Equation of Motion for Viscous Fluids
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1 2.25 Equation of Motion for Viscous Fluids Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 2001 (8th edition) Contents 1. Surface Stress …………………………………………………………. 2 2. The Stress Tensor ……………………………………………………… 3 3. Symmetry of the Stress Tensor …………………………………………8 4. Equation of Motion in terms of the Stress Tensor ………………………11 5. Stress Tensor for Newtonian Fluids …………………………………… 13 The shear stresses and ordinary viscosity …………………………. 14 The normal stresses ……………………………………………….. 15 General form of the stress tensor; the second viscosity …………… 20 6. The Navier-Stokes Equation …………………………………………… 25 7. Boundary Conditions ………………………………………………….. 26 Appendix A: Viscous Flow Equations in Cylindrical Coordinates ………… 28 ã Ain A. Sonin 2001 2 1 Surface Stress So far we have been dealing with quantities like density and velocity, which at a given instant have specific values at every point in the fluid or other continuously distributed material. The density (rv ,t) is a scalar field in the sense that it has a scalar value at every point, while the velocity v (rv ,t) is a vector field, since it has a direction as well as a magnitude at every point. Fig. 1: A surface element at a point in a continuum. The surface stress is a more complicated type of quantity. The reason for this is that one cannot talk of the stress at a point without first defining the particular surface through v that point on which the stress acts. A small fluid surface element centered at the point r is defined by its area A (the prefix indicates an infinitesimal quantity) and by its outward v v unit normal vector n . The stress exerted by the fluid on the side toward which n points on the surface element is defined as v F v = lim (1) A ® 0 A v where F is the force exerted on the surface by the fluid on that side (only one side is involved). In the limit A ® 0 the stress is independent of the magnitude of the area, but v will in general depend on the orientation of the surface element, which is specified by n . In other words, v = v (xv ,t,nv ) . (2) 3 v v The fact that depends on n as well as x, y, z and t appears at first sight to complicate matters considerably. One apparently has to deal with a quantity that depends on v six independent variables (x, y, z, t, and the two that specify the orientation n ) rather than v four. Fortunately, nature comes to our rescue. We find that because is a stress, it must v depend on n in a relatively simple way. We have seen that, in the absence of shear forces, Newton's law requires that the surface stress have the particularly simple form v = - pnv (no shear forces) (3) v where p, the magnitude of the normal compressive stress, is a function of r and t only. This is Pascal's principle, which states that in the absence of shear forces, at any point in the fluid, the stress is always normal to the surface on which it acts, and its magnitude is independent of the surface orientation. In the absence of shear stresses, therefore, the stress on any surface, anywhere in the fluid, can be expressed in terms of a single scalar field p(rv ,t) provided there are no shear forces. This gives rise to the relatively simple form of the equation of motion for inviscid flow. When shear forces are present, as they always are in practice except when the fluid is totally static in some reference frame, Newton's law imposes a somewhat more v v complicated constraint on the relationship between and n . We shall see that the stress on any surface anywhere in the fluid can in general be specified in terms of six scalar functions of x, y, z, and t. These six are the independent components of a quantity called the stress tensor. 2 The Stress Tensor The first and simplest thing that Newton's law implies about the surface stress is v that, at a given point, the stress on a surface element with an orientation n must be equal in magnitude, but opposite in direction, to that on a surface element with an opposite v orientation - n , that is, v (rv ,t,- nv ) = - v (rv ,t,nv ) (4) 4 v This result can be obtained by considering a thin, disc-shaped fluid particle at r , as shown in Fig. 2, with very small area A and thickness h. One side of the disc has an orientation v v n and the other - n . The equation of motion for this fluid particle reads Dv v h A = v (nv ) A + v (- nv ) A + h AG (5) Dt v where G is the body force per unit mass. When we let h approach zero, so that the two faces of the disc are brought toward coincidence in space, the inertial term on the left and the body force term on the right become arbitrarily small compared with the two surface force terms, and (4) follows immediately. Fig. 2: Illustration for equation (4) Newton's law also implies that the stress has a more profound attribute which leads to the concept of the stress tensor. The stress at a given point depends on the orientation of v the surface element on which it acts. Let us take as "reference stresses," at a given point r and instant t, the values of the stresses that are exerted on a surface oriented in the positive x-direction, a surface oriented in the positive y-direction, and a surface oriented in the positive z-direction (Fig. 3). We can write these three reference stresses, which of course are vectors, in terms of their components as, respectively, v v v v v (i ) = i + j + k xx yx zx v v v v v ( j ) = i + j + k (6) xy yy zy v v v v v (k ) = i + j + k xz yz zz 5 Figure 3: Reference stresses at a point in the continuum. Thus, xx , yx and zx represent the x, y, and z components of the stress acting on the surface whose outward normal is oriented in the positive x-direction, etc. (Fig. 3). The first subscript on ij identifies the direction of the stress, and the second indicates the outward normal of the surface on which it acts. In (6) the ij ' s are of course functions of position x, y, z, and time t, and the reference stresses themselves also depend on x, y, z, and t; we have simply not indicated this dependence. We shall now show, again by using Newton's law, that the stress on a surface v v having any orientation n at the point r can be expressed in terms of the reference stresses v v v v (i ) , v ( j ) , and v (k ) or, more specifically, in terms of their nine components xx , yx ,..., zz . Consider a fluid particle which at time t has the shape of a small tetrahedron v centered at x, y, z. One of its four faces has an area A and an arbitrary outward normal n , as shown in Fig. 4, and the other three faces have outward normals in the negative x, y and z directions, respectively. The areas of the three orthogonal faces are related to A by Ax = cos nx A = nx A Ay = cos ny A = ny A (7) Az = cos nz A = nz A 6 Fig. 4: Tetrahedron-shaped fluid particle at (x, y, z). where Ax represents the area of the surface whose outward normal is in the negative x- direction, is the angle between nv and the x-axis and n is the x-component of nv , and nx x so on. Consider what Newton's law tells us about the forces acting on the tetrahedron as v we let it shrink in size toward the point r around which it is centered. Since the ratio of the mass of the tetrahedron to the area of any one of its faces is proportional to the length of any one of the sides, both the mass times acceleration and the body force become arbitrarily small compared with the surface force as the tetrahedron is shrunk to a point (c.f. (5) and the paragraph that follows it). Hence, in the limit as the tetrahedron is shrunk to a point, the surface forces on the four faces must balance, that is, v v v (nv ) A + v ( j ) A + v (k ) A = 0 . (8) x z v Now we know from (4) that the stress on a surface pointing in the - i direction is the v negative of the stress on a surface in the + i direction, etc. Using this result and (7) for the areas, (8) becomes v v v v v v v v (n ) = (i )n + ( j )n + (k )n . (9) x y z Alternatively, if we use (6) to write the reference stresses in terms of their components, we obtain the components of v (nv ) as 7 (nv ) = n + n + n x xx x xy y xz z (nv ) = n + n + n (10) y yx x yy y yz z (nv ) = n + n + n . z zx x zy y zz z Thus the stress v (nv ) acting at xv ,t on a surface with any arbitrary orientation nv can be expressed in terms of the nine reference stress components xx xy xz yx yy yz xz zy zz . These nine quantities, each of which depends on position and time, are the stress tensor components.