# Integral of the substantial derivative in fluid mechanics?

I am studying the book Modern Compressible Flow by John Anderson. He writes

For steady, inviscid, adiabatic, compressible flow with no body forces, $$\rho \frac{Dh_o}{Dt}=0$$ where $$h_o$$ is the total enthalpy of the flow at a point. When integrated, this yields $$h_0=\text{const}$$ along a streamline.

The substantial derivative is defined $$\frac{D}{Dt}=\frac{\partial }{\partial t} + u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}$$

My question is, how can I show that if $$\frac{Df}{Dt}=0$$ for a function $$f(x,y,z,t)$$, if the flow is steady, adiabatic, and inviscid, then $$f=\text{const}$$ along a streamline?

The substantial derivative (also known as the material derivative) is the rate of change of a variable $$f$$ in a fluid parcel or element. The variable $$f$$ is specified in the Eulerian way, as a function of time $$t$$ and current location $$\mathbf{x}$$ of the fluid parcel. Then the total derivative is $$\mathrm{d}f\left(\mathrm{x}\left(t\right), t\right) = \frac{\partial f\left(\mathbf{x}\left(t\right), t\right)}{\partial t} \mathrm{d}t + \frac{\partial f\left(\mathbf{x}\left(t\right), t\right)}{\partial \mathbf{x}\left(t\right)} \frac{\partial \mathbf{x}\left(t\right)}{\partial t}\mathrm{d}t$$ where $$\partial \mathbf{x}\left(t\right)/\partial t = \mathbf{u}\left(t\right)$$ is just the flow velocity. Thus the material derivative is defined as $$\frac{\mathrm{D}f\left(\mathrm{x}\left(t\right), t\right)}{\mathrm{D}t} \equiv \frac{\partial f\left(\mathbf{x}\left(t\right), t\right)}{\partial t} + \mathrm{grad}\left( f\left(\mathrm{x}\left(t\right), t\right)\right) \cdot \mathbf{u}\left(t\right).$$ If the material derivative, that is the total rate of change of the variable $$f$$, is zero, then the value of $$f$$ must be constant along its streamline. This is by definition. Or you could write it as $$\frac{\mathrm{D}f}{\mathrm{D}t} =\frac{\partial f}{\partial t} + \mathrm{grad}\left( f\right) \cdot \mathbf{u}=0$$ thus $$\mathrm{d}f = \left[\frac{\partial f}{\partial t} + \mathrm{grad}\left( f\right) \cdot \mathbf{u} \right]\mathrm{d}t = 0 \, \mathrm{d}t$$ and $$f = \int \mathrm{d}f=\int 0 \, \mathrm{d}t = 0 + C$$ where $$C$$ is just the integration constant, i.e. the value of $$f$$ at the beginning of the streamline.
By the way: The conditions steady, adiabatic, and inviscid flow are not needed for this result. A variable $$f$$ does not change if $$\frac{\mathrm{D}f}{\mathrm{D}t}=0$$, but $$\frac{\partial f}{\partial t} \neq 0$$ and thus $$\frac{\partial f}{\partial t} = - \mathrm{grad}\left( f\right) \cdot \mathbf{u}$$ is still possible!