Consider the low-speed flow of air between parallel disks as...

Consider the low-speed flow of air between parallel disks as shown. Assume that the flow is incompressible and inviscid, and that the velocity is purely radial and uniform at any section. The flow speed is $V=15 \mathrm{m} / \mathrm{s}$ at $R=75 \mathrm{mm}$. Simplify the continuity equation to a form applicable to this flow field. Show that a general expression for the velocity field is $\vec{V}=V(R / r) \hat{e}_{r}$ for $r_{i} \leq r \leq R .$ Calculate the acceleration of a fluid particle at the locations $r=r_{i}$ and $r=R$.

Key Concepts

Material Derivative and Convective Acceleration

The acceleration of a fluid particle in a flow field is determined by the material derivative, which accounts for both local and convective changes in velocity. In physics, convective acceleration arises because a fluid element moving through a velocity gradient experiences a change in velocity due to its motion, even in a steady flow.

Continuity Equation

The continuity equation expresses the conservation of mass for a fluid flow. For incompressible flows, the density is constant, which simplifies the mathematical form. In cylindrical coordinates with radial symmetry, this equation reduces so that the product of the radial velocity and the circumference scales with r, leading to a 1/r dependence when the flow is purely radial.

Incompressible Flow

In incompressible flow, the fluid density remains constant throughout the flow field. This assumption simplifies the governing equations, such as the continuity equation, by removing any dependence on density variations. It is especially useful in flows where pressure changes do not lead to significant changes in density.

Inviscid Flow

An inviscid flow assumes that the fluid has no viscosity, meaning that viscous forces (internal friction) are neglected. This simplification is valid for flows where inertial forces dominate, and it allows focus on pressure and inertial effects, simplifying the momentum equations and flow analysis.

Radial Flow in Cylindrical Coordinates

Radial flow in a cylindrical coordinate system assumes that the velocity vector has only a radial component which may vary with the radial distance. This symmetry simplifies the continuity and momentum equations, leading to the general form of the velocity field as inversely proportional to the radial coordinate, reflecting the spreading of the flow area.

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