Section 10.6 Dimensional Analysis of the Equations of Change

by Kim Little and Felice Shieh

> restart;

> with(linalg):

DEFINING THE VECTOR VARIABLES

Velocity Vector in Three Dimentions

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

Direction and Gradient Vectors

> s:=vector([x,y,z]);

> m:=vector([x,y,z,t]);

Free Convection Pressure Vector in Three Dimensions

> p:=(x,y,z)->vector([px(x,y,z),py(x,y,z),pz(x,y,z)]);

Gravitational Force Vector in Three Dimensions

> G:=(x,y,z)->vector([Gx(x,y,z),Gy(x,y,z),Gz(x,y,z)]);

THE DIMENSIONAL EQUATIONS

Equation of Continuity: Eqn. 10.6-1

> eqncont:=diverge(v(x,y,z,t),s)=0;

Equation of Motion : Eqn. 10.6-2

> eqnmotion:=rho*(D[4](vx)(x,y,z,t)+D[4](vy)(x,y,z,t)+D[4](vz)(x,y,z,t))=mu*(diverge(grad(vx(x,y,z,t),s),s)+diverge(grad(vy(x,y,z,t),s),s)+diverge(grad(vz(x,y,z,t),s),s))-diverge(p(x,y,z),s)+rho*(Gx(x,y,z)+Gy(x,y,z)+Gz(x,y,z));

Equation of Energy: Eqn. 10.6-3

> eqnen:=rho*Cp*D[4](T)(x,y,z,t)=k*diverge(grad(T(x,y,z,t),m),m)+mu*((D[2](vx)(x,y,z,t)+D[1](vy)(x,y,z,t))^2+(D[3](vx)(x,y,z,t)+D[1](vz)(x,y,z,t))^2+(D[3](vy)(x,y,z,t)+D[2](vz)(x,y,z,t))^2);

TRANSFORMATION TO DIMENSIONLESS EQUATIONS USING DCHANGE

> with(PDEtools,dchange);

Transformation to Dimensionless Variables for Continuity: Eqns. 10.6-4, 10.6-8

> tr:={vx=vxstar*V,vy=vystar*V,vz=vzstar*V, x=xstar*D1,y=ystar*D1,z=zstar*D1};

Use Dchange to Tranform Continuity Equation into Dimensionless Form

> newconteqn:=dchange(tr,eqncont,[vxstar(xstar,ystar,zstar,tstar),vystar(xstar,ystar,zstar,tstar),vzstar(xstar,ystar,zstar,tstar),xstar,ystar,zstar]);

Simplify to Dimensionaless Equation of Continuity: Eqn. 10.6-9 (with gradient term expanded)

> simplify(newconteqn/V*D1);

Transformation to Dimensionless Variables for Motion: Eqns. 10.6-4, 10.6-8, 10.6-5, 10.6-6

> tr1:={vx=vxstar*V,vy=vystar*V,vz=vzstar*V, x=xstar*D1,y=ystar*D1,z=zstar*D1,px=pxstar*rho*V^2+p0,py=pystar*rho*V^2+p0,pz=pzstar*rho*V^2+p0,t=tstar*D1/V};

Use Dchange to Tranform Motion Equation into Dimensionless Form

> neweqnmot:=dchange(tr1,eqnmotion,[vxstar(xstar,ystar,zstar,tstar),vystar(xstar,ystar,zstar,tstar),vzstar(xstar,ystar,zstar,tstar),xstar,ystar,zstar,tstar,pxstar(xstar,ystar,zstar),pystar(xstar,ystar,zstar),pzstar(xstar,ystar,zstar)]);

Define Reynold's Number, Brinkman's Number, and Prandtl's Number

> mu:=D1*V*rho/Ren;D1:=V^2/(g*Fr);k:=mu*Cp/Pr;T1:=mu*V^2/(k*Br)+T0;

Sustitute in Reynold's Number, Brinkman's Number, and Prandtl's Number into Equation of Motion

> dimlessmotion:=simplify(neweqnmot);

Simplify to Dimensionless Equation of Motion: Eqn. 10.6-10...Note: Maple will not factor out and cancel (g*Fr), but the equation is the same

> simplify(dimlessmotion/(rho*g*Fr));

Transformation to Dimensionless Variables for Energy: Eqns. 10.6-4, 10.6-8, 10.6-7

> tr2:={vx=vxstar*V,vy=vystar*V,vz=vzstar*V, x=xstar*D1,y=ystar*D1,z=zstar*D1,t=tstar*D1/V,T=Tstar*(T1-T0)+T0};

Use Dchange to Tranform Energy Equation into Dimensionless Form

> neweqnen:=dchange(tr1,eqnen,[vxstar(xstar,ystar,zstar,tstar),vystar(xstar,ystar,zstar,tstar),vzstar(xstar,ystar,zstar,tstar),xstar,ystar,zstar,tstar,Tstar(xstar,ystar,zstar,tstar)]);

Simplify to Dimensionless Equation of Energy: Eqn. 10.6-11

> simplify(neweqnen*Br/Pr/(V*rho*g*Fr));

>