SOLA Calculation of Viscous Fluid Flow
	
	This plot shows how a viscous fluid is set in motion when the top wall of the container starts to move to the right.  The fluid velocity vectors (green arrows) are shown at a number of grid points (black dots).  The motion of the fluid is calculated using a flow code called SOLA.  
	
	SOLA is a Fortran program written in the 1970's by Tony Hirt and collaborators at the Los Alamos National Laboratory. It integrates the Navier-Stokes equations, the partial differential equations governing hydrodynamic flow of viscous fluids.  SOLA is a relatively simple fluid dynamics code, being restricted to two-dimensional calculations.  The two most common cases are plane geometry (parameter x = 0, coordinates x and y) and cylindrical geometry (x = 1, x now representing the radial coordinate r and y the axial coordinate z).  For details regarding this simplified Marker-and-Cell (MAC) code, see Los Alamos Scientific Laboratory Report LA-5852 (1975).

	The Navier-Stokes Equations, for those who do not know them by heart, are

	¬

where the coefficient x specifies the geometry, as indicated above, and n is the fluid's coefficient of kinematic viscosity.  The velocity components u, v are in the coordinate directions x, y.  Also, p is the ratio of the pressure to constant density and gx, gy are body accelerations (such as due to gravity).  The third equation represents the conservation of mass in the flow process.

	The numerical solution of these equations involves a finite difference method, integrating forward from time t = 0 in steps of size Dt, for given initial conditions at t = 0.  For details, see the above-cited report.

	The example presented here involves the following boundary conditions.  It is  planar geometry (x = 0), a long channel (in the z-direction) with a square cross-section in the x-y plane.  There are confining walls at x = 0 and 1, the sides, and at y = 0 and 1, the bottom and top.  At time t = 0 the top boundary wall impulsively starts to move to the right with velocity 1.0 (and continues to move in that direction indefinitely).  This velocity v = 1.0 is arbitrary, i.e.,  it is not the speed of light or even the speed of sound in the fluid.  The fluid is assumed to have viscosity n = 0.2.  Because of the sudden motion, the fluid (which was initially at rest) develops a swirling clockwise flow which rapidly approaches a steady state condition.

	The SOLA code, which is in the public domain, can be used to attack other problems.  The above-cited report also presents examples of fluid flow about a cylindrical can, flow of air (wind) over a recessed highway, and flow in a water-cooled power reactor.

		Dick Silbar
		silbar@cantina.lanl.gov