8.65 In a food industry plant, two immiscible fluids are...

8.65 In a food industry plant, two immiscible fluids are pumped through a tube such that Iluid $1\left(\mu_{1}=0.5 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)$ forms an inner core and fluid $2\left(\mu_{2}=5 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)$ forms an outer annulus. The tube has $D=5 \mathrm{mm}$ diameter and length $L=5 \mathrm{m}$. Derive and plot the velocity distribution if the applied pressure difference, $\Delta p,$ is 5 MPa.

Key Concepts

Hagen-Poiseuille Flow

This concept describes laminar, pressure?driven flow in conduits such as pipes, where the velocity profile is parabolic for a single, Newtonian fluid. It forms the basis for analyzing flows under a constant pressure gradient and is applied by solving the simplified Navier–Stokes equations under steady, incompressible, and laminar conditions.

Newtonian Fluid Behavior

Newtonian fluids have a linear relationship between shear stress and shear rate, defined by the viscosity. This simple characterization allows for the straightforward derivation of velocity profiles in pipe flows, as seen in the case where the viscosity remains constant throughout each immiscible fluid phase.

Multiphase Core-Annular Flow

In systems involving two immiscible liquids, such as a lower-viscosity fluid core surrounded by a higher-viscosity annulus, the flow must be analyzed within each region separately. This core-annular flow model is essential in many industrial applications and involves treating each region with its own viscosity while considering their geometric arrangement.

Interface Boundary Conditions

At the interface between two immiscible fluid layers, continuity in both velocity and shear stress is enforced. These conditions are critical to couple the separate governing equations for each fluid layer, ensuring that the overall solution correctly represents the physical situation of a smooth transition between the fluids.

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