Presented by:
Santosh Kumar Verma

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introduction
The problem considered is that of a general class of nonlinear free boundary problems, such as those characterized by moving boundaries whose motion is not known a priori but must be determined as part of the solution.
Specifically, the problem is concerned with axisymmetric fluid flow in tubes with surface solidification. Initially, the fluid is flowing in a tube with a fully developed velocity and a uniform temperature distribution. A segment of the tube is then given a step input in the wall temperature to a constant sub-fusion value. As a result, a twodimensional solidification start at the wall. The interface between the solid and the liquid phases moves inward. During freezing, the liquid floe rate into the cooled section is maintained constant. The inlet velocity and the temperature remain fully maintained constant. The inlet velocity and temperature remain fully developed and uniform respectively. However, the flow field in the cooled section is characterized by a boundary layer flow in the entrance region, and a fully developed flow further downstream. The inherent difficulty in the free-boundary problem isa nonlinear boundary condition that must be satisfied at the moving interface.
PREVIOUS WORKS
A lot of research and scientific work has been done and established in the field in the time gone by. The present state of work is very steady. To make note of some of the authors and scientists who have put there remarkable hard in the field are too many.
Excellent literature reviews are given by Boley and Muehlbeuer and Suderland. Most of these problems deal with a phase change without fluid flow or with external flow. Non-flow problems usually are based on two coupled conduction equations to be satisfied in the solid and liquid regions. The external flow problems ordinarily can be uncoupled, since the field variables of the external phase are not significantly affected by the motion of the free boundary.
Limited work has been done on problems involving internal flow with surface solidification. In such systems, the dynamic and thermal response of liquid phase is directly affected by the interface motion. Therefore, the field equations in both phases cannot be uncoupled unless one of the phases is assumed to be at fusion temperature.
Grigorian has considered a special one-dimensional problem of melting due to friction between two moving solid bodies. The problem was formulated in terms of the equations of continuity, motion and energy in both phases. The problem has a self-similar solution; therefore, an exact solution of the interface position was determined to within a constant which was evaluated approximately for some limiting conditions.
Bowley and Coogan considered melting of two parallel quarter-infinite solids due to an internal fluid flow between the solids. Bowley’s major restriction was that the solid region be maintained at the melting temperature throughout. This allowed uncoupling of the equations for the two regions. An integral method was used to transform the Cartesian field equations of continuity, momentum and energy to a set of first-order nonlinear partial differential equations which were then solved by quadrature.
Zerkle and Sunderlandconsidered a steady-state caseof fluid flow in tubes with surface solidification.Experimental results were obtained andused to develop a semi-empirical solution. A steady-state analytical solution was also determined. At steady state, the interface is stationary. Zerkle made use ofthis and transformed the convection equation to the classical Graetzform by assuming a parabolic velocity distribution. The coefficients of theseries solution were evaluated numerically.
Ozisik and Mulliganobtained a quasi-static solution to the freezing of liquids in forced flow inside tubes. The problem was formulated in terms of a steady-state one-dimensional conduction equation in the solid region, and a transient two-dimensional convection equation in the liquid. The method of solution was based on the integral transforms which could be used only with the assumption of slug velocity. According to the authors, the applicability of their solution is restricted to the regions where the rate of change of thickness of the frozen layers is small with respect to both time and distance, along the tube (close to steady state and away from the entrance region).
Few free boundary problems have been solved exactly. Most solutions have been obtained numerically or by approximate analytical methods. Of interesthere are the approximate variational methods. These methods, based on the minimum principle, have been successfully applied in optics, dynamics,wave, mechanics, quantum mechanics and Einstein’s law of gravitation. Helmholtz was probably the first to attempt to apply the variational principles to thermodynamics; however, the minimum principles werenot directly applicableto the dissipative systems. Biot developed amethod based on the principle of minimum rate of entropy production and applied it to several one-dimensional external flow problems. The method has also been applied by Lapadula and Mueller to an external flow problem involving freezing over a flat plate. A more general formulation of the variational principle, known as Lagrangian formalism, is usually presented without reference to any specific system. The Lagrangian formalism may be specialized to solve a diffusion orconduction equation. The variational solution presented in this paper is based on the Lagrangian formalism. The,application of the method is extended to solve the free-boundary problemsinvolving internal flow.
ANALYSIS
Problem Statement
The problem can be formulated in cylindrical coordinates in terms of a complete set of field equations in the liquid and solid region; both of these regions being coupled by a set of nonlinear boundary conditions to be satisfied at the moving liquid-solid interface. An order of magnitude analysis of such a set shows that the axial conduction, axial viscous shear, dissipation, body forces and radial pressure gradient may be neglected under the usual conditions of the boundary layer flow.
Two variational solutions of the above problem are presented in this report. The first variational solution, abbreviated as (V), is less accurate than the second, (VN), solution. The less accurate solution (V) is presented because it is more general and also because its examination permits the evaluation of several aspects of the physical problem.
Also author has used numerical solutions to solve the problem to compare the solution obtained with that of the solutions obtained from variational formulations. Authors have used this numerical solution tom plot various graphs to show different characteristics of the problem. But these numerical solutions have not been included in the paper. Also, because of the complexity of these solutions no attempt has been made to obtain them in this report.