Example 11.4-7 One Dimensional Compressible Flow > restart; > econt:=D(rhovx)(x)=0; Continuity > emot:=rhovx(x)*D(vx)(x)=-D(p)(x)+(4/3)*mu*D(D(vx))(x); Motion > een:=rhovx(x)*Cp*D(T)(x)=k*D(D(T))(x)+vx(x)*D(p)(x)+(4/3)*mu*(D(vx)(x))^2; Energy > s:=dsolve({econt,rhovx(0)=v1*rho1},rhovx(x)); rho*vx is constant > assign(s);rhovx:=unapply(rhovx(x),x); > een2:=lhs(emot)*vx(x)+lhs(een)=rhs(emot)*vx(x)+rhs(een); Getting rid of the pressure term in the energy equation by using the equation of motion. > een2:=simplify(%/(v1*rho1)); > int(een2,x); We can not integrate it. Error, (in int) wrong number (or type) of arguments > int(lhs(een2),x); We can't even integrate one side of the equation. > int(D(vx)(x)*vx(x)+Cp*D(T)(x),x); > int(rhs(een2),x); > diff(Cp*T(x)+(1/2)*vx(x)^2,x)-lhs(een2); Going the other direction to check on the validity of BS&L's answer in 10.5-57 > simplify(%); The left side checks.
> dif:=k/(Cp*rho1*v1)*diff((4/3)*Pr*D((vx^2)/2)(x)+Cp*D(T)(x),x)-rhs(een2); > simplify(%); > Cp:=Pr*k/mu; Replacing Cp with the Prandtl Number > simplify(dif); The rest of 10.5-57 checks.
> Pr:=3/4; Suggested as the only value for which we can get a solution. > een2; Here is our equation.
> T:=x->(C1+C2*exp(rho1*v1*Cp*x/k)-vx(x)^2/2)/Cp; Here is the solution in 10.5-58 > een2; Is this OK? > simplify(%); Yes. The solution given as 10.5-58 in the text satisfies the energy equation.
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