<Text-field style="Text" size="14" bold="true" underline="true" layout="Heading 1"><Font bold="true" underline="true" encoding="UTF-8" size="14">1. Equa\303\247\303\243o de momentum da camada limite</Font></Text-field>

<Text-field style="Text" size="14" bold="true" underline="true" layout="Heading 1"><Font bold="true" underline="true" encoding="UTF-8" size="14">2. Especializa\303\247\303\243o no fluxo da placa plana soprada paralela</Font></Text-field>

Com a placa plana soprada em paralelo, n\303\243o h\303\241 curvatura do fluxo. Ou seja se aplica 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 e r \342\206\222 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmluZmluO0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.

Isso simplifica a equa\303\247\303\243o de momentum para:

subs(diff(c(x),x)=0,diff(r(x),x)=0,G3); G4:=%:

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtJkkmZGVsdGFHNiI2IyIiIzYjSSJ4R0YqRi4qJkkia0dGKiIiIikqKEkjbnVHRipGMSZJImNHRio2I0kpaW5maW5pdHlHRiUhIiJGJ0Y5SSZzaWdtYUdGKkYx

A solu\303\247\303\243o geral da equa\303\247\303\243o do momento \303\251:

delta[2](x)^(sigma+1)=(1+sigma)*k*(nu/c[infinity])^sigma*x; G5:=%:

LyktJkkmZGVsdGFHNiI2IyIiIzYjSSJ4R0YnLCZJJnNpZ21hR0YnIiIiRi5GLioqRixGLkkia0dGJ0YuKSomSSNudUdGJ0YuJkkiY0dGJzYjSSlpbmZpbml0eUclKnByb3RlY3RlZEchIiJGLUYuRitGLg==

simplify(solve(G5,{delta[2](x)})[1]); G6:=%:

Ly0mSSZkZWx0YUc2IjYjIiIjNiNJInhHRiYpKiosJkkmc2lnbWFHRiYiIiJGL0YvRi9JImtHRiZGLykqJkkjbnVHRiZGLyZJImNHRiY2I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRHISIiRi5GL0YqRi8qJEYtRjk=

Resolvido depois 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 voc\303\252 obt\303\251m:

simplify(G6/x,symbolic); G7:=%:

LyomSSJ4RzYiISIiLSZJJmRlbHRhR0YlNiMiIiM2I0YkIiIiKiwpLCZJJnNpZ21hR0YlRi1GLUYtKiRGMEYmRi0pSSJrR0YlRjJGLSlJI251R0YlKiZGMUYtRjBGJkYtKSZJImNHRiU2I0kpaW5maW5pdHlHJSpwcm90ZWN0ZWRHLCRGN0YmRi0pRiRGPkYt

Voc\303\252 tamb\303\251m pode usar o n\303\272mero RE da placa LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEjUkVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictRiM2JC1GLzYlUSJ4RidGMkY1L0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRj0= escrever :

simplify(subs(c[infinity]=RE[x]*nu/x,G7),symbolic); G8:=%:

LyomSSJ4RzYiISIiLSZJJmRlbHRhR0YlNiMiIiM2I0YkIiIiKigpLCZJJnNpZ21hR0YlRi1GLUYtKiRGMEYmRi0pSSJrR0YlRjJGLSkmSSNSRUdGJUYsLCQqJkYxRi1GMEYmRiZGLQ==

A equa\303\247\303\243o resultante (2.5) pode ser usada para calcular a espessura de perda de momento para escoamentos laminares e turbulentos.

<Text-field style="Heading 2" layout="Heading 2">a) fluxo laminar</Text-field>

No caso de fluxo laminar, o seguinte se aplica aos par\303\242metros em (2.5) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmc2lnbWE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjFGJ0Y3RjdGK0Y3 , LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEmMC4yMzVGJ0Y+Rj5GK0Y+ . Isso d\303\241 a espessura da perda de pulso

evalf(subs(sigma=1,k=0.235,G8)); G9:=%:

LyomSSJ4RzYiISIiLSZJJmRlbHRhR0YlNiMiIiM2I0YkIiIiLCQqJiQiKylmYWMmbyEjNUYtKSZJI1JFR0YlRiwjRi1GK0YmRi0=

e a espessura do deslocamento 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 :

delta[1](x)/x=evalf(2.55*rhs(G9)); G9a:=%:

LyomLSZJJmRlbHRhRzYiNiMiIiI2I0kieEdGJ0YpRishIiIsJComJCIrQT4+WzwhIipGKSkmSSNSRUdGJ0YqI0YpIiIjRixGKQ==

