Chapter 9 Viscous flow

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24 CHAPTER 9. VISCOUS FLOW mass conservation equation. 1 ∂(ru r ) + ∂u z r ∂r ∂z = 0 (9.93) The axial velocity u z can be scaled by the velocity of the surface of the sphere, U, since the axial velocity varies between 0 at the plane surface and −U at the surface of the sphere. When the mass conservation equation is expressed in terms of the scaled coordinates and the scaled velocity u ∗ z = (u z/U), we get 1 1 ∂(r ∗ u r ) + U ∂u ∗ z Rǫ 1/2 r ∗ ∂r ∗ ǫR ∂z = 0 (9.94) ∗ The above equation provides the scaled for the radial velocity is u ∗ r = (u r ǫ 1/2 /U). This scaling indicates that when u ∗ r is O(1), the magnitude of the radial velocity is u r ∼ (U/ǫ 1/2 ), which is large compared to the magnitude of the axial velocity. This is because as the sphere moves downward, all the fluid displaced per unit time within a cylindrical section of radius r is πr 2 U. This fluid has to be expelled from the cylindrical surface with area 2πrh, and therefore the radial velocity scales as (Ur/h), which is proportional to (U/ǫ 1/2 ). The momentum conservation equations in the radial direction, expressed in terms of the scaled variables are ρU 2 ( ∂u ∗ r ǫ 3/2 ∂t + ∂u ∗ ∗ u∗ r r ∂r + ∂u ∗ ) ∗ u∗ r z = − 1 ( ∂p U ∂ 2 u ∗ ∂z ∗ Rǫ 1/2 ∂r ∗+µ r R 2 ǫ 5/2 ∂z + U ) 1 ∂ r ∗2 R 2 ǫ 3/2 r ∗ ∂r ∗r∗∂u∗ ∂r ∗ (9.95) In the above equation, the scaled time has been defined as t ∗ = (tǫR/U), since the relevant time scale is the ratio of the gap width and the velocity of the sphere. This ensures that the first term on the left side of equation 9.95 is of the same magnitude as the other three terms. The momentum conservation can be simplified by dividing throughout by the coefficient of the largest viscous term, which is proportional to (µU/R 2 ǫ 5/2 ), ( ρURǫ ∂u ∗ r µ ∂t + ∂u ∗ ∗ u∗ r r ∂r + ∂u ∗ ) ( ∗ u∗ r z = − Rǫ2 ∂p ∂ 2 ∂z ∗ µU ∂r + u ∗ r ∗ ∂z + ǫ ) ∂ r ∗2 r ∗ ∂r ∗r∗∂u∗ ∂r ∗ (9.96) This equation indicates that the scaled pressure should be defined as p ∗ = (pRǫ 2 /µU) for the pressure to be of the same magnitude as the viscous terms. Equation 9.96 also indicates that the appropriate Reynolds number in this case is (ρURǫ/µ), which is based on the gap thickness and the velocity of
9.3. LUBRICATION FLOW 25 the sphere. Inertial effects can be neglected when this Reynolds number is small, even though the Reynolds number based on the sphere velocity and the sphere radius and sphere velocity is large. After neglecting inertial effects, and retaining only the largest terms in the ǫ expansion, the momentum conservation equation in the radial direction becomes − ∂p∗ ∂r + ∂2 u ∗ r = 0 (9.97) ∗ ∂z∗2 The momentum conservation equation in the axial direction, expressed in terms of the scaled coordinates, velocity and pressure, is ρU 2 Rǫ ( ∂u ∗ z ∂t + ∂u ∗ ∗ u∗ z r ∂r + ∗ u∗ z ∂u ∗ ) z ∂z ∗ = − µU ∂p ∗ ( U ∂ 2 u ∗ Rǫ 3 ∂z ∗+µ z R 2 ǫ 2 ∂z + U ∗2 R 2 ǫ 1 r ∗ ) ∂ z ∂r ∗r∗∂u∗ ∂r ∗ (9.98) It is evident that the largest term in the equation is the pressure gradient, since it is multiplied by (µU/Rǫ 3 ). Dividing throughout by this factor, we get ρURǫ 2 ( ∂u ∗ z µ ∂t + ∂u ∗ ∗ u∗ z r ∂r + ∂u ∗ ) ( ∗ u∗ z z = − ∂p∗ ∂ 2 ∂z ∗ ∂z + ǫ u ∗ z ∗ ∂z + ǫ ) ∂ z ∗2 r ∗ ∂r ∗r∗∂u∗ ∂r ∗ (9.99) If all terms that are small in the ǫ expansion are neglected, the axial momentum conservation equation becomes ∂p ∗ ∂z ∗ = 0 (9.100) Equation 9.100 indicates that the pressure is independent of the z ∗ coordinate. Using this, equation 9.97 can be solved to obtain the radial velocity, u ∗ r = ∂p∗ ∂r ∗ z ∗2 2 + A 1(r ∗ )z ∗ + A 2 (r ∗ ) (9.101) The functions A 1 (r ∗ ) and A 2 (r ∗ ) are obtained from the no-slip condition u ∗ r = 0 at z∗ = 0 and the surface of the sphere z ∗ = h(r ∗ ). The radial velocity then becomes ( u ∗ r = ∂p∗ z ∗2 ∂r ∗ 2 − z∗ h(r ∗ ) ) 2 (9.102)
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Page 3 and 4: 9.1. SPHERICAL HARMONIC SOLUTIONS 3
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Page 21 and 22: 9.3. LUBRICATION FLOW 21 velocity a
Page 23: 9.3. LUBRICATION FLOW 23 R U z r r
Page 27 and 28: 9.4. INERTIAL EFFECTS AT LOW REYNOL

Chapter 9 Viscous flow

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