Velocity gradient laminar flow

Figure 48.5(a) Parabolic-shaped velocity gradient laminar flow fluid flows fastest at the center of the pipe. 

FIGURE 11.7 Two types of laminar flow, and the effect on deformation and breakup of drops at increasing velocity gradient ( ). The flow is two-dimensional, i.e., it does not vary in the z-dircction. More precisely, the flow type in (b) is plane hyperbolic flow. ... 

As described above, the viscosity of a fluid at relatively low velocities generates laminar flow. When the velocity exceeds a critical level, the inertial force becomes sig-niflcant compared to the viseosity. TTiis is called turbulent flow. A thorough analysis of turbulent flow is very difficult. However, it is recognized that the shearing stress is proportional to the square of the velocity gradient perpendicular to the solid/fluid interface as ... 

When a sample is injected into the carrier stream it has the rectangular flow profile (of width w) shown in Figure 13.17a. As the sample is carried through the mixing and reaction zone, the width of the flow profile increases as the sample disperses into the carrier stream. Dispersion results from two processes convection due to the flow of the carrier stream and diffusion due to a concentration gradient between the sample and the carrier stream. Convection of the sample occurs by laminar flow, in which the linear velocity of the sample at the tube s walls is zero, while the sample at the center of the tube moves with a linear velocity twice that of the carrier stream. The result is the parabolic flow profile shown in Figure 13.7b. Convection is the primary means of dispersion in the first 100 ms following the sample s injection. 

In configurations more complex than pipes, eg, flow around bodies or through nozzles, additional shearing stresses and velocity gradients must be accounted for. More general equations for some simple fluids in laminar flow are described in Reference 1. 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

Two approaches to this equation have been employed. (/) The scalar product is formed between the differential vector equation of motion and the vector velocity and the resulting equation is integrated (1). This is the most rigorous approach and for laminar flow yields an expHcit equation for AF in terms of the velocity gradients within the system. (2) The overall energy balance is manipulated by asserting that the local irreversible dissipation of energy is measured by the difference ... 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

Shear stresses are developed in a fluid when a layer of fluid moves faster or slower than a nearby layer of fluid or a solid surface. In laminar flow, the shear stress is equal to the product of fluid viscosity and velocity gradient or rate of shear. Under laminar-flow conditions, shear forces are larger than inertial forces in the fluid. 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

With respect to general corrosion, once a surface film is formed the rate of corrosion is essentially determined by the ionic concentration gradient across the film. Consequently the corrosion rate tends to be independent of water flow rate across the corroding surface. However, under impingement conditions where the surface film is unable to form or is removed due to the shear stress created by the flow, the corrosion rate is theoretically velocity (10 dependent and is proportional to the power for laminar flow and... 

Essentially, except for once-through boilers, steam generation primarily involves two-phase nucleate boiling and convective boiling mechanisms (see Section 1.1). Any deposition at the heat transfer surfaces may disturb the thermal gradient resulting from the initial conduction of heat from the metal surface to the adjacent layer of slower and more laminar flow, inner-wall water and on to the higher velocity and more turbulent flow bulk water. 

These observations are consistent with the proposed mechanism of the reaction being diffusion controlled in the laminar flow regime. The mass transport is aided by the velocity gradient and thus the reaction rate increases as the Reynolds number is increased. 

There is an analytical solution of the Navier-Stokes equations for the flow between two rotating cylinders with laminar flow (see e.g. [37]). The following equation applies for the velocity gradient in the annular gap in the general case of rotation of the outer cylinder (index 2) and the inner cylinder (index 1) ... 

The special flow conditions in circular (capillaries, tubes) or rectangular channels cause very different stresses depending on the position of the particles in the flow cross section. With laminar flow, for example the following applies to velocity gradient (see e.g. [37]) ... 

The Chilton-Colburn analogy can be also used to estimate the local mass transfer rate in laminar flow where the wall shear stress is related to the azimuthal velocity gradient by... 

There will be velocity gradients in the radial direction so all fluid elements will not have the same residence time in the reactor. Under turbulent flow conditions in reactors with large length to diameter ratios, any disparities between observed values and model predictions arising from this factor should be small. For short reactors and/or laminar flow conditions the disparities can be appreciable. Some of the techniques used in the analysis of isothermal tubular reactors that deviate from plug flow are treated in Chapter 11. 

The mean profiles of velocity, temperature and solute concentration are relatively flat over most of a turbulent flow field. As an example, in Figure 1.24 the velocity profile for turbulent flow in a pipe is compared with the profile for laminar flow with the same volumetric flow rate. As the turbulent fluxes are very high but the velocity, temperature and concentration gradients are relatively small, it follows that the effective diffusivities (iH-e), (a+eH) and (2+ed) must be extremely large. In the main part of the turbulent flow, ie away from the walls, the eddy diffusivities are much larger than the corresponding molecular diffusivities ... 

If a particle is moving in a fluid which is in laminar flow, the drag coefficient is approximately equal to that in a still fluid, provided that the local relative velocity at the particular location of the particle is used in the calculation of the drag force. When the velocity gradient is sufficiently large to give a significant variation of velocity across the diameter of the particle, however, the estimated force may be somewhat in error. 

It is of interest to consider first what is happening in pipe flow. Random molecular movement gives rise to a mixing process which can be described by Fick s law (given in Volume 1, Chapter 10). If concentration differences exist, the rate of transfer of a component is proportional to the product of the molecular diffusivity and the concentration gradient. If the fluid is in laminar flow, a parabolic velocity profile is set up over the cross-section and the fluid at the centre moves with twice the mean velocity in the pipe. This... 

Example 2.11. As an example of a force balance for a microscopic system, let us look at the classic problem of the laminar flow of an incompressible, newtonian liquid in a cylindrical pipe. By newtonian we mean that its shear force (resistance that adjacent layers of fluid exhibit to flowing past each other) is proportional to the shear rate or the velocity gradient. 

In Sect. 3.2, the development of the design equation for the tubular reactor with plug flow was based on the assumption that velocity and concentration gradients do not exist in the direction perpendiculeir to fluid flow. In industrial tubular reactors, turbulent flow is usually desirable since it is accompanied by effective heat and mass transfer and when turbulent flow takes place, the deviation from true plug flow is not great. However, especially in dealing with liquids of high viscosity, it may not be possible to achieve turbulent flow with a reasonable pressure drop and laminar flow must then be tolerated. 

---