Stress spherical coordinates

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 ... 

At this point, it should be again stressed that the above form for L2 appears explicitly when the kinetic energy operator - h2/2m V2 is expressed in spherical coordinates in particular,... 

Additional operations may be found on pp. xxiii to xxvi of Ref. (Hll). Most of these relations may be found in cylindrical, spherical, and other coordinate systems in standard reference works. Several of them do not, however, seem to be tabulated for handy reference these operations are given here in cylindrical and spherical coordinates. Expressions for the Newtonian stress tensor in terms of the velocity gradients and the coefficient of viscosity may be found in Ref. (G7, pp. 103-105). 

Table 5.2 presents the momentum balance in terms of deviatoric stress in the Cartesian, cylindrical and spherical coordinate systems. 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

The radial and transverse stresses can be determined from the stress-strain relationships. Owing to the orthogonality of the spherical coordinates, the formal structure of the generalized Hooke s law, given by Eq. (P4.11), is preserved, so that the nonzero components of the stress tensor are expressed in terms of the strain tensors as... 

Determine the velocity and pressure fields in the liquid as well as the velocity of the bubble by means of a full eigenfunction expansion for spherical coordinates. Does this solution satisfy the normal-stress condition on the bubble surface ... 

Coal particle intramural gas migration accords with nonsteady diffusion law in spherical coordinate frame. In the original condition, coal particle gas adsorption has been a balanced state, when external stress changing, gas will transform from adsorbed state to unbound state, and adsorbent gas will diffuse from coal particle core to surface following both mass conservation and continuity principle. In the coal particle core, adsorbent gas concentration abides by Langmuir law In the coal particle surface, the mass transfer between adsorbent gas and dissociate gas abide... 

The elastic problem for an elastic half-space contacted by a normal point force P was solved by Boussinesq in 1885. The stress field is axisymmetric around the force direction and has the general form, in spherical coordinates. 

Shear-stress components for Newtonian fluids in spherical coordinates... 

For flow past a sphere the stream function ij/ can be used in the Navier-Stokes equation in spherical coordinates to obtain the equation for the stream function and the velocity distribution and the pressure distribution over the sphere. Then by integration over the whole sphere, the form drag, caused by the pressure distribution, and the skin friction or viscous drag, caused by the shear stress at the surface, can be summed to give the total drag. 

Equations (11) and (12) follow from the definitions given by Batdorf and Heinisch (Ref. 3) associated with their equations (l)-(4). Also note that (11) and (12) employ the spherical coordinate system of Lamon and Fivans (Ref. 4). As is made clear in Batdorf and Heinisch (Ref. 3) fl is a function of both the fracture criterion selected and the applied stress state. Thus / must be a function of both as well. This is reflected in the definition given by (12). And since the coordinate system denotes the orientation of the plane of a crack, / must be a function of as shown in (12). Now set... 

In the case of spherical particles, the flow field is described by spherical coordinates originating at the particle centre. The projection of the stress tensor onto the surface normal, i.e. onto the radial coordinate, is then ... 

A shrinking cladding (matrix) around a core (inclusion) gives rise to (1) compressive stresses within the core and (2) a compressive radial stress and tangential tensile stresses within the cladding (28). If ae and are the components of the stress in spherical coordinates (Fig. 11.19), all three stress components in the... 

In this appendix are presented the equations of continuity, momentum, viscous stress, energy, and binary diffusion in the rectangular, cylindrical, and spherical coordinate systems for a Newtonian fluid [1, 2],... 

The thermal radial and tangential stresses. Or and Ocp, Oy, derived in the system of the spherical coordinates (r,(p,v) related to the Cartesian system (OX1X2X3) (Fig. 1), are investigated in the point P of a continuum along the axes Xr and x, Xy of the Cartesian system (Px XipXy), where r is the distance fi om the spherical particle centre, O. 

We next consider the formulation of the problem for the viscoelastic case. We select the PTT model again (see Eq. 3.45), which in spherical coordinates (Bird et al., 1987) leads to the following equations for the stress components ... 

In order to obtain the normal stress functions, we need to solve the equations of motion in spherical coordinates [3]. An examination of the 0 component of this equation shows dp/36 = 0. Thus, p depends on r alone, because derivatives with respect to 4> are zero. Further, because most polymer fluids are fairly viscous, we can neglect inertia and, as a result, the r component of the equation of motion yields the following [3] ... 

Note that all the stresses in Eq. (2.27) become singular at the origin where the point force is applied. To avoid this singularity, consider a small spherical cavity whose center is located at the origin of the coordinates as shown in Fig. 2.4. The coordinates are arranged such that the force is in the z-direction and is applied at the origin of the coordinates. Thus, the summation of the surface forces from the stresses in the direction of the z-axis balances the point force inside the solid medium. 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

In Eq. (25) the term v (V r) represents reversible stress work, while r Vv represents irreversible or entropy-producing stress work. The following are expressions for the latter quantity in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

The plot above has been produced with six identical distances from the centroid of the ligand (the metal center) to the amines. A non-spherical metal ion (e.g., a Jahn-Teller labile copper(II) ion) will induce different stresses to a symmetrical ligand. Alternatively, the ligand might be asymmetrical, i. e., the ligand itself might induce an asymmetry in the coordination sphere. Effects like these will be studied in Section 17.17. 

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