No slip condition

The force is direcdly proportional to the area of the plate the shear stress is T = F/A. Within the fluid, a linear velocity profile u = Uy/H is estabhshed due to the no-slip condition, the fluid bounding the lower plate has zero velocity and the fluid bounding the upper plate moves at the plate velocity U. The velocity gradient y = du/dy is called the shear rate for this flow. Shear rates are usually reported in units of reciprocal seconds. The flow in Fig. 6-1 is a simple shear flow. 

This problem requires use of the microscopic balance equations because the velocity is to he determined as a function of position. The boundary conditions for this flow result from the no-slip condition. AU three velocity components must he zero at the plate surfaces, y = H/2 and y = —H/2. 

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. 

To obtain physically meaningful solutions, a set of appropriate boundary conditions must also be specified. One obvious requirement is that no fluid should pass through the boundary (i.e. wall) itself. Thus, if we choose a reference frame in which the boundaries are at rest, we require that v fi = 0, where fi is the unit normal to the surface. Another condition, the so-called no-slip condition ([trittSS], [feyn64]), is the requirement that the fluid s tangential velocity vanishes at the surface v x n = 0. 

In fluid dynamics it is generally assumed that the velocity of flow at a solid boundary, such as a pipe wall, is zero. This is referred to as the no-slip condition. If the fluid wets the surface, this assumption can be justified in physical terms since the molecules are... 

If the fluid does not wet the wall, the no-slip condition no longer applies and the pressure gradient at a given flowrate will be lower. This effect is particularly important with the flow of molten polymers, although it does not seem to be significant in other applications. 

Considink, D. M. 272 Consistency coefficient of fluids 108 Constitutive equations 111 Contact angle and no-slip condition 670 -----boiling 483... 

The boundary conditions are as follows In Figure 3.2.2, z-axis component and r-axis component velocities are zero for (1) and (2), respectively. The gradients of other variables are zero for both the boundaries. The gradients of all variables are zero for (3) and (4). No slip condition and heat transfer from the flame kernel to the spark electrode are assumed for (5) and (6), at the surface of spark electrode. 

The discussion above that led to Eqs. (4.2.6 and 4.2.7) assumes that the no-slip condition at the wall of the pipe holds. There is no such assumption in the theory for the spatially resolved measurements. We have recently used a different technique for spatially resolved measurements, ultrasonic pulsed Doppler velocimetry, to determine both the viscosity and wall slip velocity in a food suspension [2]. From a rheological standpoint, the theoretical underpinnings of the ultrasonic technique are the same as for the MRI technique. Flence, there is no reason in principle why MRI can not be used for similar measurements. 

Fig. 3a indicates that the bubble-rise velocity measured based on the displacement of the top surface of the bubble ( C/bt) quickly increases and approaches the terminal bubble rise velocity in 0.02 s. The small fluctuation of Ubt is caused by numerical instability. The bubble-rise velocity measured based on the displacement of the bottom surface of the bubble (Ubb) fluctuates significantly with time initially and converges to Ubt after 0.25 s. The overshooting of Ubb can reach 45-50 cm/s in Fig. 3a. The fluctuation of Ubb reflects the unsteady oscillation of the bubble due to the wake flow and shedding at the base of the bubble. Although the relative deviation between the simulation results of the 40 X 40 x 80 mesh and 100 x 100 x 200 mesh is notable, the deviation is insignificant between the results of the 80 x 80 x 160 mesh and those of the 100 X 100 x 200 mesh. The agreement with experiments at all resolutions is generally reasonable, although the simulated terminal bubble rise velocities ( 20 cm/s) are slightly lower than the experimental results (21 25 cm/s). A lower bubble-rise velocity obtained from the simulation is expected due to the no-slip condition imposed at the gas-liquid interface, and the finite thickness for the gas-liquid interface employed in the computational scheme.

There are two constants of integration in equation 1.56 so two boundary conditions are required. The first is the no-slip condition at r = r, and the second is that the velocity gradient is zero at r = 0. Using the latter condition in equation 1.55 shows that A2 = 0 so that equation 1.56 becomes identical to equation 1.53. The no-slip boundary condition gives the value of B as before. 

The dynamics of the solvent and polymer are coupled in a polymer solution. Although this coupling is expressed partly through the friction and noise terms in Eq. (28), a complete description requires a boundary condition. The most common choice of this boundary condition is the no-slip condition... 

However, in the case of large Kn, the no-slip approximation cannot be applied. This implies that the mean free path of the liquid is on the same length scale as the dimension of the system itself. In such a case, stress and displacement are discontinuous at the interface, so an additional parameter is required to characterize the boundary condition. A simple technique to model this is the one-dimensional slip length, which is the extrapolation length into the wall required to recover the no-slip condition, as shown in Fig. 1. If we consider... 

