I am puzzled by an artifact of the definition of viscosity and need an intuitive picture to help explain it. I know $\tau_{yx}=-\mu{dv_x \over dy}$ but I am looking for an intuitive picture of the specific question below.
Take the prototypical definition/model of viscosity: Imagine two very large plates separated by a constant small distance with a fluid sandwiched inside. Apply a force $F$ parallel to the bottom plate so as to start moving the bottom plate at a speed of $V$. Once steady-state is reached, $F$ and $V$ are constant. The gap between plates is $Y$ and the flow is presumed/constrained to be laminar. One finds that the velocity gradient with respect to distance along $Y$ is linear with velocity at the top plate zero and the velocity at the bottom plate $V$ (no-slip condition). In equation form:
$${F \over A}=\mu{V \over Y}$$
where $\mu$ is the viscosity. So far, so good.
Now, consider the force $F$. It is proportional to $A$ which makes sense as a bigger plate will require more force to move due to the viscous contact with the liquid. It is proportional to $\mu$ which really is the definition of viscosity (higher viscosity fluid requires more force). It is proportional to $V$ which make sense since moving the plate at a higher velocity will require more force. Finally it is inversely proportional to $Y$, the plate gap. I can't quite understand this one intuitively. Why should doubling the plate gap cause the force to be halved everything else being equal? Conceptually, I get that the velocity gradient is half what it was and that this is what causes the force to be half but my intuitive picture is lacking. I just can't imagine myself exerting a smaller force on a doubled gap system. Taken to the extreme, at very large $Y$, $F$ is zero. This must be why my text Transport Phenomena by Bird, Stewart, and Lightfoot says the gap must be "small". But what really determines what "small" is?