Angular-momentum balance

Our calculations have been successful in interpreting trends that are seen in the experimentally observed rates of electron ejection. However, until now, we have not had a clear physical picture of the energy and momentum (or angular momentum) balancing events that accompany such non BO processes. It is the purpose of this paper to enhance our understanding of these events by recasting the rate equations in ways that are more classical in nature (and hence hopefully more physically clear). This is done by... 

For a flow system, streams with curved streamlines may carry angular momentum into and/or out of the system by convection. To account for this, the general macroscopic angular momentum balance applies ... 

If it is assumed that the fluid leaves the impeller tangentially at the same speed as the impeller (an approximation), then an angular momentum balance on the fluid in contact with the impeller gives ... 

Substituting the expressions found for the two velocity components (7.23) and (7.24) into the continuity (7.20) and the angular momentum balance (7.19) gives an expression for the dependence of torque on fluid pumping ... 

Following the prescription adopted above, we assert that angular momentum balance implies... 

The angular momentum balance provides a relation for the required motor torque ... 

So the angular momentum balance (for a fixed axis of rotation) becomes... 

The simple reaction turbine is analyzed most easily by the angular momentum balance, Eq. 7.65, which, as shown in Sec. 7.7 for a steady-flow turbine, reduces to Euler s turbine equation (Eq. 7.66). To find the power produced per unitjtime, we multiply both sides of Eq. 7.66 by -minus sign is used here tp agree with the thermodynamic sign convention that power... 

Begin with the equation of motion in vector-tensor notation for a generalized fluid and briefly describe a strategy to obtain the angular momentum balance. Hint angular momentum is a first-rank tensor. 

Otrly those properties Z of a system, that are conserved properties, may enter into an equation of change. Energy, hnear momentum, and angular momentum are conserved quantities and the respective equations of change are named energy balance, linear momentum balance, and angular momentum balance. 

The law of angular momentum balance requires that the stress tensor is symmetric... 

The lack of an intrinsic angular momentum balance in the classical theory has seriously handicapped application of mixture theory to problems where constituent spin is a significant factor in the overall dynamics. This would include boundary control of constituent spin or particulate saltation [4], Thus the Twiss-Eringen theory appears to be an important development for flow situations in which strong spin of one or more constituents may occur. 

Equations (11) and (12) are the linear momentum balances in the flow direction. Here the total pressure head has been partitioned into partial pressure components through the use of the solid volume distribution function . Equation (13) is the balance of linear momentum for the solid phase (the fluid phase is the same but of opposite sign) in the y direction. In this equation, we have allowed for an interphase pressure effect (incorporated in the parameter E) to ensure that the mixture remains saturated at all times [8]. Equations (14) and (15) account for the angular momentum balance in the binormal direction for the two constituents. Finally, the primes imply differentiation with respect to 3 ... 

The focus here is on inclusion of the angular momentum balances in mixture models so it would seem that micropolar theory would serve as a guide for selection of spin boundary conditions. But as shown by Kirwan [14], some care must be exercised even here to avoid trivializing the problem. The present case is much more complicated since there are more equations and they are nonlinear. 

The motion of such fluids can thus be determined by an Eulerian vector angular-momentum balance equation on the form ... 

It is perhaps worth remarking here that for isotropic fluids K and 1 are generally absent, so that the angular momentum balance law (4.45) further reduces to ijkhj = 0) indicating that for isotropic fluids the stress tensor is symmetric. Fluids in which couple stresses and body couples occur are frequently called polar fluids, and those in which they are absent are called non-polar. The forms given by equations (4.35), (4.41) and (4.45) for the balances of mass, linear momentum and angular momentum will eventually lead to the main dynamic equations for nematics. 

It is now possible to eliminate the velocity gradients and C from the angular momentum balance equations (5.211) and (5.212) by using the linear momentum... 

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