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# lecture4 - 2.20 Marine Hydrodynamics Spring 2005 Lecture 4...

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Lecture 4 - Marine Hydrodynamics Lecture 4 Introduction Governing Equations so far: Knowns Equations # Unknowns # density ρ ( ±x, t ) body force F i Continuity (conservation of mass) Euler (conservation of momentum) 1 3 velocities v i ( t ) stresses τ ij ( t ) 3 ± 6 4 9 ± 3 of the 9 unknowns of the stress tensor are eliminated by symmetry The number of unknowns (9) is > than the number of equations (4), i.e. we don’t have closure . We need constitutive laws to relate the kinematics v i to the dynamics τ ij . 1 2.20 - Marine Hydrodynamics, Spring 2005 2.20

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± 1.9 Newtonian Fluids 1. Consider a ﬂuid at rest ( v i 0). Then according to Pascal’s Law: τ ij = p s δ ij (Pascal’s Law) p s 0 0 τ = 0 p s 0 0 0 p s where p s is the hydrostatic pressure and δ ij is the Kroenecker delta function, equal to 1 if i = j and0i f i = j . 2. Consider a ﬂuid in motion. The ﬂuid stress is deﬁned as: τ ij ij + τ ˆ ij ± ²³ ´ ±²³´ isotropic components all non-isotropic components on diagonal both on/oﬀ diagonal where p is the thermodynamic pressure and ˆ τ ij are the dynamic stresses. It should be emphasized that ij includes all the isotropic components of the stress tensor on the diagonal , while ˆ τ ij represents all the non-isotropic components, which may or may not be on the diagonal (shear and normal stresses). The dynamic stresses ˆ τ ij is related to the velocity gradients by empirical relations. Experiments with a wide class of Newtonian ﬂuids showed that the dynamic stresses are proportional to the rate of strain. τ ij Newtonian Fluid Fluid u k x m 2
±² ³´ µ µ µ µ ³ ´ µ τ ˆ ij linear function of the ( rate of strain velocity gradient) ³ ´± ² ³´±² ∂X ∂u k = ∂t ∂x m u k i.e. τ ˆ ij α ijkm i, j, k, m =1 , 2 , 3 ±²³´ m 3 4 =81 empirical coeﬃcients (constants for Newtonian ﬂuids) For isotropic ﬂuids, this reduces to: i j l τ ˆ ij = μ + + λ , j i l ∇· ±v The ﬂuid properties in the previous relation are: μ - (coeﬃcient of) dynamic viscosity.

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lecture4 - 2.20 Marine Hydrodynamics Spring 2005 Lecture 4...

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