Navier-Stokes equations spherical coordinates

Thus we begin by considering the full Navier-Stokes equation expressed in terms of the streamfunction characteristic velocity and the sphere radius a as a characteristic length scale. Using spherical coordinates, with ij = cos 9, this equation is... 

The continuity and Navier-Stokes equations in cylindrical and spherical coordinate systems are given in Supplement 5. 

In chemical technology, one often meets the problem about a spherically symmetric deformation (contraction or extension) of a gas bubble in an infinite viscous fluid. In the homobaric approximation (the pressure is homogeneous inside the bubble) [306, 312], only the motion of the outer fluid is of interest. The Navier-Stokes equations describing this motion in the spherical coordinates have the form... 

The three Navier-Stokes equations can be put in very compact form by using the shorthand notation of vector calculus [6, p. 66 7 8, p. 80]. Furthermore, it is often convenient to use these equations in polar or spherical coordinates their transformations to those coordinate systems are shown in many texts [6, p. 66 8, p. 80]. The corresponding equations for fluids with variable density are also shown in numerous texts [6, p. 66 7 8, p. 80]. If we set /A = 0 in the Navier-Stokes equations, thus dropping the rightmost term, we find the Euler equation which is often used for three-dimensional flow where viscous effects are negligible. 

For flow past a sphere the stream function ij/ can be used in the Navier-Stokes equation in spherical coordinates to obtain the equation for the stream function and the velocity distribution and the pressure distribution over the sphere. Then by integration over the whole sphere, the form drag, caused by the pressure distribution, and the skin friction or viscous drag, caused by the shear stress at the surface, can be summed to give the total drag. 

When the bubble surface does not deform, it is relatively easy to solve the problem on a fixed grid. Balasubramanian and Lavery [41] used the fixed grid method to study the thermocapillary migration of a spherical bubble under microgravity for large Reynolds and Marangoni numbers. The problem they studied was a stationary bubble surrounded by the liquid with a steady state velocity field. The full Navier-Stokes equation was solved in a three-dimensional spherical (r,9,(j)) coordinate system. The origin of the coordinate is at the center of mass of the bubble. 

In order to formulate the flow equations for a fluid, for instance, for the gas in the cyclone or swirl tube, we must balance both mass and momentum. The mass balance leads to the equation of continuity the momentum balance to the Navier-Stokes equations for an incompressible Newtonian fluid. When balancing momentum, we have to balance the x-, y- and -momentum separately. The fluid viscosity plays the role of the diffusivity. Books on transport phenomena (e.g. Bird et ah, 2002 Slattery, 1999) will give the full flow equations both in Cartesian, cylindrical and spherical coordinates. 

Thus, in a manner entirely equivalent to the two-dimensional analysis, we seek rescaled equations in the inner (boundary-layer) region very near to the sphere surface within which the tangential velocity goes from the potential-flow value (3/2)sin0 to 0 at the body surface. The only difference from the previous analysis is in the detailed form of the Navier-Stokes and continuity equations for axisymmetric geometries. When expressed in terms of spherical coordinates, these equations are... 

The equations can also be written using cylindrical and spherical coordinates. The solutions to the equations are called velocity fields or flow fields. The equations were developed by Claude-Louis Navier (1785-1836) in 1822, and developed further by George Stokes (1819-1903), and find many applications including the study of the flow of fluids in pipes and over surfaces. 

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