Velocity gradient laminar

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

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. 

If at a distance a from the leading edge the laminar sub-layer is of thickness 5 and the total thickness of the boundary layer is 8, the properties of the laminar sub-layer can be found by equating the shear stress at the surface as given by the Blasius equation (11.23) to that obtained from the velocity gradient near the surface. 

It has been noted that the shear stress and hence the velocity gradient are almost constant near the surface. Since the laminar sub-layer is very thin, the velocity gradient within it may therefore be taken as constant. 

In the laminar sub-layer, turbulence has died out and momentum transfer is attributable solely to viscous shear. Because the layer is thin, the velocity gradient is approximately linear and equal to Uj,/Sb where m is the velocity at the outer edge of a laminar sub-layer of thickness <5 (see Chapter ll). 

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. 

M. K. Kim, S. H. Won, and S. H. Chung, Effect of velocity gradient on propagation speed of tribrachial flames in laminar coflow jets, Proc. Combust. Inst. 31 901-908,2007. 

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. 

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. 

Above and to the left of the criterion line is the region in which 4 < 4.-According to the Klimov-Williams criterion, the turbulent velocity gradients in this region, or perhaps in a region defined with respect to any of the characteristic lengths, are sufficiently intense that they may destroy a laminar flame. 

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. 

The results presented above clearly demonstrate the merits of the counter-current shear layer control as a flame stabilization technique. With the use of the high-resolution PIV, the near flame structure is measured with sufficient detail to obtain the velocity gradients with accuracy. Prom these measurements, it is observed that the transverse velocity gradient dU /dr assumes large values at the nozzle exit as compared to that of laminar premixed Bunsen burner flames. 

When a shear stress is applied to a suspension or liquid exhibiting laminar flow, a velocity gradient (the rate of shear) is established. When the rate of shear varies linearly with the applied shear stress, the system is termed Newtonian and the proportionality constant is termed the viscosity. Newtonian flow is usually observed in dilute... 

Any consideration of mass transfer to or from drops must eventually refer to conditions in the layers (usually thin) of each phase adjacent to the interface. These boundary layers are envisioned as extending away from the interface to a location such that the velocity gradient normal to the general flow direction is substantially zero. In the model shown in Fig. 8, the continuous-phase equatorial boundary layer extends to infinity, but the drop-phase layer stops at the stagnation ring. At drop velocities well above the creeping flow region there is a thin laminar sublayer adjacent to the interface and a thicker turbulent boundary layer between this and the main body of the continuous phase. 

The velocity of the gases is high but it is always laminar flow. Over the susceptor there will be a boundary, or stagnant, layer where the velocity gradient decreases to zero. As the gases are heated, the silane and hydrocarbon will decompose and the species will diffuse through the boundary layer to grow on the reactor walls or on the substrate. A comprehensive study of this may be found in a paper by M. Leys and H. Veenvliet [41]. 

Clearly, in the absence of a radial temperature or velocity gradient, no radial mass transfer can exist unless, of course, a reaction occurs at the bed wall. When a system is adiabatic, a radial temperature and concentration gradient cannot exist unless a severe radial velocity variation is encountered (Carberry, 1976). Radial variations in fluid velocity can be due to the nature of flow, e.g. in laminar flow, and in the case of radial variations in void fraction. In general, an average radial velocity independent of radial position can be assumed, except from pathological cases such as in very low Reynolds numbers (laminar flow), where a parabolic profile might be anticipated. 

Usually, the values of the transport coefficients for a gas phase are extremely sensitive to pressure, and therefore predictive methods specific for high-pressure work are desired. On the other hand, the transport properties of liquids are relatively insensitive to pressure, and their change can safely be disregarded. The basic laws governing transport phenomena in laminar flow are Newton s law, Fourier s law, and Fick s law. Newton s law relates the shear stress in the y-direction with the velocity gradient at right angles to it, as follows ... 

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