This is for the numerical solution of the 2D Navier Stokes Equation. The numerical solution will be in a closed two dimensional given gemoetry, called the boundary. There may be additional two dimensional objects inside and possibly attached to the boundary. Cartesian coordinates are x, y using units of meters. Time t is in units of seconds. In some solutions a cell may be at a specific time ta and rectangular xi to xi+dx, yj to yj+dy. In other solutions a cell may be a triangle or other simple two dimensional geometry. The entire area within the boundary is cells or objects. Velocity has units meters per second. Acceleration has units meters per second squared. Area has units square meters. Volume has units cubic meters Rho density has units killogram per cubic meter. mu dynamic viscosity, depends on fluid, has units kilogram per meter second nu kinematic viscosity = mu/Rho has units meters^2 per second p pressure has units killogram per square meter. F force has units newton, killogram meters per second squared. F = m a Fx, Fy below are just the acceleration ux velocity in x direction has units meters per second. For all cells in the boundary, ux(x,y) is a function value. uy velocity in y direction has units meters per second. For all cells in the boundary, uy(x,y) is a function value. Each cell has Fx, Fy on every edge of the cell. The pressure on each edge of a cell is the normal force vector divided by the length of the edge. Renolds number dimensionless Re = rho*v*l/mu = v*l/nu v velocity meters per second in direction of flow l length meters, e.g. diameter of pipe mu dynamic viscosity kilogram per meter second nu kinematic viscosity meters^2 per second less than 2000 laminar flow (can depend on shape) greater than 4000 turbulent flow mu for air = 18.27 kg/ms from 17.4 to 18.6 0 to 27 deg C mu water at 20 degrees centigrade 1.002 from 1.3 to 0.28 10 to 100 deg C rho density for air = 1.225 kg/m^3 at 15 deg C 1.204 kg/m^3 at 20 deg C rho density water = 1000 kg/m^3 air pressure at sea level 10.13 killogram per square centimeter = 101,300 killogram per square meter water pressure increases 14.5 psi 2.05 kg/cm^2 20500 kg/m^2 every 10.06 meter more depth speed of sound in air = 331.45 meter per second 100 miles per hour = 44.7 meter per second see cs455/femjava/nav1* 3D example For static in 2D u(x,y) = velocity in x direction v(x,y) = velocity in y direction p(x,y) = presseure at point x,y Conservation of mass for incompressible fluid du(x,y)/dx + dv(x,y)/dy = 0 flow non linear differential equations u(x,y)*du(x,y)/dx + v(x,y)*du(x,y)/dy = Fx - 1/rho dp(x,y)/dx + mu*(d^2u(x,y)/dx^2 + d^2u(x,y)/dy^2) u(x,y)*dv(x,y)/dx + v(x,y)*dv(x,y)/dy = Fy - 1/rho dp(x,y)/dy + mu*(d^2v(x,y)/dx^2 + d^2v(x,y)/dy^2) For dynamic in 2D d is for partial derivative d^2 second derivative u(x,y,t) = velocity in x direction v(x,y,t) = velocity in y direction p(x,y,t) = presseure at point x,y,t Conservation of mass for incompressible fluid du(x,y,t)/dx + dv(x,y,t)/dy = 0 flow non linear differential equations du(x,y,t)/dt + u(x,y,t)*du(x,y,t)/dx + v(x,y,t)*du(x,y,t)/dy = Fx - 1/rh0 dp(x,y,t)/dx + mu*(d^2u(x,y,t)/dx^2 + d^2u(x,y,t)/dy^2) dv(x,y,t)/dt + u(x,y)*dv(x,y,t)/dx + v(x,y,t)*dv(x,y,t)/dy = Fy - 1/rho dp(x,y,t)/dy + mu*(d^2v(x,y,t)/dx^2 + d^2v(x,y,t)/dy^2) For static in 3D d is for partial derivative d^2 second derivative u(x,y,z) = velocity in x direction v(x,y,z) = velocity in y direction w(x,y,z) = velocity in z direction p(x,y,z) = presseure at point x,y,z Conservation of mass for incompressible fluid du(x,y,z)/dx + dv(x,y,z)/dy + dw(x,y,z) = 0 flow non linear differential equations u(x,y,z)*du(x,y,z)/dx + v(x,y,z)*du(x,y,z)/dy + w(x,y,z)*du(x,y,z)/dz = Fx - 1/rh0 dp(x,y,z)/dx + mu*(d^2u(x,y,z)/dx^2 + d^2u(x,y,z)/dy^2 + d^2u(x,y,z)/dz^2) u(x,y,z)*dv(x,y,z)/dx + v(x,y,z)*dv(x,y,z)/dy + w(x,y,z)*dv(x,y,z)/dz = Fy - 1/rho dp(x,y,z)/dy + mu*(d^2v(x,y,z)/dx^2 + d^2v(x,y,z)/dy^2 + d^2v(x,y,z)/dw^2) u(x,y,z)*dw(x,y,z)/dx + v(x,y,z)*dw(x,y,z)/dy + w(x,y,z)*dw(x,y,z)/dz = Fz - 1/rho dp(x,y,z)/dy + mu*(d^2w(x,y,z)/dx^2 + d^2w(x,y,z)/dy^2 + d^2w(x,y,z)/dw^2) For dynamic in 3D d is for partial derivative d^2 second derivative u(x,y,z,t) = velocity in x direction v(x,y,z,t) = velocity in y direction w(x,y,z,t) = velocity in z direction p(x,y,z,t) = presseure at point x,y,z,t Conservation of mass du(x,y,z,t)/dx + dv(x,y,z,t)/dy + dw(x,y,z,t)/dz = 0 flow non linear differential equations du(x,y,z,t)/dt + u(x,y,z,t)*du(x,y,z,t)/dx + v(x,y,z,t)*du(x,y,z,t)/dy + w(x,y,z,t)*du(x,y,z,t)/dz = Fx - 1/rh0 dp(x,y,t)/dx + mu*(d^2u(x,y,z,t)/dx^2 + d^2u(x,y,z,t)/dy^2 + d^2u(x,y,z,t)/dw^2) dv(x,y,z,t)/dt + u(x,y,z,t)*dv(x,y,z,t)/dx + v(x,y,z,t)*dv(x,y,z,t)/dy + w(x,y,z,t)*dv(x,y,z,t)/dz = Fy - 1/rho dp(x,y,z,t)/dy + mu*(d^2v(x,y,z,t)/dx^2 + d^2v(x,y,z,t)/dy^2 + d^2v(x,y,z,t)/dw^2) dw(x,y,z,t)/dt + u(x,y,z,t)*dw(x,y,z,t)/dx + v(x,y,z,t)*dw(x,y,z,t)/dy + w(x,y,z,t)*dw(x,y,z,t)/dz = Fz - 1/rho dp(x,y,z,t)/dz + mu*(d^2w(x,y,z,t)/dx^2 + d^2w(x,y,z,t)/dy^2 + d^2w(x,y,z,t)/dw^2)