Hence C f f x f y f x f y 4 3 d l H k 1077 where H k k ln 1 k 1 k 3 k 2 ln 1 k

Hence c f f x f y f x f y 4 3 d l h k 1077 where h k

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Hence, C f = f x f y = f x f y = 4 3 d 0 l H ( k ) , (10.77) where H ( k ) = k ln 1 + k 1 k 3 k 2 ln 1 + k 1 k 2 k . (10.78) The function H ( k ) is a monotonically decreasing function of k in the range 0 < k < 1. In fact, H ( k 0) 3 / (4 k ), whereas H ( k 1) 1. Thus, if k ∼ O (1) [i.e., if ( d 1 d 2 ) / d 1 ∼ O (1)] then C f ∼ O ( d 0 / l ) 1. In other words, the e ff ective coe cient of friction between two solid bodies in relative motion that are separated by a thin fluid layer is independent of the fluid viscosity, and much less than unity. This result is significant because the coe cient of friction between two solid bodies in relative motion that are in direct contact with one another is typical of order unity. Hence, the presence of a thin lubricating layer does indeed lead to a large reduction in the frictional drag acting between the bodies.
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Incompressible Viscous Flow 299 10.8 Stokes Flow Steady flow in which the viscous force density in the fluid greatly exceeds the ad- vective inertia per unit volume is generally known as Stokes flow , in honor of George Stokes (1819–1903). Because, by definition, the Reynolds number of a fluid is the typical ratio of the advective inertia per unit volume to the viscous force density (see Section 1.16), Stokes flow implies Reynolds numbers that are much less than unity. In the time independent, low Reynolds number limit, Equations (10.1) and (10.2) reduce to ∇ · v = 0 , (10.79) 0 = −∇ P + µ 2 v . (10.80) It follows from these equations that P = µ 2 v = µ ∇ × ( ∇ × v ) = µ ∇ × ω , (10.81) where ω = ∇ × v , and use has been made of Equation (A.177). Taking the curl of this expression, we obtain 2 ω = 0 , (10.82) which is the governing equation for Stokes flow. Here, use has been made of Equa- tions (A.173), (A.176), and (A.177). 10.9 Axisymmetric Stokes Flow Let r , θ , ϕ be standard spherical coordinates. Consider axisymmetric Stokes flow such that v ( r ) = v r ( r , θ ) e r + v θ ( r , θ ) e θ . (10.83) According to Equations (A.175) and (A.176), we can automatically satisfy the in- compressibility constraint (10.79) by writing (see Section 7.4) v = ϕ × ∇ ψ, (10.84) where ψ ( r , θ ) is the Stokes stream function (i.e., v · ∇ ψ = 0). It follows that v r ( r , θ ) = 1 r 2 sin θ ∂ψ ∂θ , (10.85) v θ ( r , θ ) = 1 r sin θ ∂ψ r . (10.86)
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300 Theoretical Fluid Mechanics Moreover, according to Section C.4, ω r = ω θ = 0, and ω ϕ ( r , θ ) = 1 r ( r v θ ) r 1 r ∂v r ∂θ = L ( ψ ) r sin θ , (10.87) where (see Section 7.4) L = 2 r 2 + sin θ r 2 ∂θ 1 sin θ ∂θ . (10.88) Hence, given that |∇ ϕ | = 1 / ( r sin θ ), we can write ω = ∇ × v = L ( ψ ) ϕ. (10.89) It follows from Equations (A.176) and (A.178) that ∇ × ω = ϕ × ∇ [ −L ( ψ )] . (10.90) Thus, by analogy with Equations (10.84) and (10.89), and making use of Equa- tions (A.173) and (A.177), we obtain ∇ × ( ∇ × ω ) = −∇ 2 ω = −L 2 ( ψ ) ϕ. (10.91) Equation (10.82) implies that L 2 ( ψ ) = 0 , (10.92) which is the governing equation for axisymmetric Stokes flow. In addition, Equa- tions (10.81) and (10.90) yield P = µ ϕ × ∇ [ L ( ψ )] . (10.93) 10.10 Axisymmetric Stokes Flow Around a Solid Sphere Consider a solid sphere of radius a that is moving under gravity at the constant ver- tical velocity V e z
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  • Fall '06
  • MOHAMEDATTIA
  • Fluid Dynamics, The Land, stress tensor, Fluid Motion, theoretical fluid mechanics

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