Algebraic equations such as Bernoulli's relation, Bq. (1) of Example 1.3, are dimensionally consistent, but what about differential equations? Consider, for example, the boundary-layer x-momentum equation, first derived by Ludwig Prandtl in 1904: ρ u ∂ u ∂ x + ρ v ∂ u ∂ y = − ∂ p ∂ x + ρ g x + ∂ τ ∂ y where t is the boundary-layer shear stress and g x is the component of gravity in the x direction. Is this equation dimen-sionally consistent? Can you draw a general conclusion?
Algebraic equations such as Bernoulli's relation, Bq. (1) of Example 1.3, are dimensionally consistent, but what about differential equations? Consider, for example, the boundary-layer x-momentum equation, first derived by Ludwig Prandtl in 1904: ρ u ∂ u ∂ x + ρ v ∂ u ∂ y = − ∂ p ∂ x + ρ g x + ∂ τ ∂ y where t is the boundary-layer shear stress and g x is the component of gravity in the x direction. Is this equation dimen-sionally consistent? Can you draw a general conclusion?
Algebraic equations such as Bernoulli's relation, Bq. (1) of Example 1.3, are dimensionally consistent, but what about differential equations? Consider, for example, the boundary-layer x-momentum equation, first derived by Ludwig Prandtl in 1904:
ρ
u
∂
u
∂
x
+
ρ
v
∂
u
∂
y
=
−
∂
p
∂
x
+
ρ
g
x
+
∂
τ
∂
y
where t is the boundary-layer shear stress and gx is the component of gravity in the x direction. Is this equation dimen-sionally consistent? Can you draw a general conclusion?
Explain in your own understanding how Bernoulli’s principle works in aviation.
A sea-level automobile tire is initially at 32 lbf/in2 gagepressure and 75 °F. When it is punctured with a hole that resemblesa converging nozzle, its pressure drops to 15 lbf/in2gage in 12 min. Estimate the size of the hole, in thousandthsof an inch. The tire volume is 2.5 ft2.
Example 1.10 gave an analysis that predicted that theviscous moment on a rotating disk M = πμ V R 4 /(2 h ). Ifthe uncertainty of each of the four variables ( μ , Ω,, R , h )is 1.0 percent, what is the estimated overall uncertaintyof the moment M ?( a ) 4.0 percent ( b ) 4.4 percent ( c ) 5.0 percent( d ) 6.0 percent ( e ) 7.0 percent
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