BVPs from conservation principles

Let (p(r, t) be some time-varyingi.e., a function tiiat assigns to each position r and time t a unique value t). Common examples of fields in chemical engineering include 

The left-hand side is the rate of change of the total amount of within f2. The first term on the right-hand side is the net convective transport of across 9f2 into by the velocity field of the medium v(r, t ), n being the outward normal vector at the boundary. The second term is the net diffusive transport of across 9f2 into f2, with the flux vector /d often being related to the local field gradient by a constitutive equation of the form of Pick s law, 

The third term on the right-hand side is a source term, s(r, t, p) being the amount of generated per unit volmne per unit time at (r, t). Equation (6.2) is known as a macroscopic balance, as it applies to a control volume of finite size. A corresponding microscopic balance is obtained through use of the divergence theorem. 

As all integrals are over the same domain, we can combine them, 

This balance must hold for any arbitrary fixed control volmne, requiring the field to satisfy everywhere the partial differential equation 

---

Boundary value problems (BVPs) involve the solution of ODEs or partial differential equations (PDEs) on a spatial domain, subject to boundary conditions that hold on the domain boundary. Many problems from sobd and fluid mechanics, electromagnetics, and heat and mass transfer are expressed naturally as BVPs. The forms of these differential equations often resemble each other because they arise from similar conservation principles. Here the emphasis is upon BVPs that arise from problems in transport phenomena. 

---