Laminar motion

For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

Equation (11-48) is applicable to burdens in the solid, liquid, or gaseous phase, either static or in laminar motion it is apphcable to solidification equipment and to divided-solids equipment such as metal belts, moving trays, stationaiy vertical tubes, and stationaiy-shell fluidizers. 

Betvveen R,. of 2000 and 4000, the flow is considered unsteady or unstable or transitional where laminar motion and turbulent mixing flows may alternate randomly [3]. K values can still be calculated from the... 

The velocity of the fluid at the surface of a rigid sphere, held in an infinite expanse of a fluid moving in laminar motion past it, must be zero. Using spherical coordinates, the solutions then become (B4, L2, Sll)... 

The purpose of this chapter is to provide a comprehensive discussion of some simple approaches that can be employed to obtain information on the rate of heat and mass transfer for both laminar and turbulent motion. One approach is based on dimensional scaling and hence ignores the transport equations. Another, while based on the transport equations, does not solve them in the conventional way. Instead, it replaces them by some algebraic expressions, which are obtained by what could be called physical scaling. The constants involved in these expressions are determined by comparison with exact asymptotic solutions. Finally, the turbulent motion is represented as a succession of simple laminar motions. The characteristic length and velocity scales of these laminar motions are determined by dimensional scaling. It is instructive to begin the presentation with an outline of the basic ideas. 

Third, turbulent transport is represented as a succession of simple laminar flows. If the boundary is a solid wall, then one considers that elements of liquid proceed short distances along the wall in laminar motion, after which they dissolve into the bulk and are replaced by other elements, and so on. The path length and initial velocity in the laminar motion are determined by dimensional scaling. For a liquid-fluid interface, a roll cell model is employed for turbulent motion as well as for interfacial turbulence. 

In order to familiarize the reader with Prandtl s evaluation procedure [3-5] on which the algebraic method is based, let us examine the steady laminar motion of a fluid along a plate and denote by x the coordinate along the plate, by y the distance to the plate, by u and v the x and y components of the velocity, and by p the pressure. The equations of motion of an incompressible fluid of density p and kinematic viscosity v have the form... 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

The transport equations for laminar motion can be formulated, in general, easily and difficulties may lie only in their solution. On the other hand, for turbulent motion the formulation of the basic equations for the time-averaged local quantities constitutes a major physical difficulty. In recent developments, one considers that turbulence (chaos) is predictable from the time-dependent transport equations. However, this point of view is beyond the scope of the present treatment. For the present, some simple procedures based on physical models and scaling will be employed to obtain useful results concerning turbulent heat or mass transfer. 

C. Model Based on Steady Laminar Motion along a Plate... 

In this model, the velocity and concentration fields are described by the equations valid for the steady laminar motion of a semiinfinite fluid along a plate. Therefore, the velocity components in each of the paths satisfy the... 

E. Unsteady Laminar Motion Used to Describe Turbulent Transport... 

The eddies which give rise to short laminar motions along the wall are generated in the vicinity of the boundary in a region, located at the distance y0 from the interface, dominated by turbulence. The intensity of these eddies can be characterized by their kinetic energy E0 at that distance. The path length x is expected to depend on E0 and on the physical constants r and p. Consequently, dimensional analysis leads to... 

If only rotational viscometric data are available, the design of pipe lines is somewhat more complex. Such data are usually expressed in the form of a relationship between shear stress and shear rate. The shear stress on a cylindrical element of fluid of radius r flowing through a pipe in laminar motion is equal to rAP/2L. If the corresponding shear rate — du/dr) can be expressed analytically, i.e., if the functional relationship... 

Clearly the assumption of a flat velocity profile is not correct. For a film in steady, laminar motion one may obtain an expression for the velocity distribution from the Navier-Stokes equations of motion [Eq. (9)]. For this case the Navier-Stokes equations simplify to... 

Because the quantitative analysis of transport processes in terms of the microscopic description of turbulence is difficult, Kdrmdn suggested (K2) the use of a macroscopic quantity called eddy viscosity to describe the momentum transport in turbulent flow. This quantity, which is dimensionally and physically analogous to kinematic viscosity in the laminar motion of a Newtonian fluid, is defined by... 

