The pressure difference between the downstream end

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Unformatted text preview: by dτ over each layer. The pressure difference between the downstream end and the upstream end is dp. The pressure change is needed to overcome the shear stress. The total force on a layer must be zero so balancing forces on one layer (assumed 1 m wide) we get the following. dp dy + dτ dL = 0 dτ dp =− dy dL It is normally assumed that the pressure declines uniformly with distance downstream so the pressure gradient dp is assumed constant. The minus sign indicates that the pressure falls with distance. dL Integrating between the no slip surface (y = 0) and any height y we get ⎛ du ⎞ d⎜ µ ⎟ ⎜ dy ⎟ dp dτ ⎠ − = =⎝ dy dL dy − dp d 2u = µ 2 ..............(1.1) dL dy Integrating twice to solve u we get the following. −y dp du =µ +A dL dy y 2 dp − = µu + Ay + B 2 dL A and B are constants of integration that should be solved based on the known conditions (boundary conditions). For the flat surface considered in figure 1.1 one boundary condition is that u = 0 when y = 0 (the no slip surface). Substitution reveals the following. 0 = 0 +0 +B hence B = 0 At some height δ above the surface, the velocity will reach the mainstream velocity uo. This gives us the second boundary condition u = uo when y = δ. © D.J.DUNN 3 Substituting we find the following. δ 2 dp = µu o + Aδ 2 dL δ dp µu o A=− hence − 2 dL δ y 2 dp ⎛ δ dp µu o ⎞ − = µu + ⎜ − − ⎟y 2 dL δ⎠ ⎝ 2 dL ⎛ δ dp u o ⎞ u = y⎜ ⎟ ⎜ 2µ dL + δ ⎟ ⎠ ⎝ − Plotting u against y gives figure 1.2. BOUNDARY LAYER. The velocity grows from zero at the surface to a maximum at height δ. In theory, the value of δ is infinity but in practice it is taken as the height needed to obtain 99% of the mainstream velocity. This layer is called the boundary layer and δ is the boundary layer thickness. It is a very important concept and is discussed more fully in later work. The inverse gradient of the boundary layer is du/dy and this is the rate of shear strain γ. Fig.1.2 © D.J.DUNN 4 1.2. UNITS of VISCOSITY 1.2.1 DYNAMIC VISCOSITY µ The units of dynamic viscosity µ are N s/m2. It is normal in the international system (SI) to give a name to a compound unit. The old metric unit was a dyne.s/cm2 and this was called a POISE after Poiseuille. The SI unit is related to the Poise as follows. 10 Poise = 1 Ns/m2 which is not an acceptable multiple. Since, however, 1 Centi Poise (1cP) is 0.001 N s/m2 then the cP is the accepted SI unit. 1cP = 0.001 N s/m2. The symbol η is also commonly used for dynamic viscosity. There are other ways of expressing viscosity and this is covered next. 1.2.2 KINEMATIC VISCOSITY ν This is defined as : ν = dynamic viscosity /density ν = µ/ρ The basic units are m2/s. The old metric unit was the cm2/s and this was called the STOKE after the British scientist. The SI unit is related to the Stoke as follows. 1 Stoke (St) = 0.0001 m2/s and is not an acceptable SI multiple. The centi Stoke (cSt),however, is 0.000001 m2/s and this is an acceptable multiple. 1cSt = 0.000001 m2/s = 1 mm2/s 1.2.3...
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This document was uploaded on 02/07/2014.

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