Agora voc\303\252 pode ver a espessura da camada limite LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJmRlbHRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== calcular:

delta(x)/x=delta[1](x)/x/0.3; G10:=%:

LyomLUkmZGVsdGFHNiI2I0kieEdGJiIiIkYoISIiLCQqKCQiK0xMTExMISIqRiktJkYlNiNGKUYnRilGKEYqRik=

subs(G9a,G10); G11:=%:

LyomLUkmZGVsdGFHNiI2I0kieEdGJiIiIkYoISIiLCQqJiQiKzFrSUZlISIqRikpJkkjUkVHRiZGJyNGKSIiI0YqRik=

Em desvio da solu\303\247\303\243o exata (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSgmZGVsdGE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiRRIjVGJ0Y3RjdGK0Y3) a espessura da camada limite \303\251 determinada um pouco maior aqui.A abordagem agora se aplica \303\240 tens\303\243o de cisalhamento da parede

tau[W]/rho/c[infinity]^2=0.235/RE[x]/(delta[2](x)/x); G12:=%:

LyooJkkkdGF1RzYiNiNJIldHRiYiIiJJJHJob0dGJiEiIikmSSJjR0YmNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiI0YrLCQqKiQiJE4jISIkRikmSSNSRUdGJjYjSSJ4R0YmRitGO0YpLSZJJmRlbHRhR0YmNiNGMkY6RitGKQ==

H1:=G9*x: subs(H1,G12); G13:=%:

LyooJkkkdGF1RzYiNiNJIldHRiYiIiJJJHJob0dGJiEiIikmSSJjR0YmNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiI0YrLCQqJiQiKyt0I3lVJCEjNUYpKSZJI1JFR0YmNiNJInhHRiYjRilGMkYrRik=

O coeficiente de fric\303\247\303\243o da camada limite agora resulta de

c[f]=subs(x=l,tau[W]/rho/c[infinity]^2)*2*(sigma+1); G14:=%:

LyZJImNHNiI2I0kiZkdGJSwkKiwiIiMiIiImSSR0YXVHRiU2I0kiV0dGJUYrSSRyaG9HRiUhIiIpJkYkNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkR0YqRjEsJkkmc2lnbWFHRiVGK0YrRitGK0Yr

subs(sigma=1,x=l, subs(G13*rho*c[infinity]^2,G14));

LyZJImNHNiI2I0kiZkdGJSwkKiYkIis/NDhyOCEiKiIiIikmSSNSRUdGJTYjSSJsR0YlI0YtIiIjISIiRi0=

Na solu\303\247\303\243o exata da teoria da camada limite, o valor num\303\251rico \303\251 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 .

P1:=loglogplot(rhs(G9),RE[x]=1e2..1e4,axes=boxed): #P1; P2:=loglogplot(rhs(G9a),RE[x]=1e2..1e4,color=blue,axes=boxed): #P2; P3:=loglogplot(rhs(G11),RE[x]=1e2..1e4,color=green,axes=boxed): #P3; P4:=loglogplot(rhs(G13),RE[x]=1e2..1e4,color=brown,axes=boxed): #P4;

display([P1,P2,P3,P4],title=`laminare Grenzschicht`,titlefont=[TIMES,17],gridlines=true,thickness=2);

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As curvas no diagrama significam:

- vermelho - espessura de perda de pulso relacionada de acordo com (2.1.1)

- azul - espessura de deslocamento relacionada de acordo com (2.1.2)

- verde - espessura da camada limite relacionada de acordo com (2.1.4)

- marrom - tens\303\243o de cisalhamento da parede adimensional de acordo com (2.1.6)

<Text-field style="Heading 2" layout="Heading 2">b) fluxo turbulento em RE ~ 100000</Text-field>

Para fluxo turbulento (RE ~ 100000), os par\303\242metros se aplicam

sigma=2/(n+1);H2:=%: k=(1/C^(n)*n/((1+n)*(2+n)))^(2/(n+1));H3:=%:

L0kmc2lnbWFHNiIsJComIiIjIiIiLCZJIm5HRiRGKEYoRighIiJGKA==

L0kia0c2IikqKilJIkNHRiRJIm5HRiQhIiJGKSIiIiwmRilGK0YrRitGKiwmIiIjRitGKUYrRiosJComRi5GK0YsRipGKw==

n \303\251 o expoente do perfil de velocidade turbulenta. Para n\303\272meros RE em torno de 100.000, n = 7.

subs(n=7,H2),subs(C=8.74,n=7,H3); G15:=%:

