Consider fully developed laminar flow in the annular space...

Consider fully developed laminar flow in the annular space formed by the two concentric cylinders shown in the diagram for Problem $8.53,$ but with pressure gradient, $\partial p / \partial x$ and the inner cylinder stationary. Let $r_{0}=R$ and $r_{i}=k R$ Show that the velocity profile is given by $$u=-\frac{R^{2}}{4 \mu} \frac{\partial p}{\partial x}\left[1-\left(\frac{r}{R}\right)^{2}+\left(\frac{1-k^{2}}{\ln (1 / k)}\right) \ln \frac{r}{R}\right]$$ Obtain an expression for the location of the maximum velocity as a function of $k$. Plot the location of maximum velocity $(\alpha=r / R)$ as a function of radius ratio $k$. Compare the limiting case, $k \rightarrow 0,$ with the corresponding expression for flow in a circular pipe.

Key Concepts

Limiting Case Analysis

Limiting case analysis involves studying the behavior of the velocity profile and the maximum velocity location as the geometric parameters approach particular limits (for example, the inner cylinder radius approaching zero). This method is useful for verifying the derived expressions by comparing them against known results, such as the classical solution for flow in a circular pipe, thereby confirming the consistency and validity of the derived annular flow model.

Maximum Velocity Location

The location of maximum velocity within the flow is determined by differentiating the derived velocity profile with respect to the radial coordinate and setting the derivative equal to zero. This analysis yields an expression, often in the form of a transcendental equation, that relates the position of peak velocity to the geometric parameter (such as the radius ratio), providing insights into how the flow distribution changes with the geometry.

Velocity Profile Derivation

The derivation of the velocity profile in a fully developed, pressure-driven, laminar flow involves solving the Navier–Stokes equations under simplifying assumptions (steady flow, no acceleration in the flow direction, etc.) and applying the appropriate boundary conditions at the inner and outer walls. The resulting expression characterizes how the velocity varies radially through the annular region.

Annular Geometry

Annular geometry describes a flow domain bounded by two concentric circular cylinders, one forming the inner boundary and the other the outer boundary. This configuration significantly influences the velocity profile, as the curvature and differing radii of the boundaries require careful consideration when applying boundary conditions and solving the governing equations.

Laminar Flow

Laminar flow refers to a flow regime in which the fluid moves in smooth, parallel layers, with little or no mixing between them. This concept is crucial because the analysis and resulting velocity profile assume that viscous forces dominate inertial effects, allowing for a steady, predictable flow pattern modeled by the Navier–Stokes equations under low Reynolds number conditions.

Pressure-Driven Flow

Pressure-driven flow, often termed Poiseuille flow, refers to the movement of fluid under the influence of a pressure gradient. In this context, the imposed pressure gradient is the driving force that overcomes viscous resistance, establishing the velocity distribution across the annulus as dictated by the balance between pressure and viscous forces.

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