Stokes’ first problem

Transient motion of a flat plate. One of these problems (known as Stokes first problem) describes the flow near an infinite flat plate instantaneously set in motion at a velocity Uq in the plate plane. In this case, the initial and boundary conditions for Eq. (1.9.1) are written as follows ... 

Viscoelastic fluids. In the monograph [181], the exact solution of Stokes first problem (6.10.1)—(6.10.3) was obtained for Maxwellian fluids with the rheological law (6.1.10), which has the following form for the problem in question ... 

To solve this problem, we first convert the measured terminal velocity to the equivalent velocity which would be achieved by the sphere in a fluid of infinite extent. Assuming Stokes law we can determine the fluid viscosity. Finally we check the validity of Stokes law. 

The first of these problems involves relative motion between a rigid sphere and a liquid as analyzed by Stokes in 1850. The results apply equally to liquid flowing past a stationary sphere with a steady-state (subscript s) velocity v or to a sphere moving through a stationary liquid with a velocity -v the relative motion is the same in both cases. If the relative motion is in the vertical direction, we may visualize the slices of liquid described above as consisting of... 

Because the Navier-Stokes equations are first-order in pressure and second-order in velocity, their solution requires one pressure bound-aiy condition and two velocity boundaiy conditions (for each velocity component) to completely specify the solution. The no sBp condition, whicn requires that the fluid velocity equal the velocity or any bounding solid surface, occurs in most problems. Specification of velocity is a type of boundary condition sometimes called a Dirichlet condition. Often boundary conditions involve stresses, and thus velocity gradients, rather than the velocities themselves. Specification of velocity derivatives is a Neumann boundary condition. For example, at the boundary between a viscous liquid and a gas, it is often assumed that the liquid shear stresses are zero. In numerical solution of the Navier-... 

We assume that with the help of leveling we solved our first problem and found the separation between the geoid and the points of the physical surface of the earth. Our next step is to determine the position of the geoid with respect to the reference ellipsoid. The solution of this fundamental problem was given by Stokes. To begin,... 

In the previous section we described the Stokes method, which allows us to find the distance between the reference ellipsoid and the physical surface of the earth. The ellipsoid, given by its semi-major axis a, flattening a, and elements of orientation inside of the earth can be considered as the first approximation to a figure of the earth. In order to perform the transition to the real earth we have to know the distance along the normal from each point of the spheroid to the physical surface of the earth. Earlier we demonstrated that this problem includes two steps, namely,... 

Investigation of the velocity profiles obtained for the first case (crossection (x, y = 70 nm)) indicates a strong decrease of the maximum fluid velocity with increasing electrolyte concentration (see Figure 4). Furthermore, a transition of the velocity curves from a parabolic curve classically obtained from the Navier-Stokes problem (i.e., c = 0 mol/m3) to a very flattend curve for high electrolyte concentration (c = 1000 mol/m3) can be seen. 

As I have indicated, this presentation will be divided into two parts. In the first part we will discuss the development of Coherent Anti-Stokes Raman Spectroscopy, the problems inherent in applications to combustion sources, recent developments which address operational problems, and the state-of-the-art today. This will be followed by a similar discussion involving the use of saturated laser-induced fluorescence spectroscopy as a combustion diagnostic. 

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