Assume the liquid film in Example 5.9 is not isothermal, but instead has the following distribution:

\[T(y)=T_{0}+\left(T_{w}-T_{0}\right)\left(1-\frac{y}{h}\right)\]

where \(T_{0}\) and \(T_{w}\) are, respectively, the ambient temperature and the wall temperature. The fluid viscosity decreases with increasing temperature and is assumed to be described by

\[\mu=\frac{\mu_{0}}{1+a\left(T-T_{0}\right)}\]

with \(a>0\). In a manner similar to Example 5.9, derive an expression for the velocity profile.

Data From Example 5.9

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Example 5.9 ANALYSIS OF FULLY DEVELOPED LAMINAR FLOW DOWN AN INCLINED PLANE SURFACE A liquid flows down an inclined plane surface in a steady, fully developed laminar film of thickness h. Simplify the continuity and Navier-Stokes equations to model this flow field. Obtain expressions for the liquid velocity profile, the shear stress distribution, the volume flow rate, and the average velocity. Relate the liquid film thickness to the volume flow rate per unit depth of surface normal to the flow. Calculate the volume flow rate in a film of water h = 1 mm thick, flowing on a surface b = 1 m wide, inclined at 15 to the horizontal.