Momentum flux tensor

Since in real fluids, some of the energy of fluid flow is typically converted into heat by viscous forces, it is convenient to generalize equation 9.7 so that it allows for dissipation. Consider the momentum of fluid flowing through the volume dT (= pv). Since its time rate of change is given by d pv)/dt = dp/dt)v -f p dv/dt), we can use equations 9.3 and 9.7 to rewrite this expression as follows  

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As an illustrative example, focus on the element in the first row and second column, pVjcVy, for the matrix representation of the convective momentum flux tensor. The subscript x on the first velocity component indicates that pv Vy is a force per unit area acting across a simple surface oriented with a unit normal vector in the x direction. The subscript y on the second velocity component reveals that this force acts in the y direction. If we perform this analysis for all nine components in the matrix (8-6) the three entries in the first row represent X, y, and z components, respectively, of the vector force per unit area that is transmitted across the simple surface defined by a constant value of the x coordinate, which means that the unit normal vector to the surface is colinear with the X direction. In vector-tensor notation, this is... 

Summary of Forces Due to Total Momentum Flux (Le., 2, 3, and 4). In the preceding three sections, a total of 21 scalar quantities has been identified nine from pvv, nine from r, and three from the pressure contribution to momentum flux. They represent all the possible surface force components that can be generated from the total momentum flux tensor. When each of these scalars is multiplied by the surface area across which the stress acts, a quantity with units of momentum per time is obtained. If n represents the outward-directed unit normal vector at every point on surface S that encloses the system within an arbitrary control volume V, then the total force acting on the system across S (i.e., in the direction of —n) due to total momentum flux is given by... 

Answer Begin by identifying the unit normal vector from the solid to the fluid across the surface at r = / n = Sr. Then (1) take the dot product of n with the total momentum flux tensor, (2) evaluate this vector-tensor operation at the fluid-solid interface, and (3) multiply the result by the differential surface element, dS = R sind d9 d, to generate a differential vector force. Hence,... 

Answer. Prior to solving this problem, it is instructive to consider the underlying fnndamentals related to the hint provided above. In terms of the total momentum flux tensor, the total differential vector force exerted by the fluid on the tube wall is... 

U is the specific internal energy, thus /2pv + pU is the total energy per unit volume as the sum of internal and kinetic energies. Further, q is the heat flux relative to the motion, x is the momentum flux tensor. The enthalpy can be introduced by the relation U = H — pV = H — pjp. 

Time constant for Hookean dumbbell model Time constants for Rouse chain model Solvent contnbution to thermal conductivity Tensor virial multiplied by 2 Momentum space distribution function Integration variable in Taylor series Stress tensor (momentum flux tensor) External force contribution to stress tensor Kinetic contribution to stress tensor Intramolecular contribution to stress tensor Intermolecular contribution to stress tensor Fluid density... 

It can be concluded, then, that when the external forces per unit mass are different for the various beads of a particular species, the term — [e it] in Eq. 9.2 will in general not vanish Note also that the quantity itxr = rxn represents the angular momentum flux tensor. 

We recall from Chap. 5 that the momentum flux tensor consists of both a kinetic and a potential component given in dimensionless form as ... 

Now let s consider the molecular transport of momentum. The molecular mechanism is given by the stress tensor or molecular momentum flux tensor, r. Each element Ty can be interpreted as the component of momentum flux transfer in the direction. We are therefore interested in the terms tix- The rate at which the x component of momentum enters the volume element at face x is XxxAyAx Ij, the rate at which it leaves at face x + Ax is XxxAyAx i+ax, and the rate at which it enters at face y is TyxAxAz y. The net molecular contribution is therefore... 

