1 The total inflow rate of momentum into the control volume is thus MOMENTUM CONSERVATION: CAUCHY EQUATION Consider the illustrated control volume, which is fixed in space and through which momentum can freely flow in and out. dA nini dV The convective flux in the x j direction of momentum in the i direction is given as  u i u j. The control volume has outward normal n j. The net inflow velocity into the control volume is thus = - u j n j, and the net discharge of momentum dQ mom,inflow,i into the volume across elemental surface area dA is

2 There are two types of forces that can operate on a continuous mass: body forces, which act throughout the body of a control volume surface forces which act on the surface of any control volume The body force considered in this course is gravity. Where g i denotes the vector of gravitational acceleration, the gravitational force F gi acting on the mass in a control volume fixed in space is given as If x 3 is upward vertical than g i = - g  i3. Where m is the mass in the control volume, then, The surface forces act through the stress tensor. MOMENTUM CONSERVATION: CAUCHY EQUATION A body force corresponds to a net source of momentum within a control volume. As will be shown later, a surface force corresponds to a net outflow of momentum across the surface of a control volume. dA nini dV

3 MOMENTUM CONSERVATION: CAUCHY EQUATION Again, the control volume is fixed in space, and the fluid is allowed to freely flow in and out. The surface force dF Si acting on an elemental surface area dA the control volume is given as dA nini dV Remember the convention First Face Second Stress in the stress tensor  ji. Thus the index j refers to the face and the index i refers to the stress (surface force per unit area). The outward normal vector n j makes sure that the sign is correctly accounted for. The total surface force is thus given as

4 MOMENTUM CONSERVATION: CAUCHY EQUATION To see how the sign convention using n j works, we consider the elemental cubical control volume illustrated below. Consider only the component F S1. x1x1 x2x2 x3x3 x1x1 x3x3 x2x2

5 MOMENTUM CONSERVATION: CAUCHY EQUATION On the indicated face at x 1, n j = (-1, 0, 0). x1x1 x2x2 x3x3 x1x1 x3x3 x2x2 njnj Thus the contribution from this face is:

6 MOMENTUM CONSERVATION: CAUCHY EQUATION On the indicated face at x 1 +  x 1, n j = (1, 0, 0). x1x1 x2x2 x3x3 x1x1 x3x3 x2x2 njnj Thus the contribution from this face is:

7 MOMENTUM CONSERVATION: CAUCHY EQUATION On the indicated face at x 2 n j = (0, -1, 0). x1x1 x2x2 x3x3 x1x1 x3x3 x2x2 njnj Thus the contribution from this face is:

8 MOMENTUM CONSERVATION: CAUCHY EQUATION On the indicated face at x 2 +  x 2 n j = (0, 1, 0). x1x1 x2x2 x3x3 x1x1 x3x3 x2x2 njnj Thus the contribution from this face is:

9 MOMENTUM CONSERVATION: CAUCHY EQUATION On the indicated face at x 3 n j = (0, 0, -1). x1x1 x2x2 x3x3 x1x1 x3x3 x2x2 njnj Thus the contribution from this face is:

10 MOMENTUM CONSERVATION: CAUCHY EQUATION On the indicated face at x 3 +  x 3 n j = (0, 0, 1). x1x1 x2x2 x3x3 x1x1 x3x3 x2x2 njnj Thus the contribution from this face is:

11 MOMENTUM CONSERVATION: CAUCHY EQUATION Thus over the entire surface of the control volume, x1x1 x2x2 x3x3 x1x1 x3x3 x2x2 Note how this accounts correctly for the signs on every face.

12 MOMENTUM CONSERVATION: CAUCHY EQUATION dA nini dV Again consider the control volume, which is fixed in space and through which fluid can freely flow in and out. In words, conservation of momentum can be stated as:  /  t(momentum in control volume) = net inflow rate of momentum + surface force + body (gravitational) force or Using the divergence theorem,

13 MOMENTUM CONSERVATION: CAUCHY EQUATION dA nini dV Reducing with results in or thus This relation is known as the Cauchy equation. It is generally valid for any fluid. Further progress is predicated on a specification of the stress tensor  ij.