Understanding material continuity I am prepping for an exam in large scale fluid mechanics, and I struggle to understand what material conservation really means. In standard mechanics conservation would mean the variable in question does not change, the derivative of it is zero, but what does it mean that a variables material derivative is zero, conseptually?
I am using the Vallis book, and the first reference of material conservation states that is somewhat similar to "adiabatically conserved".
Since the material derivative includes both the change in time and the spatial coordinates (I think, I do not really follow the wikipedia explaination of it. That is that the material derivative is the change in time plus the covariant derivative of the tensor (https://en.wikipedia.org/wiki/Material_derivative)), could I simply say that the variable is constant in both time and space?
Thank you in advance!
 A: Meaning of material derivative
Material derivative represent the variation in time of a quantity, as seen by material particles. Imagine you label a fluid particle with a label $\mathbf{x_0}$ (experimentally you can do it with a tracer), and imagine you have an instrument mounted on that particle that can measure all the physical quantities, $f^0(\mathbf{x_0}; t)$; thus, material derivative is the time derivative of the function $f^0(\mathbf{x_0}; t)$.
You can link material derivative and partial derivatives via derivatives of composite functions. Particle labelled with $\mathbf{x_0}$ moves on the trajectory $\mathbf{x}(\mathbf{x_0}, t)$ and thus the Eulerian representation (using physical space coordinate $\mathbf{x}$ and time) can be related with the Lagrangian representation as follows
$f(\mathbf{x}, t) = f(\mathbf{x}(\mathbf{x_0};t), t) =: f^0(\mathbf{x_0},t)$.
Now, material derivatives means taking time derivative following a labelled particle, and thus taking time derivative at constant $\mathbf{x_0}$
$\dfrac{D f}{Dt} = \dfrac{\partial }{\partial t}\bigg|_{\mathbf{x_0}} f(\mathbf{x}(\mathbf{x_0},t),t) = \dfrac{\partial \mathbf{x}}{\partial t} \bigg|_{\mathbf{x_0}} \cdot \dfrac{\partial f}{\partial \mathbf{x}}\bigg|_{t} + \dfrac{\partial f}{\partial t}\bigg|_{\mathbf{x}} = \mathbf{u} \cdot \nabla f + \dfrac{\partial f}{\partial t}$.
Balance equations in differential convective form
For balance equation in integral and differential forms, in Eulerian, Lagrangian and arbitrary description, try to have a look here: https://basics.altervista.org/test/Physics/CM/BalanceEquations/main.html and the links therein.
Only few words here. Principles of physics can be written in differential balance equations in convective form as

*

*mass: $\dfrac{D \rho}{D t} = -\rho \nabla \cdot \mathbf{u}$

*momentum: $\rho\dfrac{D \mathbf{u}}{D t} = \rho \mathbf{g} + \nabla \cdot \mathbb{T}$

*total energy: $\rho\dfrac{D e^{tot}}{Dt} = \rho \mathbf{g}\cdot \mathbf{u} + \nabla \cdot (\mathbb{T} \cdot \mathbf{u}) - \nabla \cdot \mathbf{q}$
Interpretation of the equations.
Mass equation can be easily interpreted as the mass conservation for a material particle, meant as a "small" material volume: being a closed volume, it exchanges no mass with the external environment and thus its mass is constant. Being the mass density times product, $\Delta m = \rho \Delta V$, and taking the material derivative
$0 = \dfrac{D \Delta m}{D t} = \dfrac{D \rho}{D t} \Delta V + \rho \dfrac{D \Delta V}{D t}$,
and recalling the meaning of the divergence of the velocity field in terms of velocity of volume expansion, namely
$\frac{D \Delta V}{Dt} = \nabla \cdot \mathbf{u} \Delta V$, and thus
$0 = \dfrac{D \Delta m}{D t} = \Delta V \left( \dfrac{D \rho}{D t} + \rho \nabla \cdot \mathbf{u} \right)$,
Old answer: it's "conservation" not "continuity".
If I get your question right, I guess you need to study much harder...
Continuity means that it's possible to describe the physical quantities of a medium as continuous fields, i.e. continuous functions of the space and time, neglecting the microscopic molecular discrete description. As an example, in fluid dynamics using a Eulerian description, it's possible to represent velocity, pressure as $\mathbf{u}(\mathbf{r},t)$, $p(\mathbf{r}, t)$, $\dots$
