A semi-inverse problem of flows of fluids with pressure-dependent viscosities
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A flow is said to be controllable if such a flow can be engendered by the application of appropriate surface tractions. The question, in which fluids are such flows possible leads to the inverse problem of determining constitutive relations, as such flows in general will not be possible in all materials. Thus the determination of the appropriate constitutive class in which certain types of deformations are possible is an inverse problem. A very special subclass of controllable flows is defined through semi-inverse methods. Such semi-inverse solutions are not possible in all bodies but only in special bodies and thus yet constitute an inverse problem. Here, we use a semi-inverse method to consider the flow of fluids whose viscosity depends on the pressure, glowing down an inclined plane, and a flow between two parallel plates. We illustrate that we have an inverse problem with regard to choices of constitutive relations by showing that such flows are only possible in fluids with special viscosity-pressure relationships. We also show, with the help of the simple flow problem that boundary layers can develop in virtue of the viscosity depending on the pressure, a rather novel feature.