By J. A. Fox (auth.)
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Extra resources for An Introduction to Engineering Fluid Mechanics
Au ax 8x 8t B ,~L B 1f ___ k8y8t ay C' c Y I I 8y I 8x 1 o x u Fig. S In Fig. 4 normal strain is occurring: point B moves relative to point 0 (also a moving point) by an amount of (av/ay) 8y 8t in time 8t. Similarly point A moves relative to point 0 an amount (au/ax) l)x [)t. 36 An Introduction to Engineering Fluid Mechanics If the fluid is incompressible, area liy lix must be constant (considering a two dimensional case) av au ay ay at ax =- ax ax at ay au + av ax ay =0 which is another proof of the continuity equation.
U+ ax oX ~ y v x Fig. 1 29 30 An Introduction to Engineering Fluid Mechanics Fluid entering face AEGC will have a mass density Px and when it leaves across face BFHD it will have a mass density of Px + (apx/ax) 3x. Similarly for faces ABDC and EFHG the densities will be pz and pz + (apz/az) oz and for faces CGHD and AEFB Py and Py + (apy/ay) oy. The difference between the amount of fluid entering and leaving the element per second must be stored within the element and this is possible if the fluid within the element is compressible.
If the coordinate system is taken so that x and z are located in an horizontal plane then the X and Z body forces are zero and the Y body force is -wf, the specific weight of the fluid. 4) Also au au au 8u =-8x +-8y +-8z ax ay az when the flow is steady. 5) as u 2 + v2 + w2 = V2 where V is the local velocity. 5) is Bernoulli's equation. 3 Nonnal strain and deformation of a fluid element It is necessary to describe, mathematically, the displacements, distortions and rotations of any fluid element.
An Introduction to Engineering Fluid Mechanics by J. A. Fox (auth.)