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Module 6: Navier-Stokes Equation

Lecture 18: Macroscopic momentum balance for pressure-drop in a tubular flow

Newtonian and non-Newtonian fluid - macroscopic pressure-drop balance

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Module 6: Navier-Stokes Equation

Lecture 18: Macroscopic momentum balance for pressure-drop in a tubular flow

Newtonian and non-Newtonian fluid - macroscopic pressure-drop balance

In the previous lecture, we applied the NS equation to obtain an expression for the pressuredrop in

a tube for the condition of the steady-state laminar flow of a Newtonian fluid. We can also derive the

same expression, the Hagen- poiseuille equation, by making an integral (macroscopic) force balance.

(Fig. 18a)

If the flow is laminar (Re

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Module 6: Navier-Stokes Equation

Lecture 18: Macroscopic momentum balance for pressure-drop in a tubular flow

From the above expression, one can also determine viscosity of a fluid by measuring

pressure- drop across a tube for different flowrates:

(Fig. 18b)

The above-plot is known as pseudo-shear diagram, which is generally used to determine theviscosity of a slurry-mixture. A similar approach can be followed to determine the viscosity or

effective viscosity of Non-Newtonian fluids.

Bingham plastic fluid

Such a fluid has the following shear-stress vs strain rate characteristics:

-----------(1)

where is known as the yield stress. Fluid is supposed to be frozen if

or,

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Module 6: Navier-Stokes Equation

Lecture 18: Macroscopic momentum balance for pressure-drop in a tubular flow

If the flow is laminar, the velocity profile is parabolic near the walls and is uniform in the central core of

the tube. Shear-stress in such fluid also varies linearly with r:

(Fig. 18c)

Equation (1) can be integrated with to obtain:

Therefore,

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Module 6: Navier-Stokes Equation

Lecture 18: Macroscopic momentum balance for pressure-drop in a tubular flow

The uniform velocity within the core of the tube

Again,

Integrate as an exercise to obtain

(Fig. 18d)

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