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Heat Transfer: Governing Equations

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In this section the Navier - Stokes, energy and continuity equations for steady-state, 2D, incompressible flow with constant thermal properties are illustrated. A meaning of all symbols used for physical quantities are given and terms associated with fluid inertia, viscous force, pressure gradient, body force, convection, conduction and viscous dissipation are clearly explained. Furthermore, an explanation of the Stokes Hypothesis is given.

1.1 Navier-Stokes equation


The Navier-Stokes equation (1) is composed by the inertial terms (l.h.s) which arise from the momentum changes and are opposed by the pressure gradient в?‚p/в?‚x, viscous forces (in the square brackets) which always act to retard the flow, and if present, body forces, Fx.

1.2 Energy equation


The energy equation (2) on the left hand side term represents the convection of energy as a result of a fluid motion which is in equilibrium with the conduction of energy through the fluid (r.h.s-1st term) and the viscous dissipation (r.h.s-2nd term) in which Ој is the viscosity and О¦ the dissipation, given by:


It is important to note that for a stationary fluid (u=v=0) the energy equation becomes Laplace’ equation where conduction is the sole mechanism for fluid motion. Therefore, the engineer needs to think about the context of the problem and apply some logical engineering thought prior to solving as in this case there is an absence of fluid motion.

1.3 Continuity (Momentum) equation





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