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
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
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
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…
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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