520 ADV FLUID MECH I  
                      Fall Term 2003
                  
Hwk 4 (Due Dec. 2)
1. Find the velocity field of a Newtonian fluid between two concentric cylinders of radii Ri and Ro if the outer is stationary and the  inner is spinning with velocity V and there is an axial pressure gradient given by pz . Show that there are no nonlinear inertial terms, find the torque required to move the inner cylinder, and find the axial volumetric flow rate.

    Well, not exactly. There is a nonlinear term v2/r that only affects the pressure gradient term and that does not affect the velocity and hence superposition still holds.. Setting u=¶/¶z=¶/¶q=0 deletes all the rest. This means that the theta (v-) component remains as in class, while the axial (w-) component is modified from the pipe analysis by the inner bc to: w=(pz/4m)(r2+(Ro2-Ri2)/ln(Ro-Ri)(ln r-ln Ri)-Ri2)

2. Find the periodic velocity field for a horizontal film of thickness delta on a plate oscillating horizontally with velocity V sin(w t).

    u=Im{[cosh a(d-y)/ cosh ad]eiwt}, where  a= V/((1+i)Ö(w/2n) d)      
The imaginary part is because the forcing is sin instead of cos. Can also be written in terms of positive and negative exponetials.

3. Repeat the numerical integration for the Hiemenz problem with a suction wall velocity of a/2.

    Same as in class, but f(0)= 0.5.  Shooting parameter is f"(0)=1.542.

4. Perform simple experiments with various fluids (soapy water, honey...)   to find criteria when flow down an inclined plate becomes unstable (for  instance, the surface becomes rough).  Is there a limit to the inclination?   Make a rough comparison of the volumetric flow rate to the theory when a fairly uniform flow is observed.

    The flow relationship can be written in dimensionless form in terms of Re, Fr, and the angle. The flow becomes greater as the angle of inclination increases. The tendency for the flow to become unstable increases as the angle increases and continues to increase as the inclination goes past vertical (goes upside-down.) The stability is a very strong function of the film thickness.

5. Show that the circulation and net mass flux are zero for a closed contour made of four segments: two of constant radii  (Ri and Ro) and two constant theta (0 and p/2) when the flow field is given by a source/vortex at the origin of the coordinate system.

    The two curved parts of the contour cancel each other for the vorticity component in circulation while the straight parts vanish. The same is true for the source term. Superposition is possible.

6. Sketch the Joukowski airfoil for the case for a cylinder of radius a=1 if the origin of the circle is placed m=0.1 off center at an orientation of 135 degrees from the origin. Identify the stagnation points when the airfoil flies at 0 and 10 degrees angle of attack.  Extra credit for plotting streamlines and/or comparison of the mapping to the approximation given by (4.25a) of Currie.

    The circle must go through c on the positive real axis.  Hence (0.1Ö2)2+(c+0.1Ö2)2 =1. The curved Joukowski airfoil. By plotting sufficient points on the circle, the airfoil can be plotted.  To find the stagnation point, the proper uniform flow at infinity must be chosen.  Then the proper circulation must be added to the cylinder flow so the stagnation point is brought to the critical point c on the positive real axis. Making sure the stagnation point moves to the cusp in the mapped plane is the Kutta condition.