402project.html

Interrelations between The Equation of Energy in Terms of Q (Eq B.8), & The Equation of Energy for Pure Newtonian Fluids with Constant Density and Thermal Conductivity (Eq B.9) (The equation are from Transport Phenomena (ed. 2), by Bird, Stewart, and Lightfoot [2002])

By: Justin Yanosik and Robert Gillette

> restart;

Below is the right hand side of Equation B.9-1,

> eq1:=k*(diff(diff(T(x,y,z),x),x)+diff(diff(T(x,y,z),y),y)+diff(diff(T(x,y,z),z),z))+mu*Iov

> Here we have the value of Iov, the dissipation function for Newtonian Fluids, as given in equation B.7-1 in BS&L

Iov:=2*(diff(vx(x,y,z),x)^2+diff(vy(x,y,z),y)^2+diff(vz(x,y,z),z)^2)+(diff(vy(x,y,z),x)+diff(vx(x,y,z),y))^2+(diff(vz(x,y,z),y)+diff(vy(x,y,z),z))^2+(diff(vx(x,y,z),z)+diff(vz(x,y,z),x))^2-(2/3)*(diff(vx(x,y,z),x)+diff(vy(x,y,z),y)+diff(vz(x,y,z),z))^2;

>

> Below are the value for Q, heat flux, in the x, y, and z direction, using Fourier's Law

> qx:=-k*diff(T(x,y,z),x);

> qy:=-k*diff(T(x,y,z),y);

> qz:=-k*diff(T(x,y,z),z);

Below is the right hand side of Equation B.8-1 in BS&L, where constant density is assumed for the fluid, and therefore the term goes to 0

> eq2:=-(diff(qx,x)+diff(qy,y)+diff(qz,z))-taudelv;

Below is the eqation for Tau-dotted with a gradient of velocity, this is equation C, in Table A.7-1 in BS&L

> taudelv:=tauxx*diff(vx(x,y,z),x)+tauxy*(diff(vx(x,y,z),y))+tauxz*diff(vx(x,y,z),z)+tauyx*diff(vy(x,y,z),x)+tauyy*diff(vy(x,y,z),y)+tauyz*diff(vy(x,y,z),z)+tauzx*diff(vz(x,y,z),x)+tauzy*diff(vz(x,y,z),y)+tauzz*diff(vz(x,y,z),z);

Below is the right hand side of Equation 1 in expanded form

> eq1;

Below is the right hand side of Equation 2 in expanded form

> eq2;

Here is the value of the gradient of the velocity, this is equation B.1-7 in BS&L

> delv:=diff(vx(x,y,z),x)+diff(vy(x,y,z),y)+diff(vz(x,y,z),z);

>

Here is the value of Tauxx from equation B.1-1 in BS&L

> tauxx:=-mu*(2*diff(vx(x,y,z),x))+2/3*mu*delv;

Here is Tauyy from equation B.1-2 in BS&L

> tauyy:=-mu*(2*diff(vy(x,y,z),y))+2/3*mu*delv;

Here is Tauzz from equation B.1-3 in BS&L

> tauzz:=-mu*(2*diff(vz(x,y,z),z))+2/3*mu*delv;

Below is Tauxy, from equation B.1-4 in BS&L, as a rule, Tauxy=Tauyx

> tauxy:=-mu*(diff(vy(x,y,z),x)+diff(vx(x,y,z),y));

> tauyx:=tauxy;

Below is Tauyz, from equation B.1-5 in BS&L, and again Tauyz=Tauzy

> tauyz:=-mu*(diff(vz(x,y,z),y)+diff(vy(x,y,z),z));

> tauzy:=tauyz;

Below is Tauzx, from equation B.1-6 in BS&L, and again Tauzx=Tauxz

> tauzx:=-mu*(diff(vx(x,y,z),z)+diff(vz(x,y,z),x));

> tauxz:=tauzx;

Here is equation B.9-1 in a further expanded form

> eq1;

Here is equation B.8-1 in its final expanded form

> eq2;

Subtract the right hand side of equation B.8-1 from the right hand side of equation B.9-1 to determine if they are the same

> mayber:=eq1-eq2;

Simplifying...

> simplify(mayber);

>

When our expanded versions of Equation B.8-1 is subtracted from Equation B.9-1, the end result is 0, therefore the equations can be determined to be the same under the given condition of constant density.

Last Updated April 17, 2002 by Justin Yanosik and Robert Gillette