Solution

Problem 1.1: Show the following:

$$\frac{D}{Dt}(\rho \phi \Delta V) = \rho \Delta V \frac{D \phi}{Dt}$$

We start with the definition of the material derivative for a scalar field (eq 1.7 pg 5):

$$\frac{D \phi}{Dt} = \frac{\partial \phi}{\partial t} + \vec{v}\cdot\nabla \phi$$

Recall that this relation is really just the total derivative of $\phi$ with respect to time.

$$\frac{D \phi}{Dt} = \frac{d}{dt} \phi(\vec{x}(t),t) = \frac{\part... ...al z}\frac{dz}{dt} = \frac{\partial \phi}{\partial t} + \vec{v}\cdot\nabla \phi$$

We start with the left hand side and apply the definition of the material derivative and the conservation of mass ( $m = \rho \Delta V$, so $\frac{Dm}{Dt} = 0$):

$$\frac{D}{Dt} \rho \phi \Delta V = \frac{\partial}{\partial t}(\rho \phi\Delta V) + \vec{v}\cdot\nabla(\rho \phi \Delta V)$$

$$= \rho \Delta V \left(\frac{\partial \phi}{\partial t} + \vec{v}\... ...Delta V \left(\frac{\partial}{\partial t} \rho + \vec{v}\cdot\nabla \rho\right)$$

$$= \rho \Delta V \frac{D \phi}{Dt} + \phi \frac{\partial}{\partial t}(\rho \Delta V) + \phi \vec{v}\cdot\nabla(\rho \Delta V)$$

$$= \rho \Delta V \frac{D \phi}{Dt} + \phi \frac{D}{Dt}(\rho \Delta V)$$

$$= \rho \Delta V \frac{D \phi}{Dt}$$

Problem 1.2: Deduce that

$$\frac{D}{Dt} \int_{V} \rho \phi dV = \int_{V} \rho \frac{D \phi}{Dt} dV$$

We start by using Reynolds transport theorem. Then, Green's theorem is used to convert the surface integral to a volume integral. The next few steps are applications of the product rule. We collect terms of $\phi$ and $\rho$ and then apply the definition of the material derivative. Finally, we use the mass continuity equation (eq 1.33b pg 10) to simplify and obtain the result.

$$\frac{D}{Dt} \int_{V(t)} \rho \phi dV = \int_{V(t)} \frac{\partial}{\partial t}(\rho \phi) dV + \int_{\partial V(t)} \rho \phi \vec{v}\cdot \vec{dS}$$

$$= \int_{V(t)} \frac{\partial}{\partial t}(\rho \phi) dV + \int_{V(t)} \nabla \cdot (\rho \phi \vec{v}) dV$$

$$= \int_{V(t)} \frac{\partial}{\partial t}(\rho \phi) + \vec{v}\cdot\nabla(\rho \phi) + \rho \phi (\nabla \cdot \vec{v}) dV$$

$$= \int_{V(t)} \rho \frac{\partial \phi}{\partial t} + \phi \frac{... ...nabla \phi + \phi \vec{v}\cdot\nabla \rho + \rho \phi (\nabla \cdot \vec{v}) dV$$

$$= \int_{V(t)} \rho \left(\frac{\partial \phi}{\partial t} + \vec{... ...}{\partial t} + \vec{v}\cdot\nabla \rho + \rho (\nabla \cdot \vec{v})\right) dV$$

$$= \int_{V(t)} \rho \frac{D \phi}{Dt} + \phi \left(\frac{D \rho}{Dt} + \rho (\nabla \cdot \vec{v})\right) dV$$

$$= \int_{V(t)} \rho \frac{D \phi}{Dt} dV$$