While the general form of the generalized Euler s equation (equation 9.9) allows for dissipation (through the term Hifc) expression for the momentum flux density as yet contains no explicit terms describing dissipation. Viscous stress forces may be added to our system of equations by appending to a (momentarily unspecified) tensor [Pg.467]

We begin by describing the HPP model, which satisfies all of the above requirements except for the isotropy of the momentum flux density tensor. As we shall, however, this early model nonetheless has some very interesting and suggestive properties, despite not being able to reproduce Navier-Stokes-like behavior exactly. 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

We make two additional comments. First, notice that when u 0, the momentum flux density tensor reduces to the diagonal term p5ij, where the pressure p = Cgp and Cg is the speed of sound. We thus conclude that the speed of sound in the FHP-I LG is given by... 

Isotropy of the Momentum Flux Density Tensor If we trace back our derivation of the macroscopic LG Euler s and Navier-Stokes equations, we see that the only place where the geometry of the underlying lattice really enters is through the form for the momentum flux density tensor, fwhere cp = x ) + y ), k = 1,..., V... 

Now, in order for us to recover standard hydrodynamical behavior, we require that the momentum flux density tensor be isotropic i.e. invariant under rotations and reflections. In particular, from the above expansion we see that must be isotropic up to order... 

The situation with regard to convective (turbulent) momentum transport is somewhat more complex because of the tensor (dyadic) character of momentum flux. As we have seen, Newton s second law provides a correspondence between a force in the x direction, Fx, and the rate of transport of x-momentum. For continuous steady flow in the x direction at a bulk... 

The momentum flux vector, which is the divergence of the stress tensor, appears in the Navier-Stokes equation (3.53) ... 

The physical quantities commonly encountered in polymer processing are of three categories scalars, such as temperature, pressure and time vectors, such as velocity, momentum and force and tensors, such as the stress, momentum flux and velocity gradient tensors. We will distinguish these quantities by the following notation,... 

For the momentum conservation of a single-phase fluid, the momentum per unit volume / is equal to the mass flux pU. The momentum flux is thus expressed by the stress tensor i/r = (pi — t). Here p is the static pressure or equilibrium pressure / is a unit tensor and r is the shear stress tensor. Since <1> = —pf where / is the field force per unit mass, Eq. (5.12) gives rise to the momentum equation as... 

Thus, an equation, which has the sense of a law of conservation of momentum has been obtained. There is an expression for the momentum flux pviVj — Uij under the derivation symbol, which allows one to write down the expression for the stress tensor... 

The possible development of gradients in the components of the interfacial stress tensor due to flow of an adjacent fluid implies that the momentum flux caused by the the flow of liquid at one side of the interface does not have to be completely transported across the interface to the second fluid but may (partly or completely) be compensated in the interface. The extent to which this is possible depends on the rheological properties of the interface. For small shear stresses the interface may behave elastically or viscoelastically. For an elastic interfacial layer the structure remains coherent the layer will only deform, while for a viscoelastic one it may or may not start to flow. The latter case has been observed for elastic networks (e.g. for proteins) that remciln intact, but inside the meshes of which liquid can flow leading to energy dissipation. At large stresses the structure may yield or fracture (collapse), leading to an increased flow. 

The structure of the expression for totai is that of a bilinear form it consists of a sum of products of two factors. One of these factors in each term is a flow quantity (heat flux q, mass diffusion flux jc, momentum flux expressed by the viscous stress tensor o, and chemical reaction rate rr)- The other factor in each term is related to a gradient of an intensive state variable (gradients of temperature, chemical potential and velocity) and may contain the external force gc or a difference of thermodynamic state variables, viz. the chemical affinity A. These quantities which multiply the fluxes in the expression for the entropy production are called thermodynamic forces or affinities. 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Re3molds stress tensor The first and second terms on the RHS denote the production... 

The stress tensor a (r), which is the momentum current or momentum flux density [9], is given by... 

Schweitz s representation of the quantum stress tensor a (r) in terms of the flux density operator acting on the momentum in Equation (19) makes clear its interpretation as a momentum flux density. Schweitz does not, however, consider how the surface flux virial in the quantum case, Zs or i ,s, may be related to the pv product. This, as demonstrated in the following section, has been accomplished using the atomic statement of the virial theorem [12]. 

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