ch3 - Chapter 3 Balance Laws In this chapter we explore the...

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Unformatted text preview: Chapter 3 Balance Laws In this chapter we explore the governing field equations: the conservation of mass and linear momentum, the balance of angular momentum, and the conservation of energy. Although we refer to these principles as ’laws’, they are postulated and no processes have been shown to violate them. We begin with the global balance law , which in words state that: the material time rate of change of the total amount of property φ , over a given body of material, is balanced by the sum of the fluxes, q , which cross the surface plus the sum of the body forces, f . Consider the body of material in Figure 3.1. We let V ( X ) be a volume of material at reference time t = 0. This may be an entire body of material, or a part of a larger body. The volume that this body takes at a later time is v ( x , t ). What the global balance law states is that the only way the total amount of φ can change is through an amount crossing its boundary (flux), or by a body force acting throughout the material. For each balance law we have different quantities representing { φ, q , f } . For the conservation of mass, the quantity being conserved is mass and φ is density; the flux is zero (the definition of the body includes all the material), and the body force is zero. For the balance of linear momentum, the quantity being conserved is linear momentum, the flux is force, and the body force is gravity. For the conservation of energy the quantity being conserved is energy, the flux includes heat flux, and the body force is a heat source due to e.g. radiation or an electric field. ) t , x v( n ) =V( X ) , x v( n Figure 3.1: Sample Body of Material Undergoing Deformation 65 66 CHAPTER 3. BALANCE LAWS Mathematically the Global Balance Law is: D Dt Z v ( t ) φ dv ( x ) =- Z S ( t ) q · d a ( x ) + Z v ( t ) f dv ( x ) , (3.1) where v ( x , t ) is the spatial location of the body at time t , S is the surface of the body, and d a = n da , the elemental area with direction normal to the surface (recall that n is a unit vector). Note that integration is defined using the Eulerian framework over the body (because the quantities φ and f are defined per unit of spacial volume and q · n is defined per unit spatial area), but the time derivative is material (keeping X fixed). Since the body may move, the spatial volume occupied by the material, v ( x , t ), may change in time. The material over which the global balance law is defined may consist of the entire body (solid lines of Figure 3.1) or may be just part of the body, for example the portion of the body enclosed by the dashed lines. The boundary however must enclose the same material throughout the deformation. Now consider the first term on the right-hand side, which represents the flux of a quantity across the boundary. Since n is defined to be the unit vector out, a positive surface integral means that there is a net flux out, and thus the total amount of φ is decreasing....
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ch3 - Chapter 3 Balance Laws In this chapter we explore the...

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