Differential Equations of Momentum Transfer or Motion

The equation of motion is really the equation for the conservation-of-momentum equation (2.8-3), which we can write as 

We will make a balance on an element as in Fig. 3.6-2. First we shall consider only the x component of each term in Eq. (3.6-30). The y and z components can be described in an analogous manner. 

Momentum flows in and out of the volume element by the mechanisms of convection or bulk flow as given in Eq. (3.7-2) and also by molecular transfer (by virtue of the velocity gradients in laminar flow). The rate at which the x component of momentum enters the face at x by molecular transfer is Txx)x Ay Az, and the rate at which it leaves the surface at x - - Ax is tyy) +Ax Ay Az. The rate at which it enters the face at y is (lyjy Ax Az, and it leaves at y-I-Ay at a rate of Tyy)y+ y Ax Az. Note that Tyy is the flux of x momentum through the face perf endicular to the y axis. Writing a similar equation for the remaining faces the net x component of momentum by molecular transfer is 

These molecular fluxes of momentum may be considered as shear stresses and normal stresses. Hence, is the x direction shear stress on the y face and the shear stress on the z face. Also, is the normal stress on the x face. 

The net fluid pressure force acting on the element in the x direction is the difference between the force acting at x and x -I- Ax. 

---

DIFFERENTIAL EQUATIONS OF MOMENTUM TRANSFER OR MOTION 3.7A Derivation of Equations of Momentum Transfer... 

We should note that Eqs. (3.7-7) to (3.7-13) are valid for any continuous medium. Sec, 3.7 Differential Equations of Momentum Transfer or Motion 171... 

---