Figure 5 shows that the leading exponents are positive and thus the 4>4 MTRS is chaotic. In addition, they varies depending on the time intervals We see relatively small instability regions, for example, around the intervals 9000-12,000 and 70,000-75,000. The variations of finite-time Lyapunov exponents have been related to the alternations between qualitatively different motions, such as (a) chaotic and quasi-regular, laminar motions in two-dimensional systems [11] and (b) random and cluster motions in high-dimensional systems [12], and they have been utilized for detecting these ordered motions. 

The latter takes the limit forms cF = in the case of the linear force law (laminar motion Re 1, very small droplets) or cF = -0.0653/2 0.40 in the case of the quadratic force law (fully turbulent streamlining, Re >> 1, droplets of a significant size 3 mm < 2r < 7 mm). To find the limit speed value v such that v(t) —> v, one needs to assign j-t = 0 in equation (3.71). Analytical solutions with the latter assumptions for cF = cF(Rev) yield the formulas... 

In principle, the simplest theoretical approach is to examine the stability of the solution to arbitrary disturbances, that is, arbitrary departures from the laminar solution, (3-44). If these disturbances increase in magnitude with time, the basic laminar motion is said to be unstable, and it is clear that a new form for the velocity field will eventually be realized. If they decrease in magnitude, the flow will revert back to its undisturbed laminar form. It is generally possible to carry out such an analysis if the disturbance is assumed to be infinitesimal in magnitude, since then the equations of motion governing the disturbance can be linearized and an arbitrary disturbance analyzed by consideration of the growth or... 

Intermitteney, in the context of chaotic dynamical systems, is characterized by long periods of nearly periodic or laminar motion interspersed by chaotic bursts of random duration [28]. Within this broad phenomenological... 

We saw in Chap. 1 that for newtonian fluids in laminar motion the shear stress is equal to the product of the viscosity and the velocity gradient. Substituting in Eq. 6.3, we find... 

The mechanism of coagulation in a laminar flow has limited practical application since in the majority of applications the motion of liquid has a turbulent character. At turbulent motion, the collision frequency of particles increases very considerably in comparison with a quiescent environment or with laminar motion. Now consider the mechanism of particle coagulation in a turbulent flow, following the work [52]. 

The laminar regime in a channel holds until the Reynolds number reaches a critical value above which the laminar motion becomes unstable and a transition to a turbulent flow will generally occur. It has been demonstrated experimentally that for a microchannel the critical value of the Reynolds number depends on the entrance conditions, on the cross-sectional geometry, and on the wall roughness. The effect of the roughness on the laminar-to-turbulent transition is very important, and it can be evidenced by observing Fig. 8 in which the experimental values of the Poiseuille number (f Re) are shown as a function of the Reynolds number for two microtubes made in stainless steel and in fused silica. The stainless steel microtube has an internal... 

In Section 15.2.1 we noted that Pick derived his model for mass transfer pardy by analogy to Fourier s law of heat transfer and that one reason Pick s model was rapidly accepted was this close analogy to Fourier s law. Shordy after Pick s developments, Osborne Reynolds (yes, the Reynolds number is named after him) stated that heat or mass transfer in a moving fluid should be the result of both normal diffusion processes and eddies caused by the fluid motion. At the time, he had not yet discovered the difference between laminar motion (only normal diffusion operates) and turbulent motion (both molecular and eddy diffusion occur). We now know that Reynolds was correct only for turbulent flow. Since eddies depend on fluid velocity, the easiest functional form is to assume that eddy diffusion is linearly dependent on velocity. Then the equation for mass transfer becomes... 

Phase average models apply to well mixed multiphase flows, i.e. when the exact shape of the interfaces is not known, or not relevant e.g. bubbly flows. The principle could be applied under the two-fluid, six-equation model, where separate eonservation equations are required for each phase with appropriate exehange forces, or the homogeneous. Algebraic Slip model. Under sithermal, incompressible flow conditions, the equations of laminar motion for phase A are expressed in the two-fluid formalism as follows ... 

A liquid is flowing in laminar motion down a vertical wall. The wall consists of a species that is slightly soluble in the liquid. Show that the governing equation for species diffusing into the liquid from the wall can be written as... 

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