Viscous dissipation of energy

Let us calculate the rate of change of the kinetic energy at any instant t 0, after the over-pressure has been removed. Any influence of gravitational and electrical forces is neglected. The kinetic energy of the fluid can be expressed as an integral over the space V occupied by the channel as 

The time-derivative can be obtained by using the N-S equation for incompressible flow as 

Without the pressure force and neglecting the gravitational force, we have  

Substituting equation (2.92) in equation (2.90), the rate of change of the kinetic energy can be written as 

Using incompressibility flow assumption (V p) = 0 and Gauss divergence theorem  

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Consider laminar, steady, two-dimensional free convection with no viscous dissipation of energy of an incompressible fluid. The concept used here is that the fluid has constant properties, but the body force is produced by a difference in density caused by the temperature distribution. For the continuity equation, we have... 

The Taylor microscale does not represent any distinct group of eddies, but the ratios and are fair estimates of the rate of strain characterizing viscous dissipation of energy (e.g., [5], p. 148). 

Real liquids are viscous and are characterized by internal shear stresses and viscous dissipation of energy. The processes occurring in a viscous liquid are thermodynamically irreversible and have spatial heterogeneity. 

The second term in the right part (5.31) is called dissipative function O, because it characterizes the rate of viscous dissipation of energy in unit volume of fluid. 

As a result of the velocity gradients, there will be heat generation in the fluid from the viscous dissipation of energy. In rectilinear flow, the rate of energy dissipation per unit volume is given by ... 

The relaxation of PoiseuiUe flow to obtain an expression for the rate of viscous dissipation of energy, — is presented in the previous section. In this section, let us analyze the steady-state PoiseuiUe flow. Here, the pressure P dSi) to the left-hand side on dS is higher than the pressure P dS2) to the right-hand side on dS2 given as... 

The second term in equation (2.100) is the viscous dissipation part. The result for the viscous dissipation of energy in steady-state Poiseuille flow can be written as... 

The Taylor microscale does not represent any distinct group of eddies, but the ratios and are fair estimates of the rate of strain characterizing viscous dissipation of energy (e.g., [5], p. 148). Further details on this theory are given by Tennekes and Lumley ([168], Sect. 3.2), Hinze ([67], Chap. 1) and Pope ([122], p. 198). 

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