Fluids and Their Properties
Most people have a general idea of what a fluid is. Typically the idea of fluid is first presented as a substance that conforms to it’s container and has some kind of ability to flow. The field of fluid mechanics defines a fluid as a substance which is continuous and one which forces will deform. Contrast this to an elastic solid, which is also continuous in nature but will return to its original shape once unloaded.
Certain properties of fluids are of extreme importance, among these are:
- Density, \rho [kg/m^3]: The mass of a fluid within a given volume.
- Specific Gravity, SG [unitless]: The ratio of density of the given fluid to that of water at standard density.
- Dynamic Viscosity, \mu [Pa\cdots]: A measure of how easily a fluid will deform, and therefore flow. Highly viscous fluid will have more resistance to flow.
- Kinematic Viscosity, \nu [m^2/s]: The ratio of dynamic viscosity to density.
Introductory Concepts in Continuum Mechanics
Two important concepts define the motion of a fluid.
- Velocity Field: Describes the motion of a fluid particle at a specific point, function of location x,y,z and time t.
\bm{\overrightarrow{V}}(x,y,z,t)=u(x,y,z,t)\bm{\hat{i}}+v(x,y,z,t)\bm{\hat{j}}+w(x,y,z,t)\bm{\hat{k}}
- Stress Tensor: Describes the internal, local forces acting upon a fluid particle, also a function of location and time.
\bm{\tau}=\begin{bmatrix}\sigma_x & \tau_{xy} & \tau_{xz}\\ \tau_{yx} & \sigma_y &\tau_{yz}\\ \tau_{zx} & \tau_{zy} & \sigma_z\end{bmatrix}
Two kinds of forces can act on a continuous substance.
- Body Forces: Body forces act through the entire particle. Examples are gravity and electromagnetism.
- Surface Forces: Surface forces act on the surface of the particle. Examples are friction and stress.
Visualizations
Flow visualization is especially useful in experimental fluid mechanics. Flow visualization also provides a tangible understanding of flows. Four common flow visualizations are as follows:
- Streaklines: The set of points that particles have passed through after passing through a fixed point. Streaklines can not intersect.
- Pathlines: Traces the path of a single particle in the flow.
\dfrac{dx_p}{dt}=u(x,y,z,t), \dfrac{dy_p}{dt}=v(x,y,z,t), and \dfrac{dz_p}{dt}=w(x,y,z,t)
- Timelines: A line of particles that may change shape due to flow over time. Can be thought of as a set of pathlines.
- Streamlines: Curve which is always tangent to the velocity of the particle. Fluid can not flow across a streamline by definition.
In 3D: \dfrac{dx}{u}=\dfrac{dy}{v}=\dfrac{dz}{w}
In 2D: \dfrac{dy}{dx}=\dfrac{v}{u}