Fully-developed flow in an annular gap between two co-axial circular cylinders satisfies the following axial momentum...

  

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Fully-developed flow in an annular gap between two co-axial circular cylinders satisfies the following axial momentum equation in cylindrical coordinates: r Or Or where V = axial fluid velocity, i = dynamic viscosity of liquid, p/z = pressure gradient, r = radius. Consider the flow between two co-axial cylinders with inner radius R, = 9 mm and outer radius Ro = 18 mm (see Figure Q1, Page 5). The outer cylinder is stationary, so Vz0 = 0 and the inner cylinder velocity moves along its axis with a velocity Vz = 1 m/s. The fluid has dynamic viscosity u = 1.5 kgm's". (a) Using the finite volume method, develop discretization equations for the three control volumes of the radial grid betweenr = R, and r = Ro sketched in Figure Q1 (Page 5). Assume constant pressure throughout the flow domain. (Hint: multiply both sides of the axial momentum equation by r/uprior to carrying out the control volume integration). [10 marks] (b) Estimate the fluid velocity at the three grid nodes and compute the error in the numerical solution by comparing it with the analytical solution given by: In(r/ R.) In(R,/R.) V, = V1 [6 marks] (c) Verify if the viscous shear forces on the inner and outer wall balance in the numerical solution. Comment on your result. If appropriate, make reference to the comparison in Part (b). [4 marks] Outer cylinder; Ro = 18 mm; Vzo = 0 Fluid 1/23 Fluid Inner cylinder; R = 9 mm; Vzı = 1 m/s %3D Control volumes Figure Q1 Fully-developed flow in an annular gap between two co-axial circular cylinders satisfies the following axial momentum equation in cylindrical coordinates: r Or Or where V = axial fluid velocity, i = dynamic viscosity of liquid, p/z = pressure gradient, r = radius. Consider the flow between two co-axial cylinders with inner radius R, = 9 mm and outer radius Ro = 18 mm (see Figure Q1, Page 5). The outer cylinder is stationary, so Vz0 = 0 and the inner cylinder velocity moves along its axis with a velocity Vz = 1 m/s. The fluid has dynamic viscosity u = 1.5 kgm's". (a) Using the finite volume method, develop discretization equations for the three control volumes of the radial grid betweenr = R, and r = Ro sketched in Figure Q1 (Page 5). Assume constant pressure throughout the flow domain. (Hint: multiply both sides of the axial momentum equation by r/uprior to carrying out the control volume integration). [10 marks] (b) Estimate the fluid velocity at the three grid nodes and compute the error in the numerical solution by comparing it with the analytical solution given by: In(r/ R.) In(R,/R.) V, = V1 [6 marks] (c) Verify if the viscous shear forces on the inner and outer wall balance in the numerical solution. Comment on your result. If appropriate, make reference to the comparison in Part (b). [4 marks] Outer cylinder; Ro = 18 mm; Vzo = 0 Fluid 1/23 Fluid Inner cylinder; R = 9 mm; Vzı = 1 m/s %3D Control volumes Figure Q1