Question

Let an arbitrary region in a continuous medium be denoted by R and the bounding, closed surface of this region be continuous and denoted by S. Let each point on the bounding surface move with the velocity vs​. It can be shown that the time derivative of the volume intcgral over some continuous function Q(r,t) is given by dtd​∫R​Q(r,t)dV≡∫R​∂t∂Q​dV+∮S​Qvs​⋅n^dS This expression for the differentiation of a volume integral with variablc limits is sometimes known as the three-dimensional Leibniz rule. Let each element of mass in the medium move with the velocity v(r,t) and consider a special region R such that the bounding surface S is attached to a fixed set of matcrial elements. Then cach point of this surface moves itself with the matcrial velocity, that is, vs​=v, and the region R thus contains a fixed total amount of mass, since no mass crosses the boundary surface S. To distinguish the time rate of change of an integral over this material region, we replace d/dt by D/Dt and write DtD​∫R​Q(r,t)dV≡∫R​∂t∂Q​dV+∮S​Qv⋅n^dS which holds for a material region, that is, a region of fixed total mass. Show that the relation between the time derivative following an arbitiary region and the time derivative following a material rcgion (fixed total mass) is dtd​∫R​Q(r,t)dV≡DtD​∫R​Q(r,t)dV+∮S​Q(vs​−v)⋅n^dS The velocity difference v−vs​ is the velocity of the material measured relative to the velocity of the surface. The surface integral ∮S​Q(v−vs​)⋅n^dS thus measures the total outflow of the property Q from the region R.

Answer 1

The relation between the time derivatives following an arbitrary region and a material region with a fixed total mass is given by DtD​∫R​Q(r,t)dV≡DtD​∫R​Q(r,t)dV+∮S​Q(vs​−v)⋅n^dS. It measures the total outflow of the property Q from the region R.

Step-by-step explanation:

The relation between the time derivative following an arbitrary region and the time derivative following a material region with a fixed total mass is given by:

dtd∫R​Q(r,t)dV ≡ DtD∫R​Q(r,t)dV + ∮S​Q(vs​−v)⋅n^dS

Here, Q represents a continuous function over the region R and S is its bounding surface. The term Dt/D represents the time rate of change of an integral over a material region.