Chapter2 - 57:020 Fluid Mechanics Professor Fred Stern Fall...

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57:020 Fluid Mechanics Chapter 2 Professor Fred Stern Fall 2008 1 Chapter 2: Pressure and Fluid Statics Pressure For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deform and undergo motion. Normal stresses are referred to as pressure p. For the general case, the stress on a fluid element or at a point is a tensor For a static fluid, τ ij = 0 i j shear stresses = 0 τ ii = p = τ xx = τ yy = τ zz i = j normal stresses =-p Also shows that p is isotropic, one value at a point which is independent of direction, a scalar. τ ij = stress tensor = τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz i = force j = direction
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57:020 Fluid Mechanics Chapter 2 Professor Fred Stern Fall 2008 2 Δ x Δ z Definition of Pressure: A0 Fd F p Ad A lim δ→ δ == δ N/m 2 = Pa (Pascal) F = normal force acting over A As already noted, p is a scalar, which can be easily demonstrated by considering the equilibrium of forces on a wedge-shaped fluid element Geometry Δ A = Δ l Δ y Δ x = Δ l cos α Δ z = Δ l sin α Σ F x = 0 p n Δ A sin α - p x Δ A sin α = 0 p n = p x Σ F z = 0 -p n Δ A cos α + p z Δ A cos α - W = 0 y ) sin )( cos ( 2 W Δ α Δ α Δ γ = l l W = mg = ρ V g = γ V V = ½ Δ x Δ z Δ y nz 2 py c o s p y c o s cos sin y 0 2 ycos pp s i n 0 2 Δ Δα + Δ γ −Δ α α Δ= ÷Δ Δ α γ −+ Δ α = ll l l l
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57:020 Fluid Mechanics Chapter 2 Professor Fred Stern Fall 2008 3 −+− = pp nz γ α 2 0 Δ l sin p p for =→ Δ l 0 i.e., p n = p x = p y = p z p is single valued at a point and independent of direction. A body/surface in contact with a static fluid experiences a force due to p = B S p dA n p F Note: if p = constant, F p = 0 for a closed body. Scalar form of Green's Theorem: s fnds fd =∇∀ ∫∫ f = constant f = 0
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57:020 Fluid Mechanics Chapter 2 Professor Fred Stern Fall 2008 4 Pressure Transmission Pascal's law: in a closed system, a pressure change produced at one point in the system is transmitted throughout the entire system. Absolute Pressure, Gage Pressure, and Vacuum For p A >p a , p g = p A – p a = gage pressure For p A <p a , p vac = -p g = p a – p A = vacuum pressure p A < p a p g < 0 p g > 0 p A > p a p a = atmospheric pressure = 101.325 kPa p A = 0 = absolute zero
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57:020 Fluid Mechanics Chapter 2 Professor Fred Stern Fall 2008 5 Pressure Variation with Elevation Basic Differential Equation For a static fluid, pressure varies only with elevation within the fluid. This can be shown by consideration of equilibrium of forces on a fluid element Newton's law (momentum principle) applied to a static fluid Σ F = ma = 0 for a static fluid i.e., Σ F x = Σ F y = Σ F z = 0 Σ F z = 0 pdxdy p p z dz dxdy gdxdydz −+ = () ρ 0 ργ p z g =− Basic equation for pressure variation with elevation 1 st order Taylor series estimate for pressure variation over dz
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57:020 Fluid Mechanics Chapter 2 Professor Fred Stern Fall 2008 6 0 y p 0 dxdz ) dy y p p ( pdxdz 0 F y = = + = 0 x p 0 dydz ) dx x p p ( pdydz 0 F x = = + = For a static fluid, the pressure only varies with elevation z and is constant in horizontal xy planes.
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This note was uploaded on 12/08/2011 for the course MECHANICS 57:020 taught by Professor Fredrickstern during the Fall '10 term at University of Iowa.

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Chapter2 - 57:020 Fluid Mechanics Professor Fred Stern Fall...

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