Equation of Continuity and the Derivative of a Scalar following the Motion:

Section 3.1 in BS&L

> restart;

> with(linalg):

> Ds:=(s,v)->diff(s,t)+v[1]*diff(s,x)+v[2]*diff(s,y)+v[3]*diff(s,z); The derivative of a scalar: s following the motion of a fluid moving

with velocity v.

> v:=vector([vx(x,y,z,t),vy(x,y,z,t),vz(x,y,z,t)]);

> Ds(s(x,y,z,t),v); This gives the derivative following the fluid motion for a scalar s

> DDv:=v->diverge(v,vector([x,y,z]));

> DDv(v); The divergence of the vector v wrt position vector [x,y,z]

> eqcont1:=diff(rho(x,y,z,t),t)+DDv(evalm(rho(x,y,z,t)*v)); eq 3.1-4

> eqcont2:=Ds(rho(x,y,z,t),v)+rho(x,y,z,t)*DDv(v); eq 3.1-6

> simplify(eqcont1-eqcont2); Showing the 3.1-4 and 3.1-6 are identical.

> eqcon2a:=(rho,v)->Ds(rho,v)+rho*DDv(v); A more general form of the continuity equation that can be used for simplified cases.

> eqcon2a(rho(x,y,z,t),vector([vx(x,y,z,t),vy(x,y,z,t),vz(x,y,z,t)]));

> simplify(%-eqcont2);

> eqcon2a(rho(x,y),vector([vx(x,y),vy(x,y),0])); For steady flow in the x and y directions

> eqcon2a(rho,vector([vx(x,y),vy(x,y),0])); For the same case, but with rho constant.

>