Abstract
We consider flows of a viscous fluid in a domain on the boundary of which ether velocity or stress is specified. It is shown that the solutions of boundary-value problems for the Navier-Stokes and linear Stokes equations with the same stresses on the boundary satisfy an inequality, which makes it possible to obtain an estimate from above for the functionals of the velocity fields generated by the stresses applied on the boundary. For a flow in a closed domain, generated by the shear stress on the boundary, for which at high Reynolds numbers Batchelor’s theorem is valid and the condition of monotonic dependence of the velocity functional on the Reynolds number holds, two-sided estimates for the average velocity value on the boundary are obtained for all Reynolds numbers.
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