Publication: Local and Nonlocal Symmetry Analysis on Complex Flow Problems
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2025-02-25
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Indian Institute of Technology, Jodhpur
Abstract
Fluid flow problems are prevalent in many areas of real life, including engineering,edical sciences, environmental sciences, and geophysics. A suitable set of differential equations is usually used to model a fluid flow system. These problems often exhibit complex behaviors due to the nonlinear nature of the governing equations and intricate boundary conditions. Traditional analytical and numerical methods often struggle to manage these complexities effectively in many cases. Lie symmetry analysis, a mathematical technique developed by Sophus Lie provides a robust method for simplifying and solving these kinds of complicated differential equations by exploiting the intrinsic infinitesimal transformations. Symmetry analysis enhances our ability to solve fluid flow problems explicitly and to address dynamic challenges by simplifying the governing equations and improving computational efficiency. This technique is effective for finding exact solutions that are useful to analyze various critical physical insights of the flow systems. Throughout this thesis, we examine three aspects of symmetry methods for the systems of partial differential equations (PDEs), which govern different complex flow models. Specifically, we demonstrate: (i) how symmetry methods can be applied and adapted to systematically solve the general as well as the boundary value flow problems (BVPs), (ii) how to obtain local symmetries (classical and nonclassical) and local conservation laws using symbolic manipulation software, and (iii) how to use them for identifying nonlocallyrelated systems that may reveal nonlocal symmetries and nonlocal conservation laws for the governing system of PDEs. The classical symmetry analysis, based on one parameter Lie groups, is a powerful tool to identify the symmetries that reduce the complexity of the governing equations and provide explicit solutions. However, its applicability is limited as it fails in many cases to uncover desired symmetries for certain nonlinear differential equations due to its reliance on point transformations and the strict form of the infinitesimal symmetry generator. To address these limitations, nonclassical symmetry analysis extends the framework by incorporating additional constraints, such as compatibility conditions with invariant surface equations, that allow to obtain more general invariant solutions. Additionally, the nonlocal symmetry analysis broadens the scope by considering symmetries that involve integral or global transformations, capturing hidden or nonlocal structures that classical methods overlook. These approaches are essential for studying complex systems, where the classical symmetry analysis might fail to provide results or overlook important solutions. This enhances both the theoretical understanding and practical use of nonlocal symmetry methods in nonlinear sciences. Moreover, in the context of multidimensional PDEs, the classification of subgroups and their reduction to optimal systems play a crucial role in analyzing the governing systems. Optimal classifications streamline the process of identifying all inequivalent solution branches for the governing PDEs. Recognizing the significance of optimal classification in deriving a complete set of group-invariant solutions, researchers are continually working to simplify these techniques, making them more accessible and practical to apply. The application of symmetry analysis to boundary value problems has been relatively under-explored, and thus, we aim to explore the same in this thesis. In view of that, we have considered a viscous, laminar, incompressible channel flow with small-scale contractions and expansions of the weakly permeable walls, and solved this boundary value problem using the classical symmetry analysis method in Chapter 2. The use of classical symmetry analysis reduces the governing system into a fourth-order ordinary differential equation (ODE), which is solved analytically (via double perturbation and variation iteration methods) and numerically (Shooting methods). The results indicate that wall suction/contraction increases near-wall velocity, while injection/expansion slows it down. Notably, the permeation Reynolds number plays a crucial role in changing flow behavior. Moreover, we have investigated another boundary value problem involving porous media flow through a contracting and expanding channel in Chapter 3. In this study, the entire flow dynamics is modeled by the Darcy-Brinkman equations. The walls of the channel are dilating vertically and there is an inflow through the pores of the walls that develops flow within the porous media inside the channel. It is observed that the classical method is insufficient for identifying the required symmetries for the desired reduction. Consequently, nonclassical symmetry analysis is employed to uncover the hidden symmetries. The obtained nonclassical symmetry helps to find explicit solutions, which is not possible using the classical method alone. It is observed that the velocity profiles are much fuller for larger Darcy numbers, and it behaves like a Hartmann flow for the smaller Darcy numbers. The existence of a flow reversal phenomenon is also noticed for a suction flow with strong wall dilation rates. Thereafter, we move on to examine the symmetry classifications and exact solutions for a set of general flow problems. In Chapter 4, we explored the Lie symmetries and optimal classifications using specific modal approaches for the three-dimensional gas-dynamical equations. The two-dimensional optimal subalgebras offer solution ansatz that captures different physical modes, such as the Kelvin mode, normal mode, and other characteristic patterns, along with their possible combinations. The validity of the three-dimensional normal mode solution corresponding to small perturbation is confirmed through a combined ansatz function in a limiting case. Moreover, the conserved quantities of the governing model are identified using the conservation laws multiplier technique. Subsequently, we have studied the classical symmetry analysis of the three-dimensional Navier-Stokes equations corresponding to incompressible arbitrary plane shear flows (viscous and inviscid) in Chapter 5. At first, a linearization is performed concerning small perturbations, which reduces the complexity of the problem. The invariant ansatz functions are systematically derived using the complete set of classical symmetries, which are used to find the exact solutions. By taking the general symmetry, the popular three-dimensional normal modes and corresponding Orr-Sommerfeld equations for the flow system are obtained. Apart from the local symmetry analysis (classical and nonclassical), we also focus on constructing additional exact solutions by using the nonlocal symmetry analysis method for the general Trouton model in Chapter 6. The model consists of a set of quasilinear hyperbolic PDEs, which govern the dynamics of a thin film of a perfectly soluble anti-surfactant solution. In this study, first, we have computed local symmetries, onedimensional optimal systems, and local conservation laws. Then, those are used to construct nonlocally related potential and inverse potential systems, which provide nonlocal symmetries. Specifically, we obtained a new invariant solution from the nonlocal symmetry analysis of the PDEs system, which can not be obtainable through the invariant solutions of the local symmetry analysis method. Keywords: Lie group of transformations; Invariant solutions; Exact solution; Classical symmetry analysis; Nonclassical analysis; Local conservation laws; Nonlocal symmetry; Navier Stokes equations; Gas dynamical equations; Darcy Brinkman equations; Trouton model; Thin film flow; Porous media flow; Dilating channel; Cross velocity; Velocity slip; Reverse flow.
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Mandal, Sougata(2020).Local and Nonlocal Symmetry Analysis on Complex Flow Problems (Doctor's thesis).Indian Institute of Technology, Jodhpur