Introduction to Fluid Mechanics

Viscous Flow

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Solids can support a shear stress when in static equilibrium because their molecules are held together by rigid bonds. Fluids cannot support a shear stress in static equlibrium because their molecules are not held by rigid bonds. Fluids can only exert a shear stress when they move.

To an excellent approximation, the fluid molecules adjacent to a surface take the speed of that surface. One can then consider momentum diffusing through a fluid in the same way that heat and mass diffuse through a fluid: due to random molecular motion.

In a continuum, the transfer of momentum by molecular motion is modelled as a shear stress. This shear stress is proportional to the velocity gradient. The constant of proportionality is the viscosity. In high viscosity fluids, momentum is transferred quickly from one part of the flow to another.

By jostling around randomly, the molecules in a fluid transfer momentum from one part of the fluid to another. In a continuum model, the ability of a fluid to transfer momentum is measured by the viscosity. This clip explains the link between viscosity and molecular motion.

Couette flow is the steady flow between two flat plates, a fixed distance apart, in which the plates move relative to each other. Here, the velocity profile of Couette flow is derived by considering a control volume within the fluid.

When a fluid is laminar (i.e. sheets of fluid slide over each other) and is confined between flat plates, some properties (such as velocity and shear stress) only vary in one direction while other properties (such as pressure) only vary in the perpendicular direction. This means that the partial derivatives collapse to ordinary derivatives and the velocity profiles can be found by hand. This clip confirms this and also shows how to derive the governing equation for Couette and Poiseuille flow.

Poiseuille flow is the steady flow between two flat plates, a fixed distance apart, in which the plates are stationary and a pressure gradient forces the fluid to move.

The Navier-Stokes equation is Newton's second law (f=ma) applied to a viscous fluid. It is the most important equation in Fluid Mechanics

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