The Kinematics of Vorticity: Clifford Truesdell by Clifford Truesdell

Author:Clifford Truesdell
Language: eng
Format: epub
Publisher: Courier Publishing
Published: 2018-10-17T16:00:00+00:00

But since we have h.v' = hnv', ds v' = ds v', and hence

Since ds.v' = ds v', the result (63.19) becomes

Hence, in particular,

But hx, being one component of a harmonic vector, is itself a harmonic function, and since the problem of Dirichlet admits a solution for υr hx may be assigned an arbitrary continuous value upon . From (63.22) we have then v' = 0 upon , so that the field v' furnished by the second lemma must belong to thus itself the field v whose existence is asserted by the theorem.

The second linear balance condition (63.8) is as it stands sufficient for the existence of a velocity field of a motion of adherence, as is stated in the second characterization of a motion of a material which adheres to the bounding surface:5 let w, ϑ, , and υ be defined by the conditions stated in the first characterization theorem; let F be the class of twice continuously differentiable fields such that curl f = grad ψ, where ψ is a single-valued harmonic function; then in order that there exist a field v ∈ such that div v = ϑ, curl v = w, it is necessary and sufficient that

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