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2.11.2 Energy Balance
The energy balance for a continuum may be written in integral form over a control volume as where is the control surface bounding , is the specific internal energy, is the heat flux across , and is the heat supply per mass to the material in resulting from other sources. Bringing the time derivative inside the integral on the left-hand-side, and using the divergence theorem, this integral statement of the energy balance may be written as This statement must be valid for arbitrary control volumes and arbitrary processes, from which we conventionally derive the differential form of the axioms of mass, momentum and energy balance.
For the specialized conditions of a viscous fluid at constant temperature assumed in our treatment, the only state variables for the functions of state , and are and (i.e., the temperature is not a state variable since it is assumed constant). Under these conditions the entropy inequality shows that the specific entropy and the heat flux must be zero, and the Cauchy stress must have the form of (2.11.1-8) where is given by(2.11.1-9) as a function of only, leaving the residual dissipation statement as a constraint that must be satisfied by constitutive relations for . (For a Newtonian fluid, this constraint is satisfied when the viscosities and are positive.) From these thermodynamic restrictions we conclude that , where is the specific (Helmholtz) free energy, with .
For the conditions adopted here (isothermal viscous fluid), the axiom of energy balance reduces to ; since is only a function of , this expression may be further simplified using (2.11.1-6)-(2.11.1-9) to produce . In other words, isothermal conditions may be maintained only if heat dissipated by the viscous stress is emitted in the form of a heat supply density (heat leaving the system). Now, the integral form of the energy balance in (2.11.2-2) simplifies to A comparison of this statement with the statement of virtual work, presented below in (3.5.1-1), establishes a clear correspondence between the virtual velocity and , and between the virtual energy density and , with the latter representing the sum of the internal (free) and kinetic energy densities.