A shear-thinning varnish with no yield stress obeys the power law

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Non-Newtonian and viscoelastic fluids

Problem

\tau = K\dot\gamma^n

, with

K = 4.0\,\mathrm{Pa\,s^{0.5}}

n = 0.50

and density

\rho = 1200\,\mathrm{kg\,m^{-3}}

. A coat of uniform thickness

h

drains steadily down a tall vertical wall. Measure

y

outward from the wall, which is at

y = 0

; the free surface at

y = h

carries no shear stress. Take

g = 9.8\,\mathrm{m\,s^{-2}}

. ((a)) By balancing forces on the layer of varnish between a plane at distance

y

and the free surface, find the shear stress

\tau(y)

. ((b)) Derive the downward velocity

u(y)

, with

u(0) = 0

, and the flow rate per unit width of wall

q = \int_0^h u\,\mathrm{d}y

. Show that

n = 1

K = \eta

gives the Newtonian film result

q = \rho g h^3/3\eta

. ((c)) For

h = 0.50\,\mathrm{mm}

, evaluate the surface speed

u(h)

and

q

. By what factor does

q

grow if the coat is laid on twice as thick, and how does that compare with a Newtonian varnish?

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