Conservation of momentum

The conservation of momentum is another important concept of physics along with the conservation of mass. Momentum is defined to be the product of the mass of an object and its corresponding velocity. The conservation of momentum states that within some problem domain, the amount of momen-tum remains constant. In other words, momenmomen-tum is neither created nor destroyed, but only changed through the action of forces. In fact, the princi-ple of conservation of momentum is an application of Newton’s second law of motion to an object. In fluid mechanics, Newton’s second law of motion can be easily derived by an element of the fluid. That is, if we consider a given mass of fluid in a lagrangian framework, the rate of change of the momentum of the fluid mass is equal to the net external force acting on the mass. Some people prefer to think of forces only and restate this law in the form that the inertia force is equal to the net external force acting on the element.

Basically, the external forces acting on a mass of the fluid can be simply categorized into two classes. One is the class of body forces, such as gravita-tional or electromagnetic forces. The other is the class of surface forces, such as pressure forces or viscous stresses. Let f be a vector which represents the resultant of the body forces per unit mass, then the net external body force acting on a mass of volume V is R

V ρf dV . On the other hand, if a surface vector T represents the resultant surface force per unit area, then the net external surface force acting on the surface S containing V is R

STdS. Also, we assume that the mass per unit volume is ρ and its momentum is ρu, so that the momentum contained in the volume V is R

V ρudV . According to statements of Newton’s second law of motion, the rate of the change of momentum (or inertia force) is equal to the sum of the resultant forces. If the mass of the arbitrarily chosen volume V is observed in the lagrangian framework, the rate of change of momentum of the mass contained within V will be (D/Dt)R

V ρudV . Therefore, we can obtain a mathematical equation which arises from imposing the physical law of conservation of momentum in the form

In general, we can use a stress tensor [7] to represent surface forces acting on the fluid, and there are nine components of stress at any given point, one normal component and two shear components on each coordinate plane. These nine components of stress can be easily illustrated by the use of a cubical element in Fig. 2.1, then the stress components will act at a point as the length of the cube tends to zero. In Fig. 2.1 the cartesian coordinates

Figure 2.1: The diagram of nine components of stress.

(x, y, z) have been denoted by (x₁, x₂, x₃). This allows us to describe the components of stress as a double-subscript notation. In this notation, a particular component of the stress may be represented by the quantity σ_ij, in which the first subscript indicates that this stress component acts on the plane x_i = C, where C is a constant, and the second subscript indicates that it acts in the x_j-direction. The fact that the stress maybe represented by the quantity σ_ij, in which i and j may be 1, 2, or 3, means that the stress at a point may be represented by a tensor of rank 2. However, it was observed that there would be a vector force at each point on the surface of the control volume, and this force was represented by T. The surface force vector T may be related to the stress tensor σ_ij as follows: The three stress components acting on the plane x₁ = constant are σ₁₁, σ₁₂, and σ₁₃. Since the unit normal vector acting on this surface is n1, the resulting force acting in the x₁ direction is T₁ = σ₁₁n₁. Likewise, the forces acting in the x₂ direction and the x₃ direction are, respectively, T₂ = σ₁₂n₁ and T₃ = σ₁₃n₁. Then, for an arbitrarily oriented surface whose unit normal has components n1, n2, and n₃, the surface force will be given by T_j = σ_ijn_i in which i is summed from 1 to 3. That is, in tensor notation the equation expressing conservation of momentum becomes

D Dt

Z

V

ρu_jdV = Z

S

σ_ijn_idS + Z

V

ρf_jdV

Again, the left-hand side of this equation can be converted to a volume integral of only eulerian derivatives by using Eq. (2.1), meanwhile, the surface integral on the right-hand side can be changed to a volume integral by making use of Gauss’ theorem. In this way the equation which evolved from Newton’s second law becomes Collect these volume integrals to express this equation in the formR

V{}dV = 0, where the integrand is a differential equation in eulerian coordinates. As before, the arbitrariness of the control volume V implies that the integrand of the above integro-differential equation have to present the basic law of dynamics in an equivalent differential equations

∂

∂t(ρu_j) + ∂

∂x_k(ρu_ju_k) = ∂σ_ij

∂x_i + ρf_j,

If we consider ρu_ju_k as the product of ρu_k and u_j, and expand the left-hand side of the equation above, we obtain

ρ∂u_j

Note that the sum of the second and third terms on the left-hand side of this equation is zero due to the continuity equation (2.4). With this simplification, the expression of conservation of momentum becomes

ρ∂u_j

∂t + ρu_k∂u_j

∂x_k = ∂σ_ij

∂x_i + ρf_j (2.6)

It is useful to recall that this equation came from an application of Newton’s second law to an element of the fluid. The left-hand side of Eq. (2.6) rep-resents the rate of change of momentum of a unit volume of the fluid (or the inertia force per unit volume). The first term is the familiar temporal acceleration term, while the second term is a convective acceleration and ac-counts for local accelerations even when the flow is steady. Note also that this second term is nonlinear, since the velocity appears quadratically. On the right-hand side of Eq. (2.6) are the forces which are causing the acceler-ation. The first of these is due to the gradient of surface shear stresses and the second is due to body forces, such as gravity, which act on the mass of the fluid. A clear understanding of the physical significance of each of the terms in Eq. (2.6) is essential when approximations to the full governing equations must be made. In the following, we will list some assumptions and the surface-stress tensor σ_ij will be related to an expression of pressure and velocities.

1. When the fluid is at rest, the stress is hydrostatic and the pressure exerted by the fluid is the thermodynamic pressure. This implies that the stress tensor σ_ij is of the form

σ_ij = −pδ_ij + τ_ij (2.7)

where τ_ij depends on the motion of the fluid only and is called the shear-stress tensor. The quantity p is the thermodynamic pressure and δij is the Kronecker delta.

2. The stress tensor σij is linearly related to the deformation-rate tensor e_kl and depends only on that tensor. This is the distinguishing feature of newtonian fluids. There are nine elements in the shear-stress tensor τij, and each of these elements may be expressed as a linear combination of the nine elements in the deformation-rate tensor e_kl (just as a vector may be represented as a linear combination of components of the base vectors). That is, each of the nine elements of τij will in general be a linear combination of the nine elements of e_kl so that 81 parameters are needed to relate τ_ij to e_kl. This means that a tensor of rank 4 is required so that the general form of τij will be

τij = αijkl

∂u_k

∂x_l (2.8)

3. Since there is no shearing action in a solid body rotation of the fluid, no shear stresses will act during such a motion. Then

τ_ij = 1 4. There are no preferred directions in the fluid, so that the fluid properties are point functions. This condition is the so-called condition of isotropy, which guarantees that the results obtained should be independent of the orientation of the coordinate system chosen. The most general isotropic tensor of rank 4 is of the form, see appendix in [7],

λδ_ijδ_kl+ µ (δ_ikδ_jl+ δ_ilδ_jk) + γ (δ_ikδ_jl− δ_ilδ_jk) (2.10) Using the fact that δ_kl = 0 unless l = k, the expression fro the shear-stress tensor becomes Thus the constitutive relation for stress in a newtonian fluid becomes

σij = −pδij + λδij

2.2 A two-dimensional incompressible two-phase