Can light be dragged by fluid? I was reading a text on "The Speed of Light in a Moving Medium"(topic in Special Relativity from Introduction to Mechanics by Kleppner and Kolenkow). They derive the expression for the speed of light in a laboratory frame that is passing through a moving medium of refractive index $n$. In the end, they found that speed of light for an observer in the laboratory frame given by-
$$u=\frac{c}{n}+v\left(1-\frac{1}{n^2} \right)$$
Suppose the fluid is at rest so that speed of light for laboratory observe is $c/n$. That's correct as we know. Now if fluid increases its speed then lights speed also increases. The more fluid speed up , the more light (within its limit). Now one thing to say that it's a complete relativistic effect and their is no explanation (a more logical one ie. by just seeing you can't guess like time dilation). If that's the correct thing then this is end of question. But if their is then please give explanation. Is their any interaction that light make with fluid to increase its speed? I mean you assume light as array of photon particle then of course it's seems that fluid push the photon to give it momentum?
I  know their many flaw in my thinking so please give a clear explanation. Thanks

Edit: Note that I already know the proof of the above formula, I'm just confused why it seems that light is being dragged by the fluid? (or maybe it's just a relativistic effect?)

Edit 2 I tried to be more specific.
 A: In the fluid frame, the light is moving at $u' = c/n$. If the lab is moving colinearly at $\pm v$, then the velocity of the light in the lab frame is:
$$ u = \frac{u'\pm v}{1+\frac{vu'}{c^2}} = \frac{\frac c n \pm v}{1+\frac v {nc}}$$
That is exact. If we're talking about non-relativistic labs:
$$ u  \approx (\frac c n \pm v)(1-\frac v {nc}) = \frac c n \pm v - \frac v {n^2} \mp \frac{v^2} {nc} $$
Being non-relativistic, $v^2/nc \ll 1$, so that:
$$ u  \approx  \frac c n \pm v(1\mp \frac 1 {n^2})$$
So nothing is being dragged, the $v/n^2$ term is the first order correction for relativistic velocity addition.
A: 
Note that I already know the proof of the above formula, Why it seems that light is dragged by the fluid?

If you already know the proof then you already know why. The proof itself is the explanation.
It seems that light is dragged in our frame because in the medium’s frame light is slowed isotropically and the medium itself is moving in our frame. The proof shows that those two facts, taken together, logically imply the dragging-like behavior.
It may be that your real question is not why does it seem light is dragged, since you already know the answer to that, but rather how is this dragging compatible with relativity?
There are many sloppy statements of the second postulate, but the second one in the Wikipedia article is good

the speed of light in free space has the same value c in all inertial frames of reference.
https://en.m.wikipedia.org/wiki/Postulates_of_special_relativity

Note that this statement has two important qualifiers: free space and inertial frames. That the speed of light it not c in matter is not a contradiction with the postulate nor that the speed of light is not c in non-inertial frames. So dragging is ok, because it occurs only in a medium which is not required to have the speed of light equal to c.
What is prohibited is aether dragging, which is dragging in free space. The idea is that free space itself is a medium with a definite velocity, which is the idea that is contradicted by the second postulate.
