Momentum equation in three dimensions Newton’s second law states that the rate of change of momentum of a fluid

2.1.3 Momentum equation in three dimensions Newton’s second law states that the rate of change of momentum of a fluid

particle equals the sum of the forces on the particle:

Rate of increase of

Sum of forces

momentum of

= on

fluid particle

fluid particle

The rates of increase of x-, y- and z-momentum per unit volume of a fluid particle are given by

We distinguish two types of forces on fluid particles: • surface forces

– pressure forces – viscous forces – gravity force

• body forces – centrifugal force – Coriolis force – electromagnetic force

It is common practice to highlight the contributions due to the surface forces as separate terms in the momentum equation and to include the effects of body forces as source terms.

The state of stress of a fluid element is defined in terms of the pressure and the nine viscous stress components shown in Figure 2.3. The pressure,

a normal stress, is denoted by p. Viscous stresses are denoted by τ. The usual suffix notation τ ij is applied to indicate the direction of the viscous stresses. The suffices i and j in τ ij indicate that the stress component acts in the j- direction on a surface normal to the i-direction.

Figure 2.3 Stress components on three faces of fluid element

First we consider the x-components of the forces due to pressure p and stress components τ xx , τ yx and τ zx shown in Figure 2.4. The magnitude of a

2.1 GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 15

force resulting from a surface stress is the product of stress and area. Forces aligned with the direction of a co-ordinate axis get a positive sign and those in the opposite direction a negative sign. The net force in the x-direction is

the sum of the force components acting in that direction on the fluid element.

Figure 2.4 Stress components in the x-direction

On the pair of faces (E, W ) we have

δx E K δyδz + − p + H B δx E

∂x 2 E K δyδz = − B +

δxδyδz (2.12a)

F The net force in the x-direction on the pair of faces (N, S ) is

δy δxδz + τ yx

+ yx

δy δxδz = δxδyδz

F ∂y (2.12b)

Finally the net force in the x-direction on faces T and B is given by

A ∂τ zx 1 D A ∂τ zx 1 D ∂τ zx

− B τ zx −

δz δxδy + τ E B zx +

δz δxδy = δxδyδz

C ∂z 2 F C ∂z 2

F ∂z (2.12c)

The total force per unit volume on the fluid due to these surface stresses is equal to the sum of (2.12a), (2.12b) and (2.12c) divided by the volume δxδyδz:

∂(−p + τ xx ) ∂τ yx ∂τ zx

Without considering the body forces in further detail their overall effect can be included by defining a source S Mx of x-momentum per unit volume per unit time.

The x-component of the momentum equation is found by setting the rate of change of x-momentum of the fluid particle (2.11) equal to the

16 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

total force in the x-direction on the element due to surface stresses (2.13) plus the rate of increase of x-momentum due to sources:

Du ∂(−p + τ xx ) ∂τ yx ∂τ ρ

(2.14a) Dt

It is not too difficult to verify that the y-component of the momentum equation is given by

Dv ∂τ xy ∂(−p + τ yy ) ∂τ ρ

(2.14b) Dt

and the z-component of the momentum equation by Dw ∂τ

∂τ yz ∂(−p + τ ) ρ

(2.14c) Dt

The sign associated with the pressure is opposite to that associated with the normal viscous stress, because the usual sign convention takes a tensile stress to be the positive normal stress so that the pressure, which is by definition a compressive normal stress, has a minus sign.

The effects of surface stresses are accounted for explicitly; the source terms S Mx ,S My and S Mz in (2.14a–c) include contributions due to body forces only. For example, the body force due to gravity would be modelled by S Mx = 0, S My = 0 and S Mz =− ρg.