PX285 - H4 - newton's law of viscosity

considering a flow between two parallel plates, where the upper plate is moving at a constant velocity, $\vec{u}$ , and the bottom plate is stationary

considering a 1D case of a laminar flow, so $\vec{u} = u_{x} (y) \hat{i}$ , and $u_{y} = u_{z} = 0$
when steady motion is obtained, ie. no dependence on time, so, $\frac{\partial}{\partial t} = 0$
a constant force is needed to maintain the flow

F \propto \frac{u_{x} A}{h}

where, $A$ is the area of the plate perpendicular to $\hat{j}$ , ie. parallel to $\vec{u}$

\frac{u_{x}}{h} \propto \frac{\partial u_{x}}{\partial y}

F = μ A \frac{\partial u_{x}}{\partial y}

where, $μ$ is the viscosity

for air at $20 °$ C, $μ ≃ 1.8 \times 10^{- 5}$ Pa s
for water at $20 °$ C, $μ ≃$
in the neighbourhood of the moving plate, $y = h$ , the fluid acquires a certain amount of mechanical momentum
this momentum is transferred trough flow in the direction perpendicular to $\vec{u}$ , in y-direction
this is the effect of viscosity, ie. internal friction in the fluid
forces that should be considered in fluid mechanics:
- viscous force
- gravity
- negative gradient of pressure $(- \vec{\nabla} p)$
- coriolis force
- lorentz force, for electrically conductive fluids
if viscosity does not depend on speed, such fluids are known as newtonian fluids
- eg: air, water, plasmas
if viscosity does depend on speed, such fluids are known as non-newtonian fluids
- eg: oil, blood, emulsions
only newtonian fluids are discussed in this module