Introduction to Fluid Mechanics
Fluid Properties, Flow Fields and Phenomena, Dimensional Analysis
1 Properties of Fluids
When you push on a solid with a shear stress \tau, it deforms a certain amount:
Double the force on the block, and the deflection \phi doubles:
However, when you enact a shear stress on a liquid, it keeps deflecting:
The rate of deflection \frac{d\phi}{dt} doubles when you double the shear stress:
Thus, a fluid deforms continuously when a force is applied. Liquids flow if and only if something creates a shear stress (not perpendicular to the walls of the container). If a liquid is not flowing, then no such shear stresses exist.
2 Viscosity
- First property of a fluid.
Note that shear force per unit area (F/A) in a solid is the shear stress, \tau. The shear strain is an angle of deformation \phi, which (in the elastic region of deformation) is proportional to shear stress.
In a solid, \phi \propto \tau
This is Hooke’s Law.
In a liquid, the rate of change of strain is proportional to the shear stress, ie
\frac{d\phi}{dt} \propto \tau
We put a few symbols around that. For small \phi, \phi \approx \frac{x}{y}. At the start of the process, y is constant.
\begin{gather} \frac{\partial\phi}{\partial t} = \frac{\partial}{\partial t} \left( \frac{x}{y} \right) \\ = \frac{1}{y}\frac{\partial x}{\partial t} \end{gather}
\frac{\partial x}{\partial t}, the velocity in the x direction, is given its own symbol u. Generally, a flow vector \textbf{u} has flow speed components u, v, w in the x, y, z directions respectively.
So \tau \propto \frac{\partial \phi}{\partial t} \propto \frac{u}{y}.
Consider the graph of x velocity w.r.t y; there is a certain slope to that graph. The slope of that graph is directly related to viscosity by some constant \mu with units kg m^{-1} s^{-1}. We call that constant \mu the viscosity of the liquid.
On the left, a fluid with a low viscosity has a high slope (the fluid starts moving quickly as you get away from the wall); a fluid with a high viscosity has a low slope (the fluid moves slowly even a far distance from the wall).
2.1 Newton’s Law of Viscosity
\tau = \mu\frac{du}{dy}
There is a linear relationship between the distance from a wall and the rate of flow parallel to that wall.
2.2 Conclusions
Viscosity tells us how easy it is to shear the fluid; a measure of friction within the fluid. Fluids obeying Newton’s law are Newtonian Fluids: the shear stress is linearly related to the velocity in the fluid (with a slope of \mu as you get further from the wall).
Non-Newtonian fluids have a non-linear relationship between the amount of shear stress and the rate of motion within the fluid.
Picture the fluid flowing in layers, where each layer of the fluid (a plane at a certain distance from the wall) drags the layer of fluid below it along.
Viscosity is usually measured as the force created between two driven plates with a fluid between (or practically, two cylinders with a layer of fluid of known height in between).
At the wall, there is a no-slip condition: the fluid sticks to the wall as u=0.
Intuitively, \mu_{gases} << \mu_{liquids}
For gases, viscosity increases with temperature due to intermolecular interactions. With liquids, viscosity is caused by intermolecular cohesion, which decreases as temperature increases.
Kinematic viscosity \frac{\mu}{\rho} = \nu is its own thing (unit viscosity per mass density).
3 Density
- Second property of the fluid.
Density is mass per unit volume: \rho = \frac{m}{V}.
Specific Density is \frac{\rho}{\rho_{water}}. Remember that \rho_{water} is 1000 kg m^{-3}.
Flows with negligible variations in \rho are incompressible flows; flows in which variations in \rho are non-negligible are compressible.
- All liquid flows may be regarded as incompressible
- Gas flows in which the flow speed u is well below the speed of sound (c_s) may also be regarded as incompressible.
\frac{u}{c_s} = M, the Mach number. Below a Mach number of 0.3, changes in density can be expected to be less than 5%; the flow is incompressible.
4 Surface Tension
- Surface tension is the third basic property of a fluid.
Surface tension is a force in the surface of a liquid. Recall that all of the molecules within a liquid attract each other. On the surface of the liquid, there are internal molecules on one side of the surface and none on the other. Therefore, there is a net intermolecular force on the molecules at the surface.
Surface tension \sigma is defined as the force in a liquid’s surface along a line of unit length in the surface. Since it’s defined as the force divided along a line, it has units kg s^{-2}.
Surface tension decreases with temperature, as IMFs decrease.
Surface tension forces are often small & neglected; but they may dominate when dealing with droplets or bubbles.