Conservation law of fluid motions and boundary conditions

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1 Conservation laws of fluid motion and boundary conditions Governing equations of fluid flow and heat transfer The governing equations of fluid flow represent mathematical statements of the conservation laws of physics: - The mass of a fluid is conserved - The rate of change of momentum equals the sum of the forces on a fluid particle (Newton’s second law) - The rate of change of energy is equal to the sum of the rate of heat addition to and the rate of work done on a fluid particle (first law of thermodynamics) - The fluid will be regarded as a continuum. For the analysis of fluid flows at macroscopic length scales the molecular structure of matter and molecular motions may be ignored. - We describe the behaviour of the fluid in terms of macroscopic properties, such as velocity, pressure, density and temperature, and their space and time derivatives. These maybe thought of as averages over suitably large numbers of molecules. - A fluid particle or point in a fluid is then the smallest possible element of fluid whose macroscopic properties are not influenced by individual molecules. We consider such a small element of fluid with sides δx, δy and δz All fluid properties are functions of space and time so we would strictly need to write (x, y, z, t), p(x, y, z, t), T(x, y, z, t) and u(x, y, z, t) for the density, pressure, temperature and the velocity vector respectively. We will not explicitly state the dependence on space co-ordinates and time. The element under consideration is so small that fluid properties at the faces can be expressed accurately enough by means of the first two terms of a Taylor series expansion from the cnter of the element. p − p 1 x and p + p 1 x x 2 x 2 2 Net rate of flow of mass in fluid = of mass into element fluid element Mass…