulting flow has been studied by means of series expansion. Ozoe and Okkado [3] investigated flow inside and over a square cavity with vertical walls at constant temperature, horizontal walls being insulated to three dimensional situations. They found that a field horizontal but parallel to the heated surface was observed to be least effective in suppressing the circulation of flow.
As a continuation of this work Vasseur [4] has studied the effect of transverse magnetic field on the natural convection in an inclined fluid layer using 2-D Navier-Stokes equation including Lorentz's force terms. Thermal conditions are taken to be cooling through opposite walls. Resulting equations are linearized through parallel flow approximations and analytical solutions are obtained for a shallow layer. These results are verified numerically using Finite difference approach.
In the recent studies,it is observed that thermohaline circulation from equator to poles plays a major role in prediction of climate changes Veronis type of Thermal convective flows between differentially heated horizontal/vertical plates has been studied by Banerjee etal [5] . Whitehead [6] studied abrupt transitions in a thremohaline loop through a laboratory model for Stommel type of situations. A distinct abruptive transition were observed in case the cavity experiments even when no measurable hysteresis was found . Menon [7], [8] investigated the flow field characteristics and stability analysis of thermohaline fluids under sinusoidal heating which simulates the solar heating in most of naturally occurring phenomenas.
The flow field characterization of thermohaline fluids inside a cavity gives insight into various hydrodynamic problems related to fluid flow analysis across a coastal regions estuaries straits and Shelves. More over finite element method can handle complex geometries and different types of boundary conditions Therefore In this paper a Galerkin finite element method model has been developed for flow of theromohaline fliuid enclosed in an inclined shallow cavity which will be useful for many practical applications in engineering and oceanographic studies.

MATHEMATICAL MODEL AND GOVERNING EQUATIONS FOR
Modified Thermohaline equations are subjected to changes which follow Boussinesq variation in respect of ρ, µ,C v , α, η, k, according to Banerjee [5] are

(a) Equations in a body frame
The fluid is confined in a shallow square cavity and the cavity is inclined to angle 'φ 0 ' 0 < φ 0 < π 2 with x-axis . The cavity is a square shallow cavity of unit length in the X-Plane making an angle ' φ 0 ' with X-axis. Model is defined in 2-D plane. Shallow cavity is a square from both the corner, the two directions forms the body frames and normal to it depicts the shallow depth of cavity and has been neglected in the analysis. Modified Thermohaline equation in the body turns out to be as follows: Form of static solution shall remain unaltered as the state is static The variables assume the values in the new representation and are given by Perturbations over the static state are defined as Since Jacobian terms and product of perturbations give rise to non-linear terms, these equations are linearized by neglecting product of perturbations and terms of order less than 10 −2 (b) Linearized set of equations are as follows: For the sake of brevity * has been removed. As the heating is maintained at the same time invariant form, the solutions shall settle down to a steady state form.

Steady state equations are given by
Equations are now reduced to a non-dimensional form by introducing the following scales.
Closed form of the solutions satisfying the boundary conditions and equations has not feasible till date. Therefore the other approach adopted by researchers is a solution satisfying the boundary conditions but not necessarily the equations. An FEM approach has the advantage of taking care complex form of equations encompassing the geometry along with flexibility to choose solution of any degree satisfying the boundary conditions at the grids.

(d) Algorithm for Finite element method
Step 1 : Selecting global co-ordinate system for the problem under consideration Step 2: Selecting grids along with specifications and structure of the domain as given in the problem Step 3: Partitioning of domain into subdomains and identifying the subdomains where boundary conditions are specified.
Step 4: Mapping from global coordinate system into local coordinate system.
Step 5: Choice of Solutions and Formulation of Shape functions Step 6. Choice of appropriate weighting functions Step 7. Developing equation for weighted residuals Step 8. Ensemble the weighted error equations in the whole domain Step 9. Incorporate the boundary conditions into the system of weighted error functions.
Step 10. To set up procedure to determine solution at the nodes.
Step 11. Form the general solution in the domain using nodal solution and shape function.
Step 12.To experiment the solution identified above for analysis and behavior of the problem.

CO-ORDINATE TRANSFORMATION:
The main purpose of transforming an element from a global co-ordinate system to local coordinate system is for the purpose of evaluating the integral. Different elements of Mesh can be generated by assigning the global co-ordinate. The element transformation is given by R S1 = R S a 3 cos φ 0 , R S2 = R S a 3 sin φ 0 (3.38) R g1 = R g a 3 cos φ 0 , R g2 = R g a 3 sin φ 0 (3.39)

DERIVATION OF HERMITE FAMILY FINITE INTERPOLATION FUNCTION
The finite element approximation u(x, y) over an element must be differentiable, linearly independent and complete. Thus the polynomial approximation is valid and is to such as to have approximate boundary conditions over a mesh and covers the range of finite values. Depending upon the requirement of the problem, approximate functions can be taken either from Lagrange family of interpolation function or then from Hermite family of interpolation function. For this particular problem, differential equations containing terms of 4 th order derivatives in ψ and lower order in 'T and 'S . Hermite interpolating polynomial containing easy accessibility for the symmetry is preferred to the Lagrange's family. We found that cubic Hermite polynomial is not sufficient to take care of higher order terms in ψ. Therefore a sixth order polynomial in two variables ξ and η having 16 terms has been designed. To preserve symmetry and linear independence some terms are omitted. The approximate solution over an element has been taken as u e (ξ , η) = a0 + a 1 ξ + a 2 η + a 3 ξ 2 + a 4 ξ η + a 5 η 2 + a 6 ξ 3 + a 7 ξ 2 η + a 8 ξ η 2 (4.41) + a 9 η 3 + a 1 0ξ 3 η + a 11 ξ 2 η 2 + a 12 ξ η 3 + a 13 ξ 3 η 2 + a 14 ξ 2 η 3 + a 15 ξ 3 η 3 This is the higher order minimum degree polynomial which satisfies the above condition and gives approximate shape functions. Let Where N i 's are , 10, 11 (4.51) The behavior of flow variable with in an element is described by shape functions, Properties of shape functions are described along with the diagram.

PROPERTIES OF SHAPE FUNCTIONS:
• N 0 (ξ , η) = N 1 (η, ξ ) surface is symmetric in the ξ − η plane. Solution for 'Ψ e ', 'T e ' 'S e ' can be selected • Extreme values occur at (−1, −1) The above system of expressions (5.58) to (5.60) are then represented in a simple matrix notation as Where each block is of size 16 * 16.
The error function is now to be minimized by an appropriate method. This involves a proper Equivalence relation at the nodes is given as follows.
Each element is identified as the position 4 j + I and nodes as 5 j + i, j = 0, 1, 2, 3 The system of equations (5.64) is augmented over the region as follows.

CONCLUSION AND FUTURE SCOPE
In this paper a computational model for flow of thermohaline fluid enclosed in a shallow cavity using Galerkin Finite element method has been developed. This model can be extended to various cavity flows related to thermohaline flows occurring in various applications subjected to appropriate boundary conditions ranging from industrial problems to oceanic flows This work may be extended to cavities consisting of complex geometries in real time studies.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.