The relation between viscous stress and rate of strain
Since the stress at any point in the fluid is an expression of the mutual reactions of adjacent parts of fluid near that point, it is natural to consider the connection between the stress and the local properties of the fluid. In the case of a fluid at rest, this is a simple matter since the stress is determined wholly by the one scalar quantity , the hydrostatic pressure, and in turn is specified locally by the equilibrium equation of state when the values of the two parameters of state (e.g., temperature and density) are known; and if the distribution of the body force per unit volume acting on the fluid is known, there is no need to consider local variables of state at more than one point because the relative pressure is determined by hydrostatic equilibrium:
/ (6.80)The connection between stress and the local properties of a moving fluid is more complicated in two respects: first the stress tensor contains isotropic and non-isotropic parts, and second, the scalar specifying the isotropic part is no longer one of the variables of state used in equilibrium thermodynamics. The first of these non-equilibrium effects represents a transport of momentum, or internal friction, and often by far the most important.
The argument to be used in establishing a relation between the stress tensor and the local properties of the fluid is of the kind discussed under transport phenomena. The internal friction in a moving fluid is only one of several similar kinds of transport phenomena arising from a departure from equilibrium. In this case the relevant phenomenon is the molecular transport of momentum in the case of a simple shear motion. We shall seek a phenomenological approach and seek relations whose forms are independent of the nature of the molecular mechanism of the internal friction.
The part of the momentum flux across a material surface element which results from fictional interaction of the matter in relative motion on the two sides of the element and which is represented by the viscous stress tensor is assumed, to depend only on the instantaneous fluid velocities in the neighborhood of the element, or, more precisely, on the departure from uniformity of that distribution. The local velocity gradient, of which a typical component is thus the parameter of the flow field with most relevance to the viscous stress, and since is normally uniform over distances large compared with distances characteristic of the mechanism of molecular transport of momentum we assume it is the only relevant parameter. Furthermore, is zero in a stationary fluid and so vanishes with .
We have no way of deducing the dependence of on for fluids in general, and we therefore fall back on the hypothesis, that is approximately a linear function of the various components of the velocity gradient for sufficiently small magnitudes of those components. Analytically the hypothesis is expressed as
/ (6.81)where the 81 components of the fourth-order tensor depend on the local state of the fluid, but not directly on the velocity distribution, and is necessarily symmetrical in the indices and like . This equation means that each viscous stress component is linearly related to all nine components of the velocity gradient. The condition that is symmetric reduces the 81 components to two; the further condition that the viscous stress makes no contribution to the normal stress requires that reduces to a scalar quantity.
Equation 6.81 is the counterpart of the linear relation between flux and gradient of a scalar transportable quantity. As we have seen before it is convenient at this stage to write as the sum of its symmetrical part , the rate of strain tensor, and its antisymmetrical part , where is the vorticity, so that
/ (6.82)The tensor coefficient takes a simple form when the molecular structure of the fluid is statistically isotropic, that is, when the viscous stress generated in an element of the fluid by a given velocity gradient is independent of the orientation of the element. All gases have isotropic structure, as do simple liquids, although suspensions and solutions containing long chain-like molecules may exhibit some directional preferences owing to alignment of these molecules in a manner which depends on the past history of the motion. We shall restrict attention to fluids of isotropic structure, in which case is an isotropic tensor, having a form from which all directional distinction is absent.
The basic isotropic Cartesian tensor is the Kronecker delta tensor, and all isotropic tensors of even order can be written as the sum of products of delta tensors. Thus
/ (6.83)where , and are scalar coefficients, and since is symmetrical in and we require
/ (6.84)It will be observed that is now symmetrical in the indices and also, and as a consequence the term containing drops out of the expression for , giving
/ (6.85)where denotes the rate of expansion.
This expression for for a fluid of isotropic structure may be deduced from in another way which does not make explicit use of the identity for an isotropic tensor written in terms of delta tensors. Consider first a case of a fluid in pure rotation. It follows that reversal of the direction of leads to change of sign of all components of , which is impossible in an isotropic fluid because this operation is equivalent to keeping fixed and choosing a different orientation of the fluid; hence must have such a form that the term in in vanishes identically. It is taken for granted that a viscous stress cannot be generated by pure rotation, irrespective of the structure of the fluid, simply on the grounds that there is then no deformation of the fluid; however, rigorous justification for this belief is elusive. Then, for a pure straining motion, we can argue that, since the structure of the fluid does not distinguish any directions, the principal axes of must be determined by and must coincide with those of ; and thus there is only one possible linear relation between the tensor and satisfying this condition.
Finally, by definition makes no contribution to the mean normal stress, so that
/ (6.86)for all values of , implying that
/ (6.87)This result is known as the Stokes assumption, and is accurate for gases with no internal degrees of freedom.
