Unformatted text preview: till another way of stating Pascal's law is that the
pressure is the same for any two points at the same elevation in a continuous body of fluid. The following figure illustrates this point. 2 Hydrostatics.nb Stress in Hydrostatic Systems
The mathematical statement of Cauchy's First Law is a statement for the balance of linear momentum
of a fluid body
Ρv V
t m Ρb V t m t t m n (1) A t We showed in a previous set of notes that the stress vector t n at a point in a fluid can be expressed in
terms of the stress tensor T:
t nT n (2) where the stress tensor T is defined as a sum of dyadic products
T Tij ei ej For a fluid at rest Eqn. (1) becomes
Ρb V 0
m t t m n A (3) t And since a fluid at rest cannot support any shear forces, the stress vector becomes
t pn n (4) where p is the hydrostatic pressure. To show that the pressure is the same in every direction we consider a small tetrahedron shown in Figure 2 D C
A B Our tetrahedron has 3 mutually orthogonal faces (ADC, ABD, ABC) with areas x, y, z , and a slant face (BDC) with area
. The outward directed unit normals to these 4 faces are i, j, k, and n.
We apply (4) to our tetrahedron to get
Ρg V 0
m t t
m n A (5) t Since the volume of our tetrahedron is small we express the volume integral in terms of average values: Hydrostatics.nb Ρg V Ρg V (6) t m Let the stress acting on the face with outward directed normal i be t
with normals j and k be t
t
m A n t t and t j t i where we have used the fact that t
related to the area of the slant face
in
jn
kn z j t t y j py x , and in a similar way on faces A k t n A z t x i px y A j y t i . Thus the surface integral in (6) can be written as k A i x x 3 y k pz k t z z (7)
n n pn p n . The surface areas of the 3 mutually orthogonal faces rae
by simple projections nx
ny
nz (8) Thus our balance of hydrostatic forces becomes
Ρg 0 V n pn Now we divide through by
0 n pn i nx px j ny py and take the limit V/ i nx px j ny py k nz pz (9) 0 to get k nz pz (10) Recall that
n i nx j ny k nz (11) Substituting for n in (11) we get
0 i nx pn px j ny pn py k nz pn pz (12) If we take the scalar product of (13) with i , j , k respectively we get
pn px 0, pn py 0, pn pz 0 (13) Hence the stress vector on any arbitrary surface in a fluid at rest is given by
t pn n (14) Note that the stress vector t n acts in a direction opposite to that defined by the unit normal...
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 Spring '14
 Fluid Mechanics, Force, Blaise Pascal

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