1 Introduction A brief history of the Lagrangian modelling and Smoothed Particle Hydrodynamics method is presented. The advantages in relation to mesh methods, the growth of its use for the solution of the physical laws governing the flow of fluids and the transport of energy, with a view to the advancement of computational processing techniques, are presented. The importance and frequency of its use in solving problems in the continuum mechanics are also emphasized. Finally, a brief presentation of the contents of each subsequent chapter is presented. 2 Physical-Mathematical Modelling Chapter 2 presents the continuum hypothesis that enables that the physical laws of conservation (mass, momentum and energy) be written by mathematical partial differential equations. Besides these equations, the equation of state for the prediction of the dynamic pressure of a fluid flowing, the concept of the modified pressure and the modelling of the internal energy are also presented. 2.1 The Continuum Hyphotesis 2.

2 Physical Laws of Conservation 2.3 Pressure Modelling 2.3.1 Equation of State for Dynamic Pressure 2.3.2 Modified Pressure 2.4 Specific Internal Energy Modelling 3 Smoothed Particle Hydrodynamics Method In this chapter, the fundamentals of the SPH method and its application in the discretisation of the continuum domain in particles are presented. The SPH approximations to the equations of conservation are deduced and explained.

Kernels used in interpolations, temporal integration methods, the particle inconsistency problem and numerical corrections applied in simulations are also presented. A special attention is given to the presentation of commonly boundary conditions techniques applied in SPH. The use of virtual or dynamic particles and artificial repulsive forces are discussed and the reflective boundary conditions technique, which respects the laws of physics and continuum mechanics, is defended. 3.1 Fundamentals 3.2 Discretisation of the Continuum Domain 3.2.1 Approximation of the Divergent of a Vectorial Function 3.

2.2 Approximation of the Gradient of a Scalar Function 3.2.3 Approximation of the Laplacian 3.2.4 SPH Approximations for the Conservation Equations 3.2.5 Errors in SPH Approximations 3.

2.6 Smoothing Functions 3.2.7 Neighbouring Particle Search 3.2.8 Treatment of the Free Surface 3.2.9 Treatment of the Interfaces 3.

2.10 Turbulence 3.2.11 Variable Smoothing Length 3.2.12 Numerical Aspects and Corrections 3.3 Temporal Integration Methods 3.3.

1 Euler''s Integration Method 3.3.2 Leap Frog Method 3.3.3 Predictor-Corrector 3.4 Consistency ] 3.4.1 Restoration of the Consistency 3.

5 Boundary Treatment Techniques 3.5.1 Virtual Particles 3.5.2 Reflective Boundary Conditions 3.5.3 Dynamic Boundary Conditions 3.5.

4 Additional Remarks 4 Applications in Continuum Fluid Mechanics and Continuum Transport Phenomena Chapter 4 brings some applications of the SPH method in transport phenomena and continuum fluid mechanics problems. This chapter presents the implementation of the SPH particle method, show the difficulties found and discuss the results achieved. Four cases have been studied encompassing transport phenomena (diffusion in a flat plane), hydrostatic (still fluid within a immobile reservoir) and hydrodynamics (dam breaking) and oil spreading on a calm sea. New solutions to problems existing in the literature were found taking care to avoid the use of molecular concepts in the macroscopic scale of the continuum. At the final, concluding remarks to each studied case are listed. 4.1 Heat Diffusion in a Homogeneous Flat Plate 4.1.

1 Physical-Mathematical Modelling 4.1.2 Numerical Simulations 4.2 Still Liquid within an Immobile Reservoir 4.2.1 Physical-Mathematical Modelling 4.2.2 Numerical Simulations 4.

3 Dam Breaking 4.3.1 Physical-Mathematical Modelling 4.3.2 Numerical Simulations 4.4 Oil Spreading on a Calm Sea 4.4.1 Motivation 4.

4.2 Physical-Mathematical Modelling 4.4.3 Numerical Simulations 4.5 Concluding Remarks Bibliography Appendix Index.