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Extra info for Physical and Computational Aspects of Convective Heat Transfer
Example text
Therefore, RATE OF INCREASE OF X -COMPONENT MOMENTUM OF FLUID a = at(pu)dxdydz. 4) INCV The rate of flow of x-component momentum into the CV through the face whose center is R (perpendicular to the y direction) is the x-component momentum per unit mass, u, times the rate of mass flow through the face, pv dx dz, that is, puvdxdz. Note the distinction between the direction of the momentum component and the direction of the component of mass flow rate that transports the momentum, x and y, respectively, in this case.
1. Infinitesimal control volume for derivation of conservation of mass. Momentum Equations 21 Similarly the net rate of flow of mass out of the CV through the faces normal to the y axis is apu ay dxdydz. Inserting these expressions into Eq. 1), we get ap -dxdydz at = - ( -apu ax + -apu) dxdydz. 2b) in steady or unsteady flow. For a steady compressible flow, Eq. 2a) reduces to apu apu _ 0 ax + ay - . 2 Momentum Equations The momentum-conservation equations, known as the Nauier-Stokes equations, can be derived by similar use of CV concepts.
Changing x to y and y to x, u to v and v to u, throughout). 3 Show that a a a d -+u-+v-=at ax ay dt represents the rate of change, with respect to time, as seen by an observer following the motion of a fluid element. 4 By extending the arguments used to derive the x-component momentum equation for two-dimensional flow, Eq. 5 Show that the term sional flow, to a( 0iju;)/ aXj ao U ax in Eq. 25) can be expanded, in two-dimen- aovxv ax aoxl' u ay ao),),v ay ~+-'-+--'-+-- and hence, using Eqs. 6) for a Newtonian fluid, to /L + a 2v2) + au 2 aIL + av 2 aIL + ~ [( _ ~ ) "V u ] ( a2u2 ax2 ay2 ax ax ay ay dt P 3/L + a/L] +/Lv---a;-+/LUay ae ae (P- 32)/L ("Vu) +/Lexl,+exy [a/L vax+u ay 2 2 xy xy where "VU= au/ax+ av/ay, and d / dt is the transport operator defined in Eq.