Material Derivative/Lie derivative clarification from existing answer

Question

I am also reading the same book as the OP in this question - Material derivative of a material vector field namely - An Introduction To Theoretical Fluid Dynamics

and I have a couple of questions on the methodology of the derivation

Let me start with the first set of equations (1). I understand $$\dfrac{\mathbf{Dv}}{\mathbf{D}t} = \dfrac{\text{d}\mathbf{v}(\mathbf{x}(t),t)}{\text{d}t}= \dfrac{\text{d}\mathbf{v}(\mathbf{a},t)}{\text{d}t}\tag{1}$$

That is just the total derivative or the material derivative.

How is $$\dfrac{\text{d}\mathbf{v}(\mathbf{a},t)}{\text{d}t}=\dfrac{\partial\mathbf{v}(\mathbf{a},t)}{\partial t}$$

This would mean that the advection term of the material derivative is zero or is there something I am missing ?

I do not really understand how if $v$ is a material vector field it has to satisfy (2) because according to me the only way this would be true if the material derivative or the total derivative is zero.

$$\dfrac{\partial\mathbf{v}(\mathbf{a},t)}{\partial t} = \mathbf{v}\cdot\nabla \mathbf{u}\tag{2}$$

Answer 1

Notations and Context

• $\mathbf a$ — a vector, that is...

  1. selected as the initial displacement (location) of a particle. (p. 4)
  2. just a displacement, which is not chosen to be the initial displacement of any specific particle on purpose.

• $\mathbf x$ —

  1. the vector function $\mathcal X(\mathbf a,t)$, also the (Lagrangian) displacement of particle at time $t$.
     (will act as a variable, p. 4)
  2. a fixed displacement through which a stream of particles will flow past.
     (will act as a constant, p. 4)
• $\mathbf u$ — the vector field $\mathbf u(\mathbf x,t)$, flow velocity at a fixed displacement $\mathbf x$ at time $t$. (p. 4)
• $\mathbf J$ — $\left.\dfrac{\partial x_i}{\partial a_j}\right|_{t}=\mathbf J(t) $, the Jacobian of the Lagrangian map $\mathcal M_t:\mathbf a\mapsto \mathcal X(\mathbf a,t)$ evaluated at time $t$. (Note: $\mathbf J(0)=\mathbf I$, p. 7)
• $\mathbf V$ — the vector field $\mathbf V(\mathbf a)$, the initial material vector field at time $t=0$.
• $\mathbf v$ — $(\mathbf J)_{ij}(\mathbf V(\mathbf a))_j$, the material vector field at time $t$, which is determined by the Jacobian matrix of the flow. (p. 12)
• $\dfrac{\mathbf D}{\mathbf D\mathbf t}$ — material derivative in the Eulerian perspective. (p. 9)

The Confusion

\begin{align} \dfrac{\mathbf{Dv}}{\mathbf Dt} &= \dfrac{\mathbb d\mathbf v(\mathbf x(t),t)}{\mathbb dt} = \dfrac{\color{blue}{\partial}\mathbf v(\color{blue}{\mathbf x(t)},t)}{\color{blue}{\partial x_i}}\dfrac{\mathbb dx_i}{\mathbb dt} + \dfrac{\color{green}{\partial}\mathbf v(\mathbf x(t),\color{green}{t})}{\color{green}{\partial t}} = \mathbf u\cdot\color{blue}{\nabla}\mathbf v + \dfrac{\color{green}{\partial}\mathbf v}{\color{green}{\partial t}}\tag{3}\\ &= \dfrac{\mathbb d\mathbf v(\mathbf a,t)}{\mathbb dt} = \dfrac{\partial\mathbf v(\mathbf a,t)}{\partial t} \color{red}{\ne} \dfrac{\partial\mathbf v(\mathbf x(t),t)}{\partial t} = \dfrac{\partial\mathbf v}{\partial t}. \tag{4} \end{align}

So, what equation $(1)$ means is that the two perspectives below are equivalent in describing a material vector field:

• Line $(3)$: A particle starts somewhere and reaches $\mathbf a$ at time $t$. Let's study the particle at time $t$.
• Line $(4)$: Let's study all the particles that goes through a fixed displacement $\mathbf a$, and focus on the one particle that passes exactly at time $t$.

Now, let's see why $\dfrac{\text{d}\mathbf{v}(\mathbf{a},t)}{\text{d}t}=\dfrac{\partial\mathbf{v}(\mathbf{a},t)}{\partial t}$.

In the situation where the spatial dimension is $n$, We view $\mathbf v(\mathbf x,t)$ as a vector function of $n+1$ variables. Therefore, $\mathbf v:\mathbb R^{n+1}\to\mathbb R^n$ which maps $(\mathbf x,t)\mapsto\text{some vector}$. Therefore, $\mathbf v(\mathbf a,t)$ will fix the first $n$ variables. Imagine if $n=3$, then we have $\mathbf v(x,y,z,t)$. Then imagine if the displacement is fixed by a constant $\mathbf a=(a,b,c)$ to get $\mathbf v(a,b,c,t)$.

Then, what is $\dfrac{\mathbb d\mathbf v(a,b,c,t)}{\mathbb dt}$? This total derivative should be $$ \dfrac{\mathbb da}{\mathbb dt}\dfrac{\partial\mathbf v(a,b,c,t)}{\partial x} + \dfrac{\mathbb db}{\mathbb dt}\dfrac{\partial\mathbf v(a,b,c,t)}{\partial y} + \dfrac{\mathbb dc}{\mathbb dt}\dfrac{\partial\mathbf v(a,b,c,t)}{\partial z} + \dfrac{\mathbb dt}{\mathbb dt}\dfrac{\partial\mathbf v(a,b,c,t)}{\partial t}, \tag{5} $$ and you would recognize that $\dfrac{\mathbb da}{\mathbb dt}=\dfrac{\mathbb db}{\mathbb dt}=\dfrac{\mathbb dc}{\mathbb dt}=0$. On the other hand, see why this inequality is true: $$\dfrac{\mathbb d\mathbf v(a,b,c,t)}{\mathbb dt} \ne \dfrac{\mathbb d\mathbf v(x,y,z,t)}{\mathbb dt}.\tag{6}$$

It's because $\mathbf x=(x(t),y(t),z(t))$ and all $x$, $y$ and $z$ are dependent on $t$, unlike $\mathbf a=(a, b, c)$.

This observation generalizes to all spatial dimensions.

About Equation $(2)$

The process of deriving Equation $(2)$ is on page 12 of the book. Related equations are $(1.21)$ and $(1.22)$ which starts off from the partial derivative of the definition of the material vector field $$\mathbf v = (\mathbf J)_{ij}(\mathbf V(\mathbf a))_j,$$ which will eventually lead to $$\dfrac{\partial\mathbf v(\mathbf a,t)}{\partial t} = \mathbf v\cdot\nabla\mathbf u.$$

A few relations being used here are:

1. $\dfrac{\partial(\mathbf J)_{ij}}{\partial t} = \dfrac{\partial}{\partial t}\dfrac{\partial\mathbf x_i}{\partial\mathbf a_j} = \dfrac{\partial\mathbf u_i}{\partial\mathbf a_j} = \dfrac{\partial\mathbf u_i}{\partial\mathbf x_k} \dfrac{\partial\mathbf x_k}{\partial\mathbf a_j}$,

2. $\dfrac{\partial\mathbf x_k}{\partial\mathbf a_j} \mathbf V_j = \mathbf v_k$.

See if you can bridge these gaps, comment to let me know if this helps or not.