Chapter03

Chapter03 - Effects of Gravity on Pressure Effects One of...

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Effects of Gravity on Pressure Effects of Gravity on Pressure z One of the key forces acting on a fluid, real or ideal, is pressure z Even in the absence of motion, the pressure must vary to balance gravity…governed by the hydrostatic relation , which we will derive z Will use the hydrostatic relation to analyze fluid-statics problems such as the force on a dam, buoyancy, etc.
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Pressure on an Infinitesimal Element Pressure on an Infinitesimal Element - - 1 1 z Consider a differential-sized fluid element with center at point (x,y,z) z The net pressure force exerted on the fluid element by the surroundings is the sum of the pressure forces acting on all six faces
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Pressure on an Infinitesimal Element Pressure on an Infinitesimal Element - - 2 2 z If vector n denotes outer unit normal, the force exerted by the surroundings on a differential surface of area dS is – p n dS …the net pressure force is z The “O” joining the integrals denotes closed-surface integral z The 6 A m are the areas of the element faces and…
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Pressure on an Infinitesimal Element Pressure on an Infinitesimal Element - - 3 3 z So, for our infinitesimal element z Since we have infinitesimally small quantities, we approximate p as constant on each face so that
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Pressure on an Infinitesimal Element Pressure on an Infinitesimal Element - - 4 4 z Rearranging terms yields z Defining V = x y z , this simplifies to
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Pressure on an Infinitesimal Element Pressure on an Infinitesimal Element - - 5 5 z The quantity in brackets is the gradient of the pressure, i.e., z Thus, we have shown that for a differential fluid element… z We will use this mathematical fact to help derive an important result known as the hydrostatic relation
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Hydrostatic Relation Hydrostatic Relation - - 1 1 z Consider a volume of arbitrary shape z The force exerted by the surroundings on a surface element dS is p n dS . Hence, z The volume’s weight is
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Hydrostatic Relation Hydrostatic Relation - - 2 2 z Hence, for an arbitrary volume we have z To determine how pressure varies at an arbitrary point in the fluid, choose a differential-sized volume so that z Using the result derived above for the pressure integral…
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Hydrostatic Relation Hydrostatic Relation - - 3 3 z Simplifying, this tells us that z In words… The pressure varies only with depth Its rate of change is dp/dz = - ρ g z For constant density, we have the famous hydrostatic relation z Pressure increases linearly with depth ( z is more negative as depth increases) Î
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Hydrostatic Relation Hydrostatic Relation - - 4 4 z The physical meaning of the hydrostatic
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Chapter03 - Effects of Gravity on Pressure Effects One of...

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