Stress tensor cylindrical coordinates

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). 

Writing the strain-rate components in terms of the velocity field (Section 2.5) yields a general relationship between the flow field and the stress tensor in a particular coordinate system. For example, in a cylindrical coordinate system... 

The sum of the diagonal elements is an invariant of the stress tensor. That is, regardless of the particular orientation of the coordinate system, or the coordinate system itself (e.g., cartesian versus cylindrical), the sum of the diagonal elements of the stress tensor is unchanged. From Eq. 2.180 it is easily seen that... 

Working in cylindrical coordinates, and substituting the force-per-unit-volume expressions that stem from the stress tensor (Eqs. 2.137, 2.138, and 2.139), the Navier-Stokes equations can be written as... 

The (stress or strain-rate) state at a point is a physical quantity that cannot depend on any particular coordinate-system representation. For example, the stress state is the same regardless of whether it is represented in cartesian or cylindrical coordinates. In other words, the state (as represented by a symmetric second-order tensor), is invariant to the particular coordinate-system representation. 

For convenience in the following discussion, Eq. (2.17) is expressed in cylindrical coordinates. The symmetric stress tensor T is given as... 

According to these equations and using, as above, cylindrical coordinates in the stress-strain relationships, the zz component of the stress tensor can be written as... 

The stress is a rank two tensor. The relevant components here are the normal and tangential stress, namely, the components and T rs. They both enter the expression of forces exerted on the element of surface parallel to the 1/v interface. Let us first look at part. Its only contribution comes from the viscous stress. Its general expression in cylindrical coordinates is ... 

The stress tensor in cylindrical coordinate system is given by... 

The solid phase is considered to be in elastic mechanical equilibrium, which is reasonable if the characteristic time of relaxation is smaller than the characteristic time for moisture diffusion. Then, the divergence of the stress tensor must be zero in cylindrical polar coordinates, this translates into the following equations for normal stresses, a Oe and o, and shear stress,... 

Components of stress tensor in cylindrical coordinates Components of stress tensor in rectangular Cartesian coordinates Shear stress acting on mandrel surface [Eq. (46b)]... 

Let the object for uniaxial extension be a cylindrical rod with a cross-sectional area S to which an extension force F is applied. The ratio F/S = Pis the extensional stress. Alternatively, this is a component of a stress tensor in the selected coordinate system, Oyy = 032 = 2 = F. The action of the force resulted in a stressed state that is uniform in all directions (we are neglecting nonuniformities and local concentrations of stresses in certain special areas, such as the mounting points of the load or of the rod itself). The selected coordinate system in this case is the principal one. From symmetry considerations at F = Oyy, this tensor is as follows ... 

This is also calculated for the cylindrical coordinate system using equation [1.2] and the stress tensor (Table 1.2) for the velocity field [1.23] of the Poiseuille flow in a circular pipe. 

The height, H, of the apparatus is sufficiently large, and the gap width Rt R sufficiently small, to be able to neglect edge effects at the top and bottom in the force balance. The displacements and stresses can be regarded as axisyrmnetric with respect to the Oz axis and independent from z. Therefore, in the cylindrical coordinate system with the Oz axis, the different variables depend not on variables 6 and z, but only on r. Consequently, the stress tensor is written in the cylindrical coordinate system (Chapter 1, Table 1.2) as ... 

Returning to the problem of the blocked pipe, consider the flow in the cylindrical coordinate system whose Oz axis, which coincides with the axis of the pipe (see Figure 1.3 of Chapter 1), is oriented in the direction of the flow. By considering the non-zero component(s) in the velocity vector, verify that the only non-zero term in the strain rate tensor is Drz. Set out the form of this term explicitly and explain why ) 5 0. Explain also why the only non-zero shear stress in the pipe is Trz, and why Vrz <0. It will be deduced there from that the rheological relation for the Bingham fluid can be written, for the flow of that fluid in a pipe, as ... 

At a point in the filament boundary the unit outward normal vector is n and the tangential vector is t. The appropriate coordinate system for the problem is cylindrical with the axis of symmetry coinciding with the z axis (Fig. 9.3). The velocity vector is v and the total stress tensor is rr. The components... 

The anti-symmetric part of the stress tensor implies an additional torque, which becomes relevant when the boundary condition is given by forces. In cylindrical coordinates (r, 6,z), the azimuthal stress is given by [38]... 

It is instructive at this point to comment upon the constant c. The surface traction exerted by an outer cylinder upon the liquid crystal is given via equations (4.37), (4.128) and (4.129) with 1/ = e,. (notice that tzr = 0 by equations (4.121), (5.245), (5.279), (5.284), (5.285) and (5.286)). Recalling that there are surface, contributions arising from both the stress tensor (4.128) and couple stress tensor (4.129) (cf. the balance law (4.31)), the corresponding moment about the s -axis of such a cylinder of radius r is then, in cylindrical coordinates, with U = UjUj and 1% = hj ji... 

The divergence of the stress tensor T with Cartesian components T j is defined in this text by [V -T] = Tijj. The physical components of this divergence in cylindrical polar coordinates are [115, p.l44]... 

Furthermore, based on the axial symmetry of the tube, only derivatives of the stress tensor with respect to r are nonzero (Appendix 8.A). The differential momentum balance equations in cylindrical coordinates (Appendix 8.B) can be simplified as follows ... 

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