Stress sign convention

The shear-stress convention is a bit more complicated to explain. In a differential control volume, the shear stresses act as a couple that produces a torque on the volume. The sign of the torques defines the positive directions of the shear stresses. Assume a right-handed coordinate system, here defined by (z, r, 9). The shear-stress sign convention is related to ordering of the coordinate indexes as follows a positive shear xzr produces a torque in the direction, a positive xrg produces a torque in the z direction, and a positive x z produces a torque in the r direction. Note also, for example, that a positive xrz produces a torque in the negative 6 direction. 

The sign associated with the pressure is opposite to that associated with the normal viscous stress. The usual sign convention assumes that a tensile stress is the positive normal stress so that the pressure, which by definition has compressive normal stress, has a negative sign. 

The information presented in this work builds upon developments from several more established fields of science. This situation can cause confusion as to the use of established sign conventions for stress, pressure, strain and compression. In this book, those treatments involving higher-order, elastic, piezoelectric and dielectric behaviors use the established sign conventions of tension chosen to be positive. In other areas, compression is taken as positive, in accordance with high pressure practice. Although offensive to a well structured sense of theory, the various sign conventions used in different sections of the book are not expected to cause confusion in any particular situation. 

In this equation the negative sign is introduced in order to maintain a consistency of sign convention when shear stress is related to momentum transfer as in Chapter 11. Since (du,/dy)v=o must be positive (velocity increases towards the pipe centre), R0 is negative. It is therefore more convenient to work in terms of / , the shear stress exerted by the fluid on the surface (= —Rq) when calculating friction data. 

When analysing simple flow problems such as laminar flow in a pipe, where the form of the velocity profile and the directions in which the shear stresses act are already known, no formal sign convention for the stress components is required. In these cases, force balances can be written with the shear forces incorporated according to the directions in which the shear stresses physically act, as was done in Examples 1.7 and 1.8. However, in order to derive general equations for an arbitrary flow field it is necessary to adopt a formal sign convention for the stress components. 

The way to remember the conventions is as follows. For the positive sign convention, the stress component acting on the element s face at the higher y-value (the upper face) is taken as positive in the positive x-direction for the negative sign convention the same component is taken as positive in the negative x-direction. In each convention, the stress component acting on the opposite face is taken as positive in the opposite direction. 

In cylindrical coordinates, the velocity gradient dvjdr generates the shear stress component rrx and Newton s law must be expressed in the two sign conventions as ... 

It is important to appreciate that the sign conventions do not dictate the direction in which a stress physically acts, they simply specify the directions chosen for measuring the stresses. If a stress physically acts in the opposite direction to that specified in the convention being used, then the stress will be found to have a negative value, just as in elementary mechanics where a negative force reflects the fact that it acts in the opposite direction to that taken as positive. 

Both sign conventions are used in the fluid flow literature and consequently the reader should be able to work in either, as appropriate. The negative sign convention is convenient for flow in pipes because the velocity gradient dvjdr is negative and therefore the shear stress components turn out to be positive indicating that they physically act in the directions assumed in the sign convention. This is illustrated in Example 1.9. 

Determine the shear stress distribution and velocity profile for steady, fully developed, laminar flow of an incompressible Newtonian fluid in a horizontal pipe. Use a cylindrical shell element and consider both sign conventions. How should the analysis be modified for flow in an annulus ... 

Using the negative sign convention for stress components, the shear stress acting on the outer surface of the element (the higher value of r) must be measured in the negative x-direction and that on the inner surface in the positive x-direction, as indicated in Figure 1.17. 

Laminar flow in a pipe showing a typical fluid element and the velocity profile. The negative sign convention for stress components is shown... 

In general, with the different sign conventions, equations involving stress components have opposite signs in the two conventions. On substituting the appropriate form of Newton s law of viscosity, the sign difference cancels giving identical equations for the velocity profile. 

In the preceding section, only one stress component was considered and that component was the only one of direct importance in the simple flow considered. The force acting at a point in a fluid is a vector and can be resolved into three components, one in each of the coordinate directions. Consequently the stress acting on each face of an element of fluid can be represented by three stress components, as shown in Figure 1.18 for the negative sign convention. 

Negative sign convention for stress components. The diagram shows the directions in which components are taken as positive. The components acting on the faces normal to the x-axis have been omitted for clarity... 

When using the positive sign convention, the direction of each stress component that is taken as positive is the opposite of that shown in Figure 1.18. 

Independent of the sign convention used, the stress components can be classified into two types those that act tangentially to the face of the element and those that act normal to the face. Tangential components such as rxy, ryx, Tyz tend to cause shearing and are called shear stress components (or simply shear stresses). In contrast, the stress components rxx, Tyy, jzz act normal to the face of the element and are therefore called normal stress components (or normal stresses). Although there are six shear stress components, it is easily shown that ri = t, for t = j for example, ryx = rxy. Thus there are three independent shear stress components and three independent normal stress components. 

