Momentum balance Euler equations

Since the axial component of velocity near the channel/backing layer interface is zero, each consumed oxygen molecule removes momentum from the flow. Each incoming water molecule also removes momentum, since it must be accelerated to the flow velocity. The momentum balance (the Euler equation), therefore, has a form [192]... 

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... 

Momentum can be transported by convection and conduction. Convection of momentum is due to the bulk flow of the fluid across the surface associated with it is a momentum flux. Conduction of momentum is due to intermolecular forces on each side of the surface. The momentum flux associated with conductive momentum transport is the stress tensor. The general momentum balance equation is also referred to as Cauchy s equation. The Navier-Stokes equations are a special case of the general equation of motion for which the density and viscosity are constant. The well-known Euler equation is again a special case of the general equation of motion it applies to flow systems in which the viscous effects are negligible. 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

Modelling particle motion requires two things firstly, solving Newton s second law for translation and secondly, solving Euler s equations for rotation. The forces and torques which cause a change of position arise from fluid-particle interaction and are obtained by the LBM. Flow-induced forces and torques acting on particles can be determined by balancing the momentum of the fluid at the particle surface. The difference of the fluid momentum before and after a wall contact yields the local force F. Local torques Tai foUow from local forces and their distance to the centre of rotation xr ... 

The divergence is again taken with respect to the second index and y is an arbitrary scalar which absorbs all contributions parallel to the director n. Noting the first of EQNS (35) this is effectively the Euler-Lagrange equation of static theory with a dynamic term, g, added. We can also rewrite the first of EQNS (S) representing balance of linear momentum by substituting expression (17) for the stress tensor, and adding an inner product of EQN (38) with Vn to obtain... 

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