Suppose that temperature variations within the parallel-plate channel of Fig. 7.1 noticeably alter the viscosity. In particular, assume that

where α > 0. That is, there is a linear temperature variation across the channel, with T = T₀ at the center and T = T₁ at the upper wall. The viscosity at T = T₀ is μ₀ and, as is typical for liquids, μ decreases with increasing T.

(a) Determine v_x(y). It is convenient to work with a dimensionless coordinate η = y/H and a dimensionless viscosity-temperature parameter A = α(T₁ − T₀).

(b) How does the mean velocity U compare with its value for μ = μ₀?

(c) The viscosity of water (in Pa · s) is 1.139 × 10^–3 at 15 °C, 1.002 × 10^–3 at 20 °C, and 0.8904 × 10^–3 at 25 °C. What value of α best fits these data? If the wall temperatures span this range, how much would U for water differ from that for an isothermal channel at 20 °C?

Transcribed Image Text:

T(y) = To + (T₁ − To) — 17 μ = flo 1+ α (T-To) (P7.16-1)