Introduction

Consider a liquid flowing steadily over a fixed horizontal layer, with the flow speed increasing uniformly the further a layer is from that fixed bottom layer. Take any two parallel layers within the liquid, at perpendicular distances x and x+dx from the fixed layer, moving with velocities v and v+dv respectively. Newton's first law of viscous flow states that the tangential force F needed to maintain this relative sliding between the two layers is proportional to (i) the contact area A between them and (ii) the VELOCITY GRADIENT $dv/dx$ between them: $F\propto A$ and $F\propto dv/dx$, so $F=-\eta A\dfrac{dv}{dx}$. The constant of proportionality $\eta$ is the COEFFICIENT OF VISCOSITY of the liquid, and the negative sign shows that the force is frictional in nature -- it always opposes the relative motion between the layers, not drives it. The dimensional formula of the coefficient of viscosity is $[ML^{-1}T^{-1}]$. Physically, viscosity behaves much like ordinary solid friction: the kinetic energy of the moving fluid is continuously dissipated as heat by this internal friction. CAUSE OF VISCOSITY. Consider a liquid flowing over a horizontal surface in adjacent layers: the faster upper layer tends to speed up (accelerate) the slower layer just below it, while that slower lower layer simultaneously tends to slow down (retard) the layer above it -- this mutual, tangential dragging sets up a backward tangential force between every pair of adjacent layers, which continually acts to destroy the relative motion between them; this microscopic tug-of-war betwe …