We next note that it can be proven from basic physical considerations that the

We next note that it can be proven from basic

This preview shows page 80 - 82 out of 164 pages.

We next note that it can be proven from basic physical considerations that the various compo- nents of the shear stress shown in Fig. 3.9 are related as follows: τ xy = τ yx , τ xz = τ zx , τ yz = τ zy . (3.36) The interested reader is referred to more advanced treatments of fluid dynamics for such a proof; here we will provide an heuristic mathematical argument. In particular, if we consider the two shear stress components described above and imagine shrinking the fluid element depicted in Fig. 3.9 to a very small size we would see that in order to avoid discontinuities (in the mathematical sense) of τ along the edge of the cube between the x and y faces it would be necessary to require
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3.4. MOMENTUM BALANCE—THE NAVIER–STOKES EQUATIONS 75 τ xy = τ yx , and similarly for the other components listed in Eq. (3.36) along the various other edges. But, in fact, this cubic representation of a fluid element is just pictorial and should be viewed as a local projection of a more general, complicated shape. Hence, we can argue that there are such “edges” everywhere on an actual fluid element, and as we allow the size of such an element to become arbitrarily small, we must have the equalities shown in Eq. (3.36) over the entire surface of the fluid element. We next need to consider the consequences of this. We see that on an x face we have τ xy μ ∂v ∂x , while on the y face that adjoins this we have τ yx μ ∂u ∂y . But these two stresses must actually be equal, as noted above. It is completely unreasonable to expect that ∂u ∂y = ∂v ∂x in general, for this would imply an irrotational flow (recall Eqs. (2.16)), and most flows are not irrotational. The simplest way around this difficulty is to define the shear stresses acting on the surface of a 3-D fluid element as follows: τ xy = μ parenleftbigg ∂u ∂y + ∂v ∂x parenrightbigg = τ yx , (3.37a) τ xz = μ parenleftbigg ∂u ∂z + ∂w ∂x parenrightbigg = τ zx , (3.37b) τ yz = μ parenleftbigg ∂v ∂z + ∂w ∂y parenrightbigg = τ zy . (3.37c) We will often use the short-hand notations employed earlier for partial derivatives to express these as, τ xy = μ ( u y + v x ) , τ xz = μ ( u z + w x ) , τ yz = μ ( v z + w y ) . These provide the generalizations of Newton’s law of viscosity alluded to earlier. We can also provide a simple, physical argument for the form of these stresses. To understand the physics of the multi-dimensional shear stresses we again appeal to Newton’s law of viscosity by recalling that it relates shear stress to rate of angular deformation through the viscosity. Thus, we need to seek the sources of angular deformation for each face of our fluid element. We will specifically consider only one of the x faces, but the argument we use will apply to any of the faces. Figure 3.10 provides a schematic of the deformation induced by x - and y -direction motions caused by fluid elements moving past this face, viewed edgewise, such that both contribute to τ xy . In particular, in the left figure we see that changes of the v component of velocity in the x direction (caused by y -direction motion) will tend to distort the fluid element by moving the x face in a generally counter-clockwise direction. This change of
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