NiQvSSZzaWdtYUc2IiMiIiIiIiUvSSJrR0YlJCIrPXUpb0QiISM2

Para a espessura de perda de momento da camada limite turbulenta, obtemos a partir de (2.5):

evalf(subs(G15,G8)); G16:=%:

LyomSSJ4RzYiISIiLSZJJmRlbHRhR0YlNiMiIiM2I0YkIiIiLCQqJiQiKyN6J1wwTyEjNkYtKSZJI1JFR0YlRiwjRi0iIiZGJkYt

A espessura do deslocamento \303\251 ent\303\243o:

delta[1](x)/x=delta[2](x)/x*F; G17:=%: F=(n+2)/n; H4:=%:

LyomLSZJJmRlbHRhRzYiNiMiIiI2I0kieEdGJ0YpRishIiIqKC0mRiY2IyIiI0YqRilGK0YsSSJGR0YnRik=

L0kiRkc2IiomLCYiIiMiIiJJIm5HRiRGKEYoRikhIiI=

subs(subs(n=7,H4),G16,G17); G18:=%:

LyomLSZJJmRlbHRhRzYiNiMiIiI2I0kieEdGJ0YpRishIiIsJComJCIrTChRY2olISM2RikpJkkjUkVHRidGKiNGKSIiJkYsRik=

Agora, a espessura da camada limite turbulenta pode ser calculada:

delta(x)/x=delta[1](x)/x*(n+1); G19:=%:

LyomLUkmZGVsdGFHNiI2I0kieEdGJiIiIkYoISIiKigtJkYlNiNGKUYnRilGKEYqLCZJIm5HRiZGKUYpRilGKQ==

subs(n=7,G18,G19); G20:=%:

LyomLUkmZGVsdGFHNiI2I0kieEdGJiIiIkYoISIiLCQqJiQiKycpNF4zUCEjNUYpKSZJI1JFR0YmRicjRikiIiZGKkYp

A tens\303\243o de cisalhamento da parede da camada limite turbulenta resulta da rela\303\247\303\243o

tau[W]/rho/c[infinity]^2=k*RE[x]^(-sigma)*(delta[2](x)/x)^(-sigma); G21:=%:

LyooJkkkdGF1RzYiNiNJIldHRiYiIiJJJHJob0dGJiEiIikmSSJjR0YmNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiI0YrKihJImtHRiZGKSkmSSNSRUdGJjYjSSJ4R0YmLCRJJnNpZ21hR0YmRitGKSkqJkY5RistJkkmZGVsdGFHRiY2I0YyRjhGKUY6Rik=

evalf(subs(H2,H3,G16,n=7,C=8.74,G21)); G22:=%:

LyooJkkkdGF1RzYiNiNJIldHRiYiIiJJJHJob0dGJiEiIikmSSJjR0YmNiNJKWluZmluaXR5RyUqcHJvdGVjdGVkRyIiI0YrLCQqJiQiK011UiUpRyEjNkYpKSZJI1JFR0YmNiNJInhHRiYjRikiIiZGK0Yp

Finalmente, obt\303\251m-se o coeficiente de atrito da camada limite

subs(G22*rho*c[infinity]^2,H2,x=l,n=7,G14);

LyZJImNHNiI2I0kiZkdGJSwkKiYkIismZSQqNEAoISM2IiIiKSZJI1JFR0YlNiNJImxHRiUjRi0iIiYhIiJGLQ==

O valor num\303\251rico em cf fica um pouco menos se voc\303\252 usar um m\303\251todo diferente!

P5:=loglogplot(rhs(G16),RE[x]=1e5..1e8,color=red,axes=boxed): #P5; P6:=loglogplot(rhs(G18),RE[x]=1e5..1e8,color=blue,axes=boxed): #P6; P7:=loglogplot(rhs(G20),RE[x]=1e5..1e8,color=green,axes=boxed): #P7; P8:=loglogplot(rhs(G22),RE[x]=1e5..1e8,color=brown,axes=boxed): #P8;

display([P5,P6,P7,P8],title=`turbulente Grenzschicht`,titlefont=[TIMES,17],gridlines=true,thickness=2);

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As curvas no diagrama 2 significam: - vermelho - espessura de perda de pulso relacionada de acordo com (2.2.3) - azul - espessura de deslocamento relacionada de acordo com (2.2.5) - verde - espessura da camada limite relacionada de acordo com (2.2.7) - marrom - tens\303\243o de cisalhamento da parede adimensional de acordo com (2.2.9)