Total momentum balance equation

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

These equations apply to the total mass or mass density of the system, while we use moles when describing chemical reaction. Therefore, whenever we need to solve these equations simultaneously, we must transform our species mass balances into weight fraction when including momentum and total mass-balance equations. 

When writing the boundary conditions for the above pair of simultaneous equations the heat transferred to the surroundings from the reactor may be accounted for by ensuring that the tube wall temperature correctly reflects the total heat flux through the reactor wall. If the reaction rate is a function of pressure then the momentum balance equation must also be invoked, but if the rate is insensitive or independent of total pressure then it may be neglected. 

The internal energy balance equation for the fluid is based on the momentum balance equation. The assumption of local thermodynamic equilibrium will enable us to introduce the thermodynamic relationships linking intensive quantities in the state of equilibrium and to derive the internal energy balance equation on the basis of equilibrium partial quantities. By assuming that the diffusion is a slow phenomenon, 1" J/p pv2, the change of the total energy of all components per unit volume becomes... 

Mass conservation equations apply to water and air. When the porous medium is deformable, the momentum balance equation (mechanical equilibrium) is also taken into account. In non-isothermal problems, the internal energy balance for the total porous medium must be considered. The basic equations solved by the finite element code CODE BRIGHT are ... 

To calculate the pore pressure response due to a volume source we use the Green s function based on the effective Biot theory. We write the coupled system of equations directly from the constitutive relations given by Biot (1962). These are the total stress of the isotropic porous medium, the stress in the porous fluid, the momentum balance equation for total stress, and the generalized Darcy s law. Following Parra (1991) and Boutin et al. (1987), the coupled system of differential equations in the... 

The momentum balance equation can be simplified in the case of very slow fluid motion. In this case the total derivative of the velocity and the acceleration due to the gravity can be neglected in the Navier-Stokes equation ... 

The inteifacial momentum balance equation (Equation 5.35) will now be written for the case of negligible siuface excess mass and momentum and two Newtonian fluids. The stress tensor in a Newtonian fluid is written following Bird et al. (2002) as pi - pt [Vv + (Vv) ], where I is the identity tensor and the superscript T represents the transpose of a tensor. When the inertial forces are considered as well, the total force on an interface exerted by the ith phase is n (Pi(v - d )v + pj - / [VVj + (Vv )T]). The dot product with n denotes that the forces act on a surface characterized by m. It is in making a force balance on the intraface that the effects of interfacial tension make themselves felt. The balance is known to be (see Chapter 5)... 

Hence, in addition to the electromagnetic momentum-energy equation, and the total energy balance equation, we can write the following momentum-energy balance equations applicable in the 3D space ... 

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

Tlie equation of momentum transfer - more commonly called die equation of motion - can be derived from niomentmii consideradons by applying a momentum balance on a rate basis. The total monientmn witliin a system is uiicluinged by an c.xchaiige of momentum between two or more masses of the system. This is known as die principle or law of conservation of monientmn. This differendal equation describes the velocity distribution and pressure drop in a moving fluid. 

Differential momentum, mechanical-energy, or total-energy balances can be written for each phase in a two-phase flowing mixture for certain flow patterns, e.g., annular, in which each phase is continuous. For flow patterns where this is not the case, e.g., plug flow, the equivalent expressions can usually be written with sufficient accuracy as macroscopic balances. These equations can be formulated in a perfectly general way, or with various limitations imposed on them. Most investigations of two-phase flow are carried out with definite limits on the system, and therefore the balances will be given for the commonest conditions encountered experimentally. 

In the text, however, the numerical problem is formulated using momentum and total-energy balances on a finite control volume. The intent of this problem is to write a numerical simulation that is based on a finite-difference representation of the differential equations. 

Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, total energy, and mechanical energy may be found in Whitaker (ibid.), Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. 

Equation (3.90) is the mass balance or continuity equation, (3.91) the momentum balance or Cauchy s equation of motion and (3.92) is the energy balance. As a momentum balance exists for each of the three coordinate directions, j = 1, 2, 3, there are five balance equations in total. The enthalpy form (3.83) is equivalent to the energy balance (3.92). 

General physical laws often state that quantities like mass, energy, and momentum are conserved. In computational mechanics, the most important of these balance laws pertains to linear momentum (when reckoned per unit volume, linear momentum may be expressed as the material density p times velocity v). The balance equation for linear momentum may be considered as a generalization of Newton s second law, which states that mass times acceleration equals total force. As we saw in the previous section, stresses in a material produce tractions, which may be considered as internal forces. In addition, external forces such as gravity may contribute to the total force. These are commonly reckoned per unit mass and are usually referred to as body forces to distinguish them from tractions, which may be considered as surface forces. For a one-dimensional motion, balance of linear momentum requires that (37,38)... 

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