The Material Derivative in Cylindrical Coodrinates
This one a little bit more involved than the Cartesian derivation. The reason for this is that the unit vectors in cylindrical coordinates change direction when the particle is moving.
In the Lagrangian reference, the velocity is only a function of time. When we switch to the Eulerian reference, the velocity becomes a function of position, which, implicitly, is a function of time as well as viewed from the Eulerian reference. Then
(1)
\begin{align} \mathbf{V}\left(r,\theta,z,t\right) = u\left(r,\theta,z,t\right)\mathbf{e}_r + v\left(r,\theta,z,t\right)\mathbf{e}_\theta + w\left(r,\theta,z,t\right)\mathbf{e}_z \end{align}
and the material derivative is written as (with the capital D symbol to distinguish it from the total and partial derivatives)
(2)
\begin{align} \frac{\text{D}\mathbf{V}}{\text{D}t}=\frac{\partial \mathbf{V}}{\partial t}+\frac{\partial \mathbf{V}}{\partial r}\frac{\text{d}r}{\text{d}t}+\frac{\partial \mathbf{V}}{\partial \theta }\frac{\text{d}\theta }{\text{d}t}+\frac{\partial \mathbf{V}}{\partial z}\frac{\text{d}z}{\text{d}t} \end{align}
or, by making use of the definition of velocities ($u \equiv \frac{{\rm d}r}{{\rm}t}$, $\frac{v}{r} \equiv \frac{{\rm d}\theta}{{\rm d}t}$, and $w \equiv \frac{{\rm d} z}{{\rm d} t}$), we have
(3)
\begin{align} \frac{\text{D}\mathbf{V}}{\text{D}t}= \frac{\partial \mathbf{V}}{\partial t} +u \frac{\partial \mathbf{V}}{\partial r} +\frac{v}{r} \frac{\partial \mathbf{V}}{\partial \theta } +w \frac{\partial \mathbf{V}}{\partial z} \end{align}
Special attention must be made in evaluating the time derivative in Eq. 2. In dynamics, when differentiating the velocity vector in cylindrical coordinates, the unit vectors must also be differentiated with respect to time. In this case, the partial derivative is computed at a fixed position and therefore, the unit vectors are "fixed" in time and their time derivatives are identically zero. Then, we have
(4)
\begin{align} \frac{\partial \mathbf{V}}{\partial t}=\frac{\partial u}{\partial t}\mathbf{e}_{r}+\frac{\partial v}{\partial t}\mathbf{e}_{\theta }+\frac{\partial w}{\partial t}\mathbf{e}_{z} \end{align}
(5)
\begin{align} u \frac{\partial \mathbf{V}}{\partial r} = u\left(\frac{\partial u}{\partial r}\mathbf{e}_r + u \underbrace{\frac{\partial \mathbf{e}_r}{\partial r}}_0 \right ) + u\left(\frac{\partial v}{\partial r}\mathbf{e}_\theta + v \underbrace{\frac{\partial \mathbf{e}_\theta}{\partial r}}_0 \right ) + u\left(\frac{\partial w}{\partial r}\mathbf{e}_z + w \underbrace{\frac{\partial \mathbf{e}_z}{\partial r}}_0 \right ) \end{align}
or
(6)
\begin{align} u \frac{\partial \mathbf{V}}{\partial r} = u\left( \frac{\partial u}{\partial r}\mathbf{e}_r + \frac{\partial v}{\partial r}\mathbf{e}_\theta +\frac{\partial w}{\partial r}\mathbf{e}_z \right ) \end{align}
similarly
(7)
\begin{align} \frac{v}{r} \frac{\partial \mathbf{V}}{\partial r} = \frac{v}{r}\left(\frac{\partial u}{\partial \theta}\mathbf{e}_r + u \underbrace{\frac{\partial \mathbf{e}_r}{\partial \theta}}_{\mathbf{e}_\theta} \right ) + \frac{v}{r} \left(\frac{\partial v}{\partial \theta}\mathbf{e}_\theta + v \underbrace{\frac{\partial \mathbf{e}_\theta}{\partial \theta}}_{- \mathbf{e}_r} \right ) + \frac{v}{r} \left(\frac{\partial w}{\partial \theta}\mathbf{e}_z + w \underbrace{\frac{\partial \mathbf{e}_z}{\partial \theta}}_0 \right ) \end{align}
or
(8)
\begin{align} \frac{v}{r} \frac{\partial \mathbf{V}}{\partial r} = \frac{v}{r}\left[ \left(\frac{\partial u}{\partial \theta} - v \right) \mathbf{e}_r + \left( u + \frac{\partial v}{\partial \theta} \right) \mathbf{e}_\theta + \frac{\partial w}{\partial \theta}\mathbf{e}_z \right ] \end{align}
finally
(9)
\begin{align} w \frac{\partial \mathbf{V}}{\partial z} = w \left(\frac{\partial u}{\partial z}\mathbf{e}_r + u \underbrace{\frac{\partial \mathbf{e}_r}{\partial z}}_{0} \right ) + w \left(\frac{\partial v}{\partial z}\mathbf{e}_\theta + v \underbrace{\frac{\partial \mathbf{e}_\theta}{\partial z}}_{0} \right ) + w \left(\frac{\partial w}{\partial z}\mathbf{e}_z + w \underbrace{\frac{\partial \mathbf{e}_z}{\partial z}}_0 \right ) \end{align}
