Frictional dissipation

In trying to find a generalized solution of problem (43) one can come across oscillations of the grid solution and its derivatives ( ripple ), which essentially reduce the accuracy of a scheme. What is more, the lines of discontinuity of derivatives spread over several grid intervals, thus causing the difficulties in determination of proper discontinuity propagation velocity. This is the result of introducing the fictitious friction (dissipation) in the difference approximation. 

An external energy source is necessary to account for frictional dissipation. 

Note that frictional dissipation and pressure energy are neglected. 

F energy flux produced via frictional dissipation at a polymer-metal interface... 

According to Summerfield (Ref 12, p 442) The ideal rocket motor analysis rests on the following simplifications (a) the proplnt gas obeys the perfect gas law (b) its specific heat is constant, independent of temp (c) the flow is parallel to the axis of the motor and uniform in every plane normal to the axis, thus constituting a one-dimensional problem (d) there is no frictional dissipation in the chamber or nozzle (e) there is no heat transfer to the motor walls (f) the flow velocity in the chamber before the nozzle entrance is zero (g)combustion or heat addition is completed in the chamber at constant pressure and does not occur in the nozzle and (h) the process is steady in time. ... 

Frictional dissipation of mechanical energy can result in significant heating of fluids, particularly for very viscous liquids in small channels. Under adiabatic conditions, the bulk liquid temperature rise is given by AT=AP/CV p for incompressible flow through a channel of constant cross-sectional area. For flow of polymers, this amounts to about 4°C per 10 MPa pressure drop, while for hydrocarbon liquids it is about... 

In a packed bed, the fluid-particle interaction force is insufficient to support the weight of the particles. Hence, the fluid that percolates through the particles loses energy due to frictional dissipation. This results in a loss of pressure that is greater than can be accounted for by... 

Three underlying mechanisms are responsible for the nonlinearity.17,18 (1) Frictional dissipation occurs at the fiber/matrix interfaces, whereupon the sliding resistance of debonded interfaces, r, becomes a key parameter. Control of t is critical. This behavior is dominated by the fiber coating, as well as the fiber morphology.19,20 By varying r, the prevalent damage mechanism and the resultant non-linearity can be dramatically modified. (2) The matrix cracks... 

To understand the dynamics of the bending fluctuations associated with these natural modes, a force balance per unit length is required. The force per unit length associated with the bending mode of wavelength A is calculated by differentiating the energy Ux and it is resisted by the frictional dissipation ... 

To understand the frictional dissipation term, recall, that the friction coefficient of a Kuhn segment of length b is the ratio of force and velocity in a liquid. Hence, (jb is the ratio of force per unit length and the velocity dhy/dt. This equation can be solved by separation of variables and integration ... 

In order to rationalize this result for PE, a simple calculation of the energy required for scission of main-chain bonds (E c c) the frictional dissipation between chains due to mechanical action was performed. The latter was calculated from the activation energy for viscosity of the monomer repeat unit, E, determined from a low-molar-mass analogue of polyethylene. This was then used to determine the number of repeat units, n, necessary to exceed the C-C bond energy of 349 kJ/mol. From the value of 4.22 kJ/mol for E, an estimate for n of 83 was made, which was considered a good approximation to the experimental result of 100 (Sohma, 1989b). However, such a simple approach is not amenable to extension to systems of structural complexity such as chain scission during the mastication of rubber. 

The pressure recovery at the outlet is more marked in rotary valves, where the less convoluted fluid path leads to a lower frictional loss, especially at large openings. For gases, there is usually a rise in pressure over the outlet section if the valve is of the rotary type, but the greater frictional dissipation in a globe valve usually causes a fall in pressure between the throat and valve outlet. The extent of the pressure recovery between valve throat and valve outlet will be considered in the next section for liquids. 

High-Speed Flow. Frictional dissipation occurs in the turbulent boundary layer for highspeed flow because of combined viscous and turbulent shear mechanisms. To account for variations in the specific heat of the fluid, it is best to express the heat potential as enthalpy rather than temperature. It was shown by van Driest [93] that under high-speed conditions, Eq. 6.180 is replaced by... 

Variable Fluid Properties. The fluid properties within a boundary layer may vary because of large temperature differences caused by frictional dissipation at high Mach numbers or because the coolant introduced through the surface possesses properties quite different from those of the main stream. 

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