- #1
Mechdude
- 117
- 1
Homework Statement
1.a) the velocity components of a 3-d flow are
[tex] u= \frac{ax}{x^2+y^2} [/tex]
[tex] v= \frac{ay}{x^2+y^2} [/tex]
[tex] \omega = c [/tex]
where a and c are arbitrary constants . show that the streamlines of this
flow are helics
[itex] x=acos(t) [/itex] ;
[itex] y=asin(t) [/itex] ;
[itex] x=a*c*t [/itex]
a. what is the irrotational velocity filed associated with the potential
[itex] \phi = 3x^2 -3x +3y^2 + 16t^2 +12zt [/itex] ? does the flow satisfy the
incompresible continuity equation [itex] \nabla \vec{q} = 0 [/itex] where q is the velocity.
2. b) the velocity potential of a 2D incompressible flow is
[tex] \phi = \frac{1}{2} log \left( \frac{(x+a)^2 +y^2}{(x-a)^2+y^2 } \right) [/tex]
show that the stream function [itex] \psi [/itex] is given by:
[tex] \psi = \arctan \frac{y} {x-a} - \arctan \frac{y}{x-a} [/tex]
3 a)
Air obeying Boyles law [itex] p=k \rho [/itex] is in motin in a uniform tube of small
cross-sectional area. show that if [itex] \rho [/itex] is the density and u is the velocity
at a distance x from a fixed point a; and t is time , this is true:
[tex]\frac{ \partial^2 \rho}{\partial t^2} = \frac {\partial^2 (u^2 + k) \rho}{\partial x^2} [/tex]
3 b. A steam is rushing from a boiler throught a conical pipe, the diameters of the ends a
being D and d . if v and u are the corresponding velocities of the steam an if the motion is
supposed to be that of divergence from the vortex of the cone prove that
[tex] \begin{displaymath} \frac{u}{v} = \frac {D^2} {d^2} e^{\frac{u^2-v^2}{2k} } \end{displaymath}[/tex]
where k is the pressure divide by the density and its a constant ie [itex] k= \frac{p}{\rho} [/itex]
note its getting in at one end with a velocity [itex] v [/itex] and density [itex] \rho_1 [/itex]
and out the other side with [itex] u [/itex] and [itex] \rho_2 [/itex]
Homework Equations
[itex] \nabla \vec{q} = 0 [/itex]
Boyles law: [itex] p=k \rho [/itex]
The Attempt at a Solution
i do not know how to start this stuff with all honesty.