Figure 1. No-slip condition and slip condition with slip length, for one-dimensional shear flow. The slip length b is the extrapolation distance into the solid, to obtain the no-slip point. The slope of the linear velocity profile near the wall is the shear rate y.

The condition on the tangential velocity at the interface is not as obvious as that on the normal velocity. There is now ample experimental evidence that the fluid velocity at the surface of a rigid or noncirculating particle is zero relative to the particle, provided that the fluid can be considered a continuum. This leads to the so-called no-slip condition, which for a fluid particle takes the form... 

For solid particles a sufficient set of boundary conditions is provided by the no slip condition, the requirement of no flow across the particle surface, and the flow field remote from the particle. For fluid particles, additional boundary conditions are required since Eqs. (1-1) and (1-9) apply simultaneously to both phases. Two additional boundary conditions are provided by Newton s third law which requires that normal and shearing stresses be balanced at the interface separating the two fluids. 

When r = Rs, the velocity is zero. This is described as a nonslip (or no-slip) condition between the stationary surface of the particle and the layer of fluid adjacent to it. There is nothing particularly self-evident about the nonslip condition between the solid and the fluid, but it is an experimental fact. Some commonplace evidence that suggests this is the layer of dust that accumulates on the blades of a fan. However stiff the breeze may be some distance in front of the blades, the air is still and travels with the surface of the blades. 

As discussed in Section 2.6, vorticity is a measure of the angular rotation rate of a fluid. Generally speaking, vorticity is produced by forces that cause rotation of the flow. Most often, those forces are caused by viscous shearing action. As viscous fluid flows over solid walls, for example, the shearing forces caused by a no-slip condition at the wall is an important source of vorticity. The following analysis shows how vorticity is transported throughout a flow field by convective and viscous phenomena. 

The term p, V2u> reveals that vorticity (i.e., the strength of fluid rotation) can diffuse by molecular interactions throughout a flow field, with the viscosity being the diffusion coefficient. Quite often the source of vorticity is the fluid tumbling caused by the shearing action associated with a no-slip condition on a solid wall. This vorticity, once produced, is both convected and diffused throughout the flow. The relative strength of the convective and diffusive processes depends on the flow field and the viscosity. 

Because the no-slip condition requires that the velocities at the wall vanish, the axial momentum equation at the wall has a significantly reduced form. Stated in terms of vor-ticity, the incompressible Navier-Stokes equations can be written as... 

Since the radial domain in the duct ranges between 0 < r < rw, the constant Ci must be zero. Otherwise, the velocity would become unbounded at the centerline. The other constant is determined easily from a no-slip condition at the wall, rw. The solution is... 

The entry-length region is characterized by a diffusive process wherein the flow must adjust to the zero-velocity no-slip condition on the wall. A momentum boundary layer grows out from the wall, with velocities near the wall being retarded relative to the uniform inlet velocity and velocities near the centerline being accelerated to maintain mass continuity. In steady state, this behavior is described by the coupled effects of the mass continuity and axial momentum equations. For a constant-viscosity fluid,... 

Consider a long cylindrical shell whose interior is filled with an incompressible fluid. If the fluid is initially at rest when the cylinder begins to rotate, a boundary layer develops as the momentum diffuses inward toward the center of the cylinder. The fluid s circumferential velocity vu comes to the cylinder-wall velocity immediately, owing to the no-slip condition. At very early time, however, the interior fluid will be only weakly affected by the rotation, with the influence increasing as the boundary layer diffuses inward. If the shell continues to rotate at a constant angular velocity, the fluid inside will eventually come to rotate as a solid body. 

As illustrated in Fig. 5.2, the classic Jeffery-Hamel flow concerns two-dimensional radial flow in a wedge-shaped region between flat inclined walls. The flow may be directed radially outward (as illustrated) or radially inward. The flow is assumed to originate in a line source or terminate in a line sink. Velocity at the solid walls obeys a no-slip condition. In practice, there must be an entry region where the flow adjusts from the line source to the channel-confined flow with no-slip walls. The Jeffery-Hamel analysis applies to the channel after this initial adjustment is accomplished. 

This equation represents a third-order boundary-value problem, which requires three independent conditions for solution. Two of the boundary conditions are immediately evident, but the third requires a bit more care. At the centerline a = 0 there is a symmetry condition, and at the wall there is a no-slip condition,... 

Solution of Eq. 5.58 is subject to three conditions. Two follow easily from the no-slip condition at the walls,... 

Assuming that Ar is known, the governing equations represent a third-order boundary-value problem that demands three boundary conditions. Assume that z is measured from the stagnation plane. At the stagnation surface (z = 0), the no-slip condition requires that... 

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