The pressure acting on a surface in a static fluid is the normal force per unit area, ie the normal stress. The pressure of the surrounding fluid acts inwards on each face of a fluid element. Consequently, with the negative sign convention the normal stress components may be identified with the pressure. With the positive sign convention, the normal stress components may be identified with the negative of the pressure positive normal stresses correspond to tension with this convention. 

In the case of a flowing fluid the mechanical pressure is not necessarily the same as the thermodynamic pressure as is the case in a static fluid. The pressure in a flowing fluid is defined as the average of the normal stress components. In the case of inelastic fluids, the normal stress components are equal and therefore, with the negative sign convention, equal to the pressure. It is for this reason that the pressure can be used in place of the normal stress when writing force balances for inelastic liquids, as was done in Examples 1.7-1.9. 

Using the negative sign convention for stress components, Newton s law of viscosity can be written as... 

The term viscosity has no meaning for a non-Newtonian fluid unless it is related to a particular shear rate y. An apparent viscosity fia can be defined as follows (using the negative sign convention for stress) ... 

In the simplest case, that of time-independent behaviour, the shear stress depends only on the shear rate but not in the proportional manner of a Newtonian fluid. Various types of time-independent behaviour are shown in Figure 1.19(a), in which the shear stress is plotted against the shear rate on linear axes. The absolute values of shear stress and shear rate are plotted so that irrespective of the sign convention used the curves always lie in the first quadrant. 

It should be noted that for shear thinning and shear thickening behaviour the shear stress-shear rate curve passes through the origin. This type of behaviour is often approximated by the power law and such materials are called power law fluids . Using the negative sign convention for stress components, the power law is usually written as... 

It has been assumed that the flow is incompressible so that there are no fluctuations of the density. Equation 1.91 shows that the momentum flux consists of a part due to the mean flow and a part due to the velocity fluctuation. The extra momentum flux is proportional to the square of the fluctuation because the momentum is the product of the mass flow rate and the velocity, and the velocity fluctuation contributes to both. The extra momentum flux is equivalent to an extra apparent stress perpendicular to the face, ie a normal stress component. As (v x)2 is always positive it produces a compressive stress, which is positive in the negative sign convention for stress. 

In equation 1.94, (Tyx)v is the viscous shear stress due to the mean velocity gradient dvjdy and pv yv x is the extra shear stress due to the velocity fluctuations v x and v y. These extra stress components arising from the velocity fluctuations are known as Reynolds stresses. (Note that if the positive sign convention for stresses were used, the sign of the Reynolds stress would be negative in equation 1.94.)... 

Consider a fixed element of space with unit area in the x-z plane and having its surfaces at distances y and y+5y from the plate. Using the negative sign convention for stress components (which coincides with the... 

Consider the steady, laminar flow of an incompressible fluid in a long and wide closed conduit channel subject to a linear pressure gradient, (a) Derive the equation for velocity profile, (b) Derive the equation for discharge per unit width and cross-sectional mean velocity, and compare this with the maximum velocity in the channel, (c) Derive the equation for wall shear stress on both walls and compare them. Explain the sign convention for shear stress on each wall. 

The finite control-volume dimensions as illustrated in Fig. 2.13 may be a potential source of confusion. While the stress tensor represents the stress state at a point, it is only when the differential control volume is shrunk to vanishingly small dimensions that it represents a point. Nevertheless, the control volume is central to our understanding of how the stress acts on the fluid and in establishing sign conventions for the stress state. For example, consider the normal stress xrr, which can be seen on the r + dr face in the left-hand panel and on the r face in the right-hand panel. Both are labeled rrr, although their values are only equal when the control volume has shrunk to a point. Since the stress state varies continuously and smoothly throughout the flow, the stress state is in fact a little different at the centers of the six control-volume faces as illustrated in Fig. 2.13 where the... 

The sign convention for the stress components is important. A positive normal stress is tensile (i.e., tending to expand the control volume) and a negative normal stress is compressive. Thus, for example, referring to Fig. 2.13, a positive xrr points in the positive r direction on the r + dr face while it points in the negative r direction on the r face. 

Work on a Cylindrical Differential Element Consider the cylindrical differential control volume such as the one illustrated in Fig. 3.9. A two-dimensional projection of this element is illustrated in Fig. 3.10. Recall the discussion in Section 2.8.2 on the sign convention for the stress components—the sign conventions are important. At z, r, and 0 the rates of work done on the near control-volume faces are... 

This discretization method obeys a conservation property, and therefore is called conservative. With the exception of the first element and the last element, every element face is a part of two elements. The areas of the coincident faces and the forces on them are computed in exactly the same way (except possibly for sign). Note that the sign conventions for the directions of the positive stresses is important in this regard. The force on the left face of some element is equal and opposite to the force on the right face of its leftward neighbor. Therefore, when the net forces are summed across all the elements, there is exact cancellation except for the first and last elements. For this reason no spurious forces can enter the system through the numerical discretization itself. The net force on the system of elements must be the net force caused by the boundary conditions on the left face of the first element and the right face of the last element. 

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