Basics of Fluid Kinematics

The previous chapters dealt with important basic knowledge and information of mathematics and physics as applied in the field of fluid mechanics. This knowledge is needed to describe fluid flows or derive and utilize the basic equations of fluid mechanics in order to solve flow problems. In this respect, it is important to know that fluid mechanics is primarily interested in the velocity field U _j (x _i , t), for given initial and boundary conditions, and in the accompanying pressure field P(x _i , t), i.e. fluid mechanics seeks to present and describe flow processes in field variables. This presentation and description result in the “Eulerian form” of considerations of fluid flows. This form is best suited for the solution of flow problems and is thus mostly applied in experimental, analytical and numerical fluid mechanics. Thanks to the introduction of field quantities also for the thermodynamic properties of a fluid, e.g. the pressure P(x _i , t) and the temperature T(x _i , t), the density ρ(x _i , t) and the internal energy e(x _i , t), and for the molecular transport quantities, e.g. the dynamic viscosity μ(x _i , t), the heat conductivity λ(x _i , t) and the diffusion coefficients D(x _i , t), a complete presentation of fluid mechanics is possible. With the inclusion of diffusive transport quantities, i.e. the molecular heat transport q̇ _i (x _i , t), the molecular mass transport ṁ _i (x _i , t) and the molecular momentum transport λ _ij (x _i , t), it is possible to formulate the conservation laws for mass, momentum and energy for general application. The basic equations of fluid mechanics can thus be formulated locally, as is shown in Chap. 5, and hold for all flow problems in the same form. The differences in the solutions of these equations result from the different initial and boundary conditions that define the actual flow problems. These enter into the solutions by the integration of the locally formulated basic fluid flow equations.