fluid

Examples of fluid in the following topics:

• Physics and Engineering: Fluid Pressure and Force

  • Pressure is an important quantity in the studies of fluid (for example, in weather forecast).
  • For fluids near the surface of the earth, the formula may be written as $p = \rho g h$, where $p$ is the pressure, $\rho$ is the density of the fluid, $g$ is the gravitational acceleration, and $h$ is the depth of the liquid in meters.
  • Using this expression, we can calculate the total force that the fluid pressure gives rise to:

• Surface Integrals of Vector Fields

  • Imagine that we have a fluid flowing through $S$, such that $\mathbf{v}(\mathbf{x})$ determines the velocity of the fluid at $\mathbf{x}$.
  • The flux is defined as the quantity of fluid flowing through $S$ in unit amount of time.
  • This illustration implies that if the vector field is tangent to $S$ at each point, then the flux is zero, because the fluid just flows in parallel to $S$, and neither in nor out.

• Vector Fields

  • Vector fields are often used to model the speed and direction of a moving fluid throughout space, for example, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.
  • In the case of the velocity field of a moving fluid, a velocity vector is associated to each point in the fluid.

• Curl and Divergence

  • If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid.
  • (Note that we are imagining the vector field to be like the velocity vector field of a fluid (in motion) when we use the terms flow, sink, and so on.)

• Hyperbolic Functions

  • The latter is important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.

• Green's Theorem

  • In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area.

• Calculus of Vector-Valued Functions

  • Vector calculus is used extensively throughout physics and engineering, mostly with regard to electromagnetic fields, gravitational fields, and fluid flow.

• The Divergence Theorem

  • The divergence theorem is an important result for the mathematics of engineering, in particular for electrostatics and fluid dynamics.