Pressure distribution creeping flow

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

Use the general representation of solutions for creeping flows in terms of vector harmonic functions to solve for the velocity and pressure fields in the two fluids, as well as the deformation and surfactant concentration distribution functions, at steady state. You should find... 

When the radius of the particle is very small, or the fluid viscosity is very large, or the relative velocity is very small, or the fluid density is very low, the Reynolds number becomes very small and the flow satisfies the conditions of the Stokes or creeping flow. In this limit, the inertial forces near the particle are small and can be neglected in the Navier-Stokes equations. The pressure distribution rm the drop in this limit takes the form... 

Problem 8-3. Consider the flow that is due to a point source of fluid, such that the fluid emanates from the origin at a rate Q (vol/time - a scalar). We may assume that this flow is a creeping motion. Under what conditions will this be true What is the resulting pressure and velocity distribution There is obviously more than one way to solve this problem, but the objective here is to use the singularity methods of this chapter. 

M3 in Fig. 11a). Flow here represents quasistationary creep including episodic high activity, depending on infiltration and the pore pressure (Fig. 11b). The distribution of velocities in space and time within the landslide mass justifies its macroscopic description as a ductile medium. However, as the following two examples will prove, brittle deformations or interactions between pore fluid and the rock/soil mass also exist and resemble seismic sources. 

Fig. 12 Le/i Effect of the flow configuration and methane conversion fraction (PR) on the stress. Case of an anode-supported cell with LSM-YSZ cathode and compressive gaskets, a Temperature profile and b First principal stress in the anode. The MIC is displayed in transparency, c First principal stress in the cathode (insert alxtve the symmetry line), d Contact pressure on the cathode GDL and compressive gasket and e vertical displacement along the z-axis, with an amplification factor of 2,000. Right column effect of creep in a cell based on a LSCF cathode and a temperature distribution, on b the evolution of the first principal stress in the anode support in operation and c during thermal cycling to RT and d evolution of the first principal stress in the GDC compatibility layer after thermal cycling. The profiles above and below the symmetry axis refer to different operation time [88, 89]. Reproduced here with kind permission from Elsevier 2012...

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