LT 2 UT From this it follows thatma ρL3 U T ρU 2 L 2 inertial force On the

Lt 2 ut from this it follows thatma ρl3 u t ρu 2 l

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L/T 2 U/T . From this it follows that ma ρL 3 U T ρU 2 L 2 inertial force . On the other hand, we can estimate the viscous forces based on Newton’s law of viscosity: viscous force = τA μ ∂u ∂y A μ U L · L 2 μ UL . From this it follows that inertial force viscous force ρU 2 L 2 μ UL = ρ UL μ = Re . Froude Number In a similar manner we can argue that the Froude number, Fr , represents the ratio of inertial forces to gravitational forces. In particular, recalling that ρ g is gravitational force per unit volume and using the expression just obtained for inertial force, we have inertial force gravitational force ρU 2 L 2 ρgL 3 = U 2 gL = Fr 2 . Pressure Coefficient We next recall the pressure coefficient given earlier in Eq. (3.59). It is easily checked that this is actually a ratio of pressure force to inertial force. If we consider the force that would result from a pressure difference such as p p p , we have pressure force inertial force pL 2 ρU 2 L 2 = p ρU 2 = 1 2 C p , which is equivalent to Eq. (3.59).
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3.7. SUMMARY 99 Other Dimensionless Parameters There are many other dimensionless parameters that arise in in fluid dynamics although most are associated with very specific flow situations. One that is more general than most others is the Mach number, M , that arises in the study of compressible flows, especially in aerodynamic applications. The Mach number is defined as M flow speed sound speed = U c . We can use this to somewhat more precisely define what we mean by an incompressible flow. Recall that we have previously viewed a flow as incompressible if its density is constant. But it is sometimes preferable, especially in studies of combustion, to use a different approach. In particular, it can be shown that if the Mach number is less than approximately 0.3, no more than an approximately 10% error will be incurred by treating the flow as incompressible and, in particular, invoking the divergence-free condition. Thus, a “rule of thumb” is that flow can be considered incompressible if M 0 . 3, and otherwise it must be treated as compressible. It turns out that in many combusting flows M is very low, but at the same time the density is changing quite rapidly. Our first thought in analyzing such flows is that they cannot be incompressible because of the variable density. But the Mach number rule of thumb indicates that we can treat the flow as divergence free. We leave as an exercise to the reader demonstration that this is not inconsistent with using the compressible continuity equation to handle the variable density. The final dimensionless parameter we will mention here is the Weber number, denoted We . This is the ratio of inertial forces to surface tension forces and is given by inertial force surface tension force = ρU 2 L σ = We , where σ is surface tension. (Recall that σ is a force per unit length—rather than force per unit area—which leads to the factor L in the numerator.) As would be expected, this number is important for flows having a free surface such as occur at liquid-gas interfaces, but in the present lectures we will not be providing any further treatment of these.
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