On choosing as the one independent scalar constant, we obtain for the viscous stress tensor the expression
/ (6.88)the quantity within brackets is simply the non-isotropic part of the rate-of-strain tensor. This expression for was obtained by Saint-Venant (1843) and Stokes (1845) in essentially the above way, after having been derived by Navier (1822) and Poisson (1829) from specific assumptions concerning the molecular mechanism of internal friction. There is an analogous linear relation between stress and amount of strain for isotropic elastic solids.
It will be noticed that a spherically symmetrical straining motion, for which , is associated with no viscous stress. This is a simple consequence of the symmetry of the motion and of our definition of as the departure of the stress tensor from an isotropic form, This raises the question: are there any non-equilibrium effects in an isotropic expansion? The answer is that there may be, although they are only rarely of any importance, and that they are incorporated, in the quantity defined as the mean normal stress from all causes. The manner in which the departure from equilibrium represented by an isotropic expansion may affect the mean normal stress will be examined in the next section.
The significance of the parameter , which depends on the local state of the fluid, can be seen in the special case of a pure shearing motion. With as the one non-zero velocity derivative, all components of are zero except the tangential stresses
/ (6.89)Thus is the constant of proportionality between rate of shear and the tangential force per unit area when plane layers of fluid slide over each other, and termed the viscosity of the fluid. The fact that is the only scalar constant needed in the above general expression for is associated with the result of the Cauchy-Stokes theorem that a general relative motion near any point may be represented as the superposition of two simple shearing motions, each of which gives rise to a tangential stress determined by and the corresponding velocity gradient, together with a rigid rotation and an isotropic expansion, neither of which has any effect (in a fluid of isotropic structure) on the non-isotropic part of the stress tensor; and our last expression for the viscous stress tensor may of course be regarded as the only possible linear tensorial relation, involving one scalar parameter, between and a symmetrical, traceless tensor .
It is common experience that the force between layers of fluid in relative sliding motion is always a frictional force resisting the relative motion, corresponding to , as expected from the fact that molecular transfer of momentum resulting from random movement or arrangement of the molecules of the fluid tends to smooth out spatial variations of mean velocity irrespective of the mechanism of the transfer. The relation for the viscous stress shows that a positive value of also corresponds to principal stresses arising from of such signs as to resist the principal rates of strain; that is to say, a small material sphere being deformed into an ellipsoid exerts on the surrounding fluid a frictional force whose normal component is outward (inward) at places on the surface where the surface is moving inward (outward) relative to a sphere of the same volume as the ellipsoid.
Experiments on a variety of fluids and flow fields have shown that the above linear relation between the rate of strain and the non-isotropic part of the stress may hold over a remarkably wide range of values of the rate of strain. Observations of the flux of fluid volume along a circular tube of small radius with a maintained difference between the pressures at the two ends are particularly sensitive for this purpose. Although the exclusion of all but a linear term in the velocity gradient on the right-hand side of the relation for viscous stress has been proposed purely as a hypothesis likely to be accurate only for small magnitudes of the velocity gradient, it seems from observation that small magnitudes of the velocity gradient may include those values normally encountered in practice. For water and most gases, the linear law appears to be accurate under all except possibly the most extreme conditions, such as within a shock wave. Fluids for which the linear relation between the non-isotropic parts of the stress and rate-of-strain tensors does hold accurately are usually said to be Newtonian. For liquids of elaborate molecular structure, and in particular for those consisting of long molecular chains, and for some emulsions and mixtures, the expression for the viscous stress may cease to be accurate at only moderate rates of strain ; and for some rubber-like liquids the stress evidently depends on the strain history as well as on its instantaneous rate of change. Little is known about how the expression should be modified for such liquids.
The observation that a linear relation between viscous stress and rate of strain holds over a large range of values of the rate of strain for many fluids becomes understandable when the molecular mechanism of internal friction is considered. The bulk relative motion of the fluid can cause only a small change in the statistical properties of the molecular motion when the characteristic time of the bulk motion, i.e., the reciprocal of the rate of strain, is long compared with the characteristic time of the molecular motion (which in the case of a gas would be given by the average time between collisions). These are the circumstances in which a perturbation assumption of the kind used in obtaining the linear relation might be expected to be valid. For air at normal temperature and pressure the average time between collisions is about s; and for gases at least, it is evident that common practical values of the bulk rate of strain are indeed small in the sense used above. For liquids one cannot so readily estimate the relevant characteristic time of the molecular motion, but any time associated with the molecular movement is likely to be exceedingly small when measured against the reciprocal of a common value of the bulk rate of strain.
For air at normal temperature and pressure gcms and for water gcms. In neither of these cases does vary much with pressure, but for air increases with temperature at the rate of about 0.3% per kelvin rise in temperature and for water decreases at the rate of about 3% per cent per degree rise in the neighborhood of normal temperature. The viscosities of air and water under all ordinary conditions are thus exceedingly small when expressed in units which are practical for most other mechanical quantities, and it is natural to enquire if these common fluids may be regarded, for some purposes at least, as having zero viscosity, that is, as being inviscid.
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