Lecture 3: Tensor Analysis a – scalar A i – vector, i=1,2,3 σ ij – tensor, i=1,2,3; j=1,2,3 Rules for Tensor Manipulation: 1.A subscript occurring twice is a repeated (or dummy) index and is summed over 1,2, and 3. 2.A subscript occurring once in a term is called a free index, can take on the range 1,2,3 but not summed. 3.No index can appear in a term more than twice.

Scalar and Vector Products Scalar product: Vector product: vector notations tensor notations

Kronecker delta -- Kronecker delta (or, delta-symbol) Manipulations involving delta-symbol,

Levi-Civita symbol -- even permutation -- odd permutation Properties: (even permutation) (odd permutation)

Calculus -- gradient -- divergence -- curl Example of use of tensor notations:

Levi-Civita Tullio Levi-Civita (29 March 1873 – 29 December 1941) (pronounced /'levi ʧ ivita/) was an Italian mathematician, most famous for his work on tensor calculus and its applications to the theory of relativity

Lecture 4: Hydrodynamic approach Lagrangian and Eulerian Description Material Derivative Number of governing equations

Fluid velocity 1. Hydrodynamics is a macroscopic approach when the random motion of single molecules is averaged over a macroscopic volume. 2. The smallest examined object in hydrodynamics is a ‘fluid particle’. A fluid particle consists of ‘many’ randomly moving molecules. 3. Fluid velocity is defined as the velocity of the centre mass of a fluid particle.

Lagrangian and Eulerian Description. Material Derivative Lagrangian: tracing the position and velocity of chosen fluid particles. All variables are function of time. Eulerian: tracing the fluid velocity (and other quantities) at a chosen point. All variables are functions of time and coordinate. Material derivative: as an example, consider the rate of change of density ρ of a fluid particle, full or material derivative:

Number of unknowns/governing equations To define the thermodynamic state of a single-phase fluid, two variables are needed, e.g. temperature and pressure. (only pressure would be required for isothermal flow.) + three components of the velocity field. On the whole, 5 variables (unknowns) are required to describe a single-phase fluid flow. As the number of equations should be equal to the number of unknowns, 5 governing equations should be provided. These are the continuity equation, the Navier-Stokes equation (this is the vector equation, hence, it gives 3 scalar equations), and the equation for the energy transport.