or
(10)
\begin{align} w \frac{\partial \mathbf{V}}{\partial z} = w \left(\frac{\partial u}{\partial z}\mathbf{e}_r + \frac{\partial v}{\partial z}\mathbf{e}_\theta + \frac{\partial w}{\partial z}\mathbf{e}_z \right ) \end{align}
This was a rather tedious way of deriving the material derivative as one could have used vector technology to obtain an invariant form that works for all coordinates. Nonetheless, it is interesting to see the intricacies of the derivation using chain rule differentiation. Note that if were computing the material derivative for a scalar, the extra terms in Eq. 7 (in the radial and tangential components) would disappear. These are purely reminicsent of the vectorial nature of the velocity field (or any other vector field for that matter).
It is very interesting to note the intimate link between the physical nature of the velocity and its mathematical description through vectors. One would pose the following argument: why don't we treat the material derivative of the velocity as that of three scalars, namely, u_r, u_theta, and u_z? Doing this will obviously remove the hassles of dealing with derivatives of unit vectors, but will eventually lead to inconsistent results. So what's the issue here?
The problem with that treatment is that in essense, the velocity is one quantity that we describe using vectors: a magnitude and a direction. If we are to use three scalars to describe the velocity we lose an essential ingredient which is the direction. In the end, the material derivative of the velocity can be decomposed into the material derivatives of three scalars (u_r, u_theta, and u_z) plus some correction. This correction stems from the directional nature of the velocity field. In other words, this correction can be thoguht of as being the material derivative of the direction of the velocity field.
Comments
It is a very profitable post for me. I've enjoyed reading the post. It is very informative and useful post. I would like to visit the post once more its valuable content.
curt
Nike Air Zoom Barcelona
number one choice for student credit cards
Best Student Credit Cards
can you pay off student loans with a credit card
good student credit card
eliminate student credit card debt in one month
what is the easiest student credit card to get
Grants to Pay Off Student Loans
credit card to get for a college student
what is the best credit card for a college student
paying tuition to build credit
Student Answers
instant approval student credit cards
how to choose a student credit card
Best Student Credit Card to Open
do you have a student credit card
Student Credit Cards With No Job
college student get a credit card
prepaid credit card or regular student credit card
what credit card is best for a college student
thanks
Thank you for this post. Thats all I are able to say. You most absolutely have built this blog website into something speciel. You clearly know what you are working on, youve insured so many corners.thanks
So what are you waiting for? Purchase this SparXXrX® Gold package today to attain four month’s worth of satisfying, strong, and impeccable boners!
So what are you waiting for? Purchase this SparXXrX® Gold package today to attain four month’s worth of satisfying, strong, and impeccable boners!
Its remarkable to pay a quick visit this web site and reading the views of all mates regarding this post, while I am also zealous of getting familiarity.
jbrandt dot com
You wrote: "the material derivative is written as with the capital D symbol to distinguish it from the total and partial derivatives"
What is the difference between the total derivative and the material derivative?
i need the full derivation of material time derivative in cylendrical coordinates system and also some examples related to it … if we r given a velocity in cylendrical coordinate system then hw we find its acceleraTION .. PLZ tell it to me as soon as possible , thanks
it seems that the left hand side of equations (7) and (8) has a typo. I should be partial V with respect to theta, not r
Post preview:
Close preview