Fluid, thickness profiles

Fig. 18. The simple shear geometry used to characterise the interfacial friction the fluid thickness is e. The top plate limiting the sample is translated at the velocity and transmits this velocity to the fluid. The bottom plate is immobile, and the local velocity of the fluid at the bottom interface is Vs. The fluid is submitted to a simple shear, with a shear rate y = (Vt-VsYe. The velocity profile extrapolates to zero at a distance b below the interface, with...

The simulations were performed assuming that the flow is laminar. Additionally, the contact angle is assumed to be known. The initial velocity is assumed to be zero everywhere in the domain. The initial fluid temperature profile is taken to be linear in the natural convection thermal boundary layer and the thermal boundary layer thickness, 5j, is evaluated using the correlation for the turbulent natural convection on a horizontal plate as, Jj. =1. 4(vfiCil ... 

Such evaporation singularities can be very easily observed because the thickness profile of a thin liquid film locally reflects the rate of solvent removal. For very thin films, especially near the drying line, back flow due to gravitational draining can be neglected. The mass flux h(x+dx)uo carried into a fluid element dx, as shown in Figure 1, is balanced by the flux carried out by the substrate h(x)m and the mass lost through evaporation E(x)dx this observation leads to the continuity equation... 

Optohydrodynamics Fluid Actuation by Light, Fig. 5 (a) Fluid velocity profile in a two-layer system in presence of an interfacial tension gradient notations are also illustrated, (b) Analytical resolution of the steady flow pattern and the interface deformation in a double-layer conflguration with same shear viscosities and Hi = 5 2 = 5w. (c) Interface deformation in a three-layer system composed of a thin Aim of thickness 2Hi bounded by two external liquid layers of same thickness H2 for Hi = O.lw and H2 = 2w. Top. The dald T > 0 case leads to the formation of a dimple. Bottom. The dat I < 0 case leads to the formation of a nose... 

An additional advantage to neutron reflectivity is that high-vacuum conditions are not required. Thus, while studies on solid films can easily be pursued by several techniques, studies involving solvents or other volatile fluids are amenable only to reflectivity techniques. Neutrons penetrate deeply into a medium without substantial losses due to absorption. For example, a hydrocarbon film with a density of Ig cm havii a thickness of 2 mm attenuates the neutron beam by only 50%. Consequently, films several pm in thickness can be studied by neutron reflecdvity. Thus, one has the ability to probe concentration gradients at interfaces that are buried deep within a specimen while maintaining the high spatial resolution. Materials like quartz, sapphire, or aluminum are transparent to neutrons. Thus, concentration profiles at solid interfaces can be studied with neutrons, which simply is not possible with other techniques. 

Thus, the shear stress is expressed as a function of the boundary layer thickness S and it is therefore implicitly assumed that a certain velocity profile exists in the fluid. As a first assumption, it may be assumed that a simple power relation exists between the velocity and the distance from the surface in the boundary layer, or ... 

Show that, if the Blasius relation is used for the shear stress R at the surface, the thickness of the laminar sub-layer <5, is approximately 1.07 times that calculated on the assumption that the velocity profile in the turbulent fluid is given by PrandtFs one seventh power law. 

The hydrodynamic boundary layer has an inner part where the vertical velocity increases to a maximum determined by a balance of viscous and buoyancy forces. In fluids of high Schmidt number, the concentration diffusion layer thickness is of the same order of magnitude as this inner part of the hydrodynamic boundary layer. In the outer part of the hydrodynamic boundary layer, where the vertical velocity decays, the buoyancy force is unimportant. The profile of the vertical velocity component near the electrode can be shown to be parabolic. 

FIGURE 11.32 Flow profiles in microchannels, (a) A pressure gradient, - AP, along a channel generates a parabolic or Poiseuille flow profile in the channel. The velocity of the flow varies across the entire cross-sectional area of the channel. On the right is an experimental measurement of the distortion of a volume of fluid in a Poiseuille flow. The frames show the state of the volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule, (b) In electroosmotic flow in a channel, motion is induced by an applied electric field E. The flow speed only varies within the so-called Debye screening layer, of thickness D. On the right is an experimental measurement of the distortion of a volume of fluid in an electroosmotic flow. The frames show the state of the fluorescent volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule [165], Source http //www.niherst.gov.tt/scipop/sci-bits/microfluidics.htm (see Plate 12 for color version). 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Figure 8 Left Schematic graph of the setup for the simulation of rubbing surfaces. Upper and lower walls are separated by a fluid or a boundary lubricant of thickness D. The outermost layers of the walls, represented by a dark color, are often treated as a rigid unit. The bottom most layer is fixed in a laboratory system, and the upper most layer is driven externally, for instance, by a spring of stiffness k. Also shown is a typical, linear velocity profile for a confined fluid with finite velocities at the boundary. The length at which the fluid s drift velocity would extrapolate to the wall s velocity is called the slip length A. Right The top wail atoms in the rigid top layer are set onto their equilibrium sites or coupled elastically to them. The remaining top wall atoms interact through interatomic potentials, which certainly may be chosen to be elastic.

Fig. 4.5 Nondimensional temperature profiles produced by viscous dissipation in the annular region between a moving rod and a stationary guide. The fluid is characterized by Pr = 4000, which is typical of lubricating oils. The rod and guide geometry is characterized by r lhr — 10, meaning that the rod radius is 10 gap thicknesses. The temperature profiles are parameterized by P = -j jr—. These solutions were generated by a 12-node finite-volume algorithm implemented in Excel.

Figure 4.14 illustrates the transient solution to a problem in which an inner shaft suddenly begins to rotate with angular speed 2. The fluid is initially at rest, and the outer wall is fixed. Clearly, a momentum boundary layer diffuses outward from the rotating shaft toward the outer wall. In this problem there is a steady-state solution as indicated by the profile at t = oo. The curvature in the steady-state velocity profile is a function of gap thickness, or the parameter rj/Ar. As the gap becomes thinner relative to the shaft diameter, the profile becomes more linear. This is because the geometry tends toward a planar situation. 

Unfortunately, the thinness of most liquid films makes it difficult to measure the velocity profiles experimentally, since it is practically impossible to introduce any of the usual fluid-velocity probes into a film which may be less than 1 mm. thick without grossly distorting the flow patterns. Nevertheless, film velocity profile measurements have been reported for a few special cases. 

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