Q1. A horizontal straight pipe of an impermeable solid material extends in the z-direction for a length L and has a circular cross section, of radius r = a. An incompressible Newtonian fluid is undergoing steady state, low Reynolds number, laminar, unidirectional, rectilinear pressure driven flow in the z-direction of the pipe.

(a) Write down the appropriate complete conservation of linear momentum equation.

(b) Simplify the equation of linear momentum for the given problem conditions. Indicate the basis on which you are eliminating terms.

(c) Determine the spatial shear stress distribution in the pipe.

(d) Determine the spatial velocity distribution in the pipe.

(e) Determine the average velocity in the pipe.

Q2. The erythrocyte sedimentation rate test of blood is often used as a diagnostic for immune disorders and other diseases. Assume a red blood cell (erythrocyte) is approximately a sphere of diameter D_p = 5 [µm]. Red blood cell density is ρ_p = 1125 [kg/m³] and the liquid plasma it is settling in has a density ρ_f = 1025 [kg/m³] and a viscosity µ = 1.5 [cP].

(a) Determine the sedimentation rate (velocity) for a single red blood cell.

(b) Verify the equation you used in part (a) is correct.

Q3. Chromatographic separation of proteins can be accomplished using resin ion exchange beads. Spherical Dowex 50-X8 beads of diameter D_p = 0.56 [mm] are packed in a cylindrical chromatography column of radius R = 0.1 [m] and length L = 1 [m] at a porosity ε = 0.35. The proteins are in solution in water of density ρ = 1000 [kg/m³] and viscosity µ = 1 x 10^-3 [Pa s], that is fed to the column at a volumetric flow rate of Q = 2.133 x 10^-4[m³/s]. Determine the pressure drop across the column. Verify any simplifications you make.

Q4. The following equation is a simplified θ-direction linear momentum balance for an incompressible fluid. Using the letters identify the physical type of each term.

i) The viscous force density

ii) The pressure force density

iii) The time dependent acceleration force density

iv) The inertial force density

v) The gravitational force density