Rate of viscous dissipation

In the case of laminar flow in a pipe, work is done by the shear stress component rTX and the rate of doing work is the viscous dissipation rate, that is the conversion of kinetic energy into internal energy. The rate of viscous dissipation per unit volume at a point, is given by... 

Thus the physical significance of U2 and U becomes evident. The former reflects the reduction (U2 < 1) of the rate of melt removal of the him by drag how as a result of temperature dependence and shear thinning of the viscosity, whereas U /2 is the rate of viscous dissipation (per unit width) in the melt him. The relative signihcance of conduction and dissipation for melting is obtained by comparing the two terms in the square Brackets in Eq. 5.7-55. 

The homogeneous rate of dissipation of acoustic energy per unit volume by viscosity in the gas is of the order of fi V /(a/coy, since a/cu is the wavelength that determines the velocity gradient associated with the sound field. From this estimate and equation (39) we see that the ratio of the homogeneous rate of viscous dissipation to the rate of viscous dissipation at the walls is roughly Here pUpa) is on the... 

The term Ey is the rate of viscous dissipation of mechanical energy its estimation is described in Bird, Stewart, and Lightfoot (2002). is a reference pressure, in units of M/Lt the resulting SI unit is the Pascal. 

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. 

The rate of viscous dissipation due to the contribution of the density variation... 

The rate of viscous dissipation for an incompressible fluid (Oinc)... 

Frictional heating n. (1) Heat evolved when two surfaces rub together. For a ffictional force F and relative velocity V, the rate of heat evolution is FV (SI J/s). (2) Viscous dissipation within liquids undergoing shear flow. At any point in a laminar-flowing liquid, the rate of viscous dissipation, per unit volume, is equal to the product of the shear rate times the shear stress, which is also equal to the product of the local viscosity and the square of the shear rate. Both these types of functional heating are important mechanisms of plastication in extruders and injection molders. 

The rate of viscous dissipation represents the amount of energy irreversibly transformed into heat by means of viscous friction. It is calculated in the following form from Equation 2 ... 

Further, Figure 32b illustrates that the phase shift between the cyclic stress and strain for a viscoelastic material results in a hysteresis loop, the size of which reflects the rate of viscous dissipation or heat generation during the cyclic deformation. For an ideal Hookean material, there is a zero-area loop (Fig. 32a). But as tan 5 increases, so does the size of the hysteresis loop. This notion is particularly important in applications such as the damping of vibration and soimd. 

The rate of viscous dissipation e is an important parameter that represents the energy injected from large scales to small scales of turbulence. This rate can be described by the relation [4]... 

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... 

However, the rate of viscous dissipation D must of course be positive, and this at once requires that the coefficients of terms linear in the gradients of velocity and angular velocity and in the angular velocity be zero, allowing us to deduce that stress and couple stress must take the forms... 

The rate of viscous dissipation V is necessarily positive. Given that the signs of ti/i, vji j and Vi j may be arbitrarily chosen, it follows that the coefficients of any terms linear in Wi Wij and Vij which occur in (4.55) must equate to zero so that V remains positive. This leads to the conclusion that the stress Uj and couple stress lij may take the forms [175]... 

Friction is generated at the microscopic level by relative motion between polymer molecules in flow. In a simple shear flow such as that introduced in Section 7.2.1, the relative velocity between a polymer molecule and its surroundings can be estimated as Vc = yf g, where is the polymer coil radius in the melt. If the rate of shear is so small that it does not perturb the polymer shape, the rate of viscous dissipation per molecule q is therefore given by... 

The rate of viscous dissipation per unit volume q-r is related to the rate of dissipation per molecule by... 

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