Viscous momentum flux

Reynolds number flows /vRe N -°Vp /vRe — pV2 pV/D AQp izDp PV2 Tw/8 Pipe flow rw =wall stress (inertial momentum flux)/ (viscous momentum flux) Pipe/internal flows (Equivalent forms for external flows)... 

With regard to the drag on a sphere moving in a Bingham plastic medium, the drag coefficient (CD) must be a function of the Reynolds number as well as of either the Hedstrom number or the Bingham number (7V Si = /Vne//VRe = t0d/fi V). One approach is to reconsider the Reynolds number from the perspective of the ratio of inertial to viscous momentum flux. For a Newtonian fluid in a tube, this is equivalent to... 

This represents the ratio of the momentum flux carried by the fluid axially along the tube to the viscous momentum flux transported normal to the flow in the radial direction. Laminar flows are dominated by the fluid viscosity and are stable, whereas turbulent flows are dominated by the fluid density inertia and are unstable. 

For the Bingham plastic, a corresponding expression for the Reynolds nnmber based on the ratio of inertial to viscous momentum flux is... 

The kinematic viscosity has the same dimensions as the mass diffusivity, length2/time, while the quantity up can be interpreted as a volumetric momentum concentration. The shearing stress X may also be interpreted as a viscous momentum flux toward the solid surface. Equation (1-122) is, therefore, a rate equation analogous to Fick s first law, equation (1-40). The Schmidt number is defined as the dimensionless ratio of the momentum and mass diffusivities ... 

Forces Due to Viscous Momentum Flux (i.e., 3). A molecular momentum flux mechanism exists which relates viscous stress to Unear combinations of velocity gradients via Newton s law of viscosity if the fluid is Newtonian. Viscous... 

In rectangular coordinates, the matrix representation for viscous momentum flux is written as... 

As illustrated in Chapter 8 via equation (8-42), if dynamic pressure 5 is dimen-sionalized using a characteristic viscous momentum flux (i.e., iV/L), then = lYand one obtains the following form of the dimensionless equation of motion for laminar flow ... 

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]

These turbulent momentum flux components are also called Reynolds stresses. Thus, the total stress in a Newtonian fluid in turbulent flow is composed of both viscous and turbulent (Reynolds) stresses ... 

The dimensionless parameters, such as the Nusselt and Reynolds numbers, can be thought of as measures of the relative importance of of certain aspects of the flow. For example, if the flow through an area dA is considered, as shown in Fig. 1.19, the rate momentum passes through this area is equal to the mass flow rate times the velocity, i.e., equal to mV dA, i.e., equal to pVVdA, i.e., equal to pV2dA. If, therefore, U is a measure of the velocity, the quantity pU1 is a measure of the magnitude of the momentum flux in the flow. This quantity is often termed the inertia force . Further, since the Newtonian viscosity law indicates that the viscous shear stresses... 

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. 

On pages 92 to 98 tliie distribution of velocity and its accompanying momentum flux in a flowing stream in turbulent flow through a pipe was described. Three rather ill-defined zones in the cross section of the pipe were identified. In the first, immediately next to the wall, eddies are rare, and momentum flow occurs almost entirely by viscosity in the second, a mixed regime of combined viscous and turbulent momentum transfer occurs in the main part of the stream, which occupies the bulk of the cross section of the stream, only the momentum flow generated by the Reynolds stresses of turbulent flow is important. The three zones are called the viscous sublayer, the buffer zone, and the turbulent core, respectively. 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

In the limit of vanishingly small Reynolds numbers, forces due to convective momentum flux are negligible relative to viscous, pressure, and gravity forces. Equation (12-4) is simplified considerably by neglecting the left-hand side in the creeping flow regime. For fluids with constant /r and p, the dimensionless constitutive relation between viscons stress and symmetric linear combinations of velocity gradients is... 

However, at the microscale, the performance of the microrocket is limited by viscous losses due to the low Reynolds number of the expanding flow. These losses are characterized by the thrust efficiency which compares the observed momentum flux to the momentum flux predicted for an ideal (zero viscosity) fluid. Computations and experiments (e.g., [1]) cOTifirmed supersonic exit flow and have found efficiencies ranging from 10 % to 80 %, depending on the Reynolds number of the system, which in turn depends directly on the size of the throat and the fluid temperature and pressure as it passes through the nozzle, prior to supersonic expansion. The performance of the system can be improved by raising the temperature of the gas. Examples include using an electrical heater in the plenum upstream of the throat [2] and using a chemical reaction, such as combustion [3]. 

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. 

Since viscous flow occurs at constant temperature and density, the momentum flux, with the geometry of Fig. 2.6, is... 

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 Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

The numerator is the flux of inertial momentum carried by the fluid along the tube in the axial direction. The denominator is proportional to the viscous shear stress in the tube, which is equivalent to the flnx of viscous momentum normal to the flow direction, i.e. in the radial direction. Thus, the Reynolds number is a ratio of the momentum flux due to inertia (in the flow direction) to the momentum flux due to viscous stresses (in the radial direction). 

Tyx can also be interpreted in another fashion. Xyx may be considered as the viscous flux of x momentum in the y direction. The idea here is that the plate located at y = H transmits its X momentum to the layer below, which in turn transmits momentum to the next layer. The momentum flux, tyx, is negative in this case as the momentum is transferred in the negative y direction. The sign convention follows the ideas used for heat flux in that heat flows from hot to cold or in the direction of a negative temperature gradient. This also makes the law of viscosity tit with the ideas of diffusion in which matter flows in the direction of decreasing concentration. 

Equation (74) shows that continuity (43) is automatically satisfied by any LB model. The Navier-Stokes equation (44) will be satisfied, if we succeed in ensuring that the Euler stress pc Sap + pUaUp, the Newtonian viscous sbess, a p (45), and the fluctuating sbess (47) are given correctly by the sum of the momentum fluxes in (75). Since depends only on p and j, it must be identified with the Euler sbess ... 

The nonequilibrium momentum flux (+ T p ) /2 must be connected with the sum of viscous and fluctuating stresses. 

The procedure of Mason and Evans has the electrical analog shown in Figure 2.2, where voltages correspond to pressure gradients and currents to fluxes. As the argument stands there is no real justification for this procedure indeed, it seems improbable that the two mechanisms for diffusive momentum transfer will combine additively, without any interactive modification of their separate values. It is equally difficult to see why the effect of viscous velocity gradients can be accounted for simply by adding... 

MOMEN- TUM BALANCE Rate of change of momentum per unit volume Rale of change of momenium by convection per unit volume Rale of change of momentum by molecular transfer (viscous transfer) per volume Generation per volume (External forces) (Ex gravity) Empirically determined flux specified (3)< Velocity specified (1.2b) ... 

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