Linear Momentum Equation

If we take linear momentum as parameters N which is used as a general term (vector and scalar) in the Reynold transport equation, we have N = P = mv. The term n in this case would become momentum per unit mass, or simply v, then, P = JJJ v(pdV). Then the Reynold transport equation can be written as followings:

cs cv

The equation of linear momentum is a vector equation and can be divided into its three components.

Newton's second law states that:

In above equation, F if the resultant of the all external forces acting on the system and v the time derivative are taken from the inertial references. Since, the second law of Newton is based on the Eulerian viewpoint, we are following a system. Thus, it can be written as:

There are two types of the external forces. Surface forces, T(x, y, z, t) given as forces per unit area on the boundary surface, bs. Force acting on the material inside the boundary, ib are called body force, B(x,y,z,t) given as force per unit mass. Thus, The above equation can be written as followings:

bs ib

Equation 2.8 is the Newton's second law for a finite system. Knowing that the control volume and system are coincident at time t, if we fix the control volume in inertial space, then the derivatives in the right hand side is taken from a inertial references and we may use the Newton's second law (equation 2.8) to replace it in the Reynold equation and then we get:

cv cs cv

Equation 2.9 means that: Sum of surface forces acting on the control surface and body forces acting on the control volume are equal with the sum of the rate of the efflux of the linear momentum across the control surface and the rate of increase of linear momentum inside the control volume.

Equation 9 is a vector equation and can be written for each components. For example in x direction we can write it as follows:

TxdA +111 BxpdV = 11 vx(pv.d A)+dt II I vxpdV (2.10)

In this book for simplicity, the positive direction of the reference axis xyz are the same as the positive direction of the velocity components as well as the surface and body force components. Sign of v.dA is independent of the sign consideration.

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