With the advent of high-powered computers, our ability to solve many complex problems in fluid mechanics (and elsewhere) has been greatly enhanced. The cost for this ability is often highly intensive and time consuming computational problems. Now, more than ever, we must not lose sight of the physically motivated geometrical and dynamical approximations to governing equations which can often afford much less intensive---and sometimes analytical---solutions.
This talk begins with a brief description of the typical fluid flow configuration employed in the manufacture of photographic products (referred to as ``coating flows''). The general system of equations and boundary conditions governing such flows is provided. A brief overview of the simplifications to such equations afforded by the geometrical and dynamical aspects of coating flows is provided, including the techniques of asymptotics, linearization, and integral approaches.
Next, we consider the detailed analysis of a slot flow of small
aspect ratio (a Hele-Shaw flow) and having an angled boundary
in one of its long dimensions; such a problem is relevant to
the distribution and delivery of fluids in coating flows. The
unsimplified system of equations in this geometry is formidable,
and is 3-dimensional in nature. To solve this problem, matched
asymptotics are employed in conjunction with inverse and conformal
mappings to achieve a greatly simplified system of equations;
this system is one-dimensional in nature, and is easy to solve.
We discuss the physical results obtained, and close this talk
with an emphasis on the required interplay and balance between
computation and modeling approaches in industrial settings.
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