Question

Mention the mathematics properties of elliptical and hyperbolic fluid dynamic equation.

Answer 1

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Solution - Mathematics properties of elliptical flud dynamic equation, Elliptical equations are characteristic of equilibrium problem, this inclucks many (but not all) stondy state Hows. for potential flows the velocity is expressed in terms of a velocity patential : U= 30. Beause the flow is incomprensible, J.u=0, which results in 720-0. This is known as Laplace Equation: The solution depands solely on the bandary condition. The disturbance in the interior of the solution affect the solution everywhere ele, As a result, solutions are always smooth, even when boundary condition are du continuon.

_ 242 Mathematics properties of Hyperbolic fluid dynamic equation. Huberbalic equations are typical of marching problems with negligible disipation. An example is the ware equation : 820 - (2226 This describes the transverx displacement of a string during small amplitude vibrations. If y is the displacement, x the coordinates along the string, and a the initial amplitude the solution is u y(xt) = a cos/rect An (rix ! [NOTE! - Amplitucle is independent of time, there is no disipution]. = Hyperbolic problems can have discontinuous solution. Disturbance may affect only a limited region napore Local solutions may only depends on initial conditions in the domain of dependence. L