Molecular viscosity

If the laminar sub-layer extends from y3 = 0 to y1 = 5, obtain the equation for the relation between and y+ in the buffer zone, and show that the ratio of the eddy viscosity to the molecular viscosity increases linearly from 0 to 5 through this buffer zone. 

Leaving aside the difficult question of whether this model holds for multiphase flows, we still have the problem of determining in terms of the computed properties of the flow. The reader should appreciate that choosing an effective viscosity for a multiphase flow is much more complicated than just adding a turbulence model as done in single-phase turbulent flows. Indeed, even for a case involving two fluids (e.g., two immiscible liquids) for which the molecular viscosities are constant, the choice of the effective viscosities is not obvious. For example, even if the mass-average velocity defined by... 

Here p is the molecular viscosity the second term on the right-hand side of the equation is the effect of volume dilation. 

The near-wall region is conceptually subdivided into three layers, based on experimental evidence. The innermost layer is the viscous sublayer in which the flow is almost laminar, and the molecular viscosity plays a dominant role. The outer layer is considered to be fully turbulent. The buffer layer lies between... 

The negative sign indicates that momentum is transferred down the velocity gradient. The proportionality constant n (Pa s) is called molecular viscosity or simply viscosity, which is an intensive property. The unit of viscosity in CGS (centimeter-gram-second) units is called poise (gcm s ). From Equation 2.4 we obtain... 

When representing rates of transfer of heat, mass, and momentum by eddy activity, the concepts of eddy thermal conductivity, eddy diffusivity, and eddy viscosity are sometimes useful. Extending the concepts of heat conduction, molecular diffusion, and molecular viscosity to include the transfer mechanisms by eddy activity, one can use Equations 2.13-2.15, which correspond to Equations 2.2,2.3, and 2.5, respectively. 

S, = source term for zth momentum equation V = velocity of i direction a = permeability e = void fraction p = molecular viscosity (kg/m-s) p = density (kg/m3)... 

Figure 6.8 Conceptual diagram of the different scales of the components of the benthic boundary layer (BBL). In bottom water above the sediment-water interface where the Eckman layer occurs as flow is affected by the rotation of the Earth and bottom friction, where w = friction velocity and / = Coriolis parameter the logarithmic layer predominates when the velocity profile is well described using a logarithmic function a viscous sublayer is formed by molecular viscosity a diffusive boundary layer forms, whereby solute transport is controlled by molecular diffusion. (Modified from Boudreau and Jprgensen, 2001.)...

Modeling a disk by solving the full three-dimensional Navier-Stokes equations is a complicated task. Moreover, it is still not fully understood what is the cause of frictional forces in the disk. Molecular viscosity is by orders of magnitude too small to cause any appreciable accretion. Instead, the most widely accepted view is that instabilities within the disk drive turbulence that increases the effective viscosity of the gas (see Section 3.2.5). A powerful simplification of the problem is (a) to assume a parameterization of the viscosity, the so-called a-viscosity (Shakura Syunyaev 1973) ((3-viscosity in the case of shear instabilities, Richard Zahn 1999) and (b) to split the disk into annuli, each of which constitutes an independent one-dimensional (ID) vertical disk structure problem. This then constitutes a 1+1D model a series of ID vertical models glued together in radial direction. Many models go even one step further in the simplification by considering only the vertically integrated or representative quantities such as the surface density X(r) = p(r, z.)dz... 

Molecular viscosity, which is measured in centistokes, is a nonlinear function of molecular weight and of concentration. Thus, a 2% solution of polymer in water usually does not have twice the viscosity of a 1% solution. Each batch of a commercial polymer therefore must be measured for viscosity at the appropriate concentration. The addition of salts can affect the final viscosity of some polymers. Divalent anions and cations can have a major effect on the conformation of polymers in solution, occasionally causing incompatibilities when formulations are mixed together in the eye. 

Here Sgain represents the rate (per unit mass) of energy gain from buoyancy forces and radiation mechanisms eioss represents the losses due to heat conduction and molecular viscosity and finally E,ransfcr is the energy trans-... 

Thorpe and Rodger compared viscosities at temperatures where the slope of the curve is the same (d y/d/=const.), i.e. when temperature has the same effect, on the viscosity. The ratio of viscosities measured at two temperatures of equal slope was found to be constant. They also considered the molecular viscosities, r Mv), where (Mv) is the molecular volume, and what they called the molecular viscosity work rj Mv), or the work required to move a surface equal to the molecular viscosity through the specific molecular length MvY at unit velocity. Some general results found for normal liquids were ... 

The basic theory of the evolution of an accretion disk can be derived by assuming that there is some physical mechanism operating that results in an effective viscosity of the gas. Because the intrinsic molecular viscosity of hydrogen gas is far too small... 

In laminar flow, i is the molecular viscosity in that case Eq. (3) is a second-order linear two-dimensional PDE of the Poisson type. In turbulent flow, jl also depends on the velocity gradients (hence on the lateral position), and Eq. (3) then is quasilinear or nonlinear. 

Here, is referred to as turbulent or eddy viscosity, which, in contrast to molecular viscosity, is not a fluid property but depends on the local state of flow or turbulence. It is assumed to be a scalar and may vary significantly within the flow domain, k is the turbulent kinetic energy (normal turbulent stresses) and can be expressed as... 

Substitution of Eq. (3.14) in the Reynolds-averaged momentum conservation equations (Eqs (3.11)) leads to a closed set, provided the turbulent viscosity is known. The form of the Reynolds-averaged momentum equations remain identical to the form of the laminar momentum equations (Chapter 2 and Table 2.2) except that molecular viscosity is replaced by an effective viscosity,... 

The innermost layer, called the viscous sub-layer in which flow is laminar-like and the molecular viscosity plays a dominant role ... 

When initiating the calculations, turbulent viscosity is set to a small value (usually the same as the molecular viscosity). Linearized source terms for k and s can then be written ... 

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