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MMVN10 Fluid Mechanics — Questions on Theory (March 12, 2020)

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MMVN10 Fluid Mechanics — Questions on Theory (March 12, 2020) MMVN10 Fluid Mechanics | questions on theory (March 12, 2020) Chapter 1 | Introduction 1.1 What is the fundamental difference between a solid and a fluid? 1.2 Give a practical definition of the density of a fluid. Illustrate with a figure. 1.3 (a) For newtonian fluids, shear stresses are proportional to corresponding shear strain rates. For a simple shear flow, V = u(y)^i, show that the shear stress τ is proportional to du=dy. (b) Discuss and illustrate with diagrams some possible characteristics of non-newtonian fluids (liquids). 1.4 State the SI-dimensions of dynamic and kinematic viscosity, respectively. Also provide approx- imate values (with two significant digits) for water and air at standard room conditions (20◦C, 1 atm). At around standard room conditions, what are the separate effects of changing the pressure and temperature, respectively? 1.5 Define or explain shortly: (a) incompressible flow, (b) cavitation, (c) streamline, (d) streakline. Chapter 2 | Pressure Distribution in a Fluid 2.1 Derive the hydrostatic balance equation rp = ρ g, valid for a fluid at rest. 2.2 Derive the manometer formula ∆p = (ρm −ρ)g h , where h is the manometer reading (height), ρm the density of the manometer liquid and ∆p the pressure difference across two horizontal points in the fluid of density ρ (U-tube manometer, see Example 2.3). Chapter 3 | Integral Analysis 3.1 (a) Provide general definitions of volume flow rate and mass flow rate through a surface. Illustrate with a figure. (b) Show that the mean (average) velocity for laminar fully developed flow in a pipe of circular cross section is equal to half the velocity at the centerline. The velocity profile is parabolic. 3.2 Let B be an arbitrary extensive property and b the same property but expressed per unit mass. For a fixed and non-deforming control volume with one inlet (1) and one outlet (2), both assumed one-dimensional, the Reynolds transport theorem, at steady flow conditions, reads: d B = (b ρ V A) − (b ρ V A) dt sys 2 1 State without proof the Reynolds transport theorem for an arbitrarily moving but still non- deforming control volume, at non-steady (time-dependent) conditions. Illustrate the convective part in a figure. 3.3 Assuming steady one-dimensional incompressible flow, state without proof the momentum inte- gral equation applicable to a control volume with one inlet and one outlet. Give a full account for different terms and properties. How can this equation be modified to include velocity variations across the inlet and outlet? Calculate the correction coefficient (β) for the case of fully developed laminar flow in a straight pipe (circular cross section). 3.4 Assuming steady one-dimensional incompressible flow, state without proof the energy integral equation applicable to a control volume with one inlet and one outlet (the extended Bernoulli equation). Give a full account of different terms and properties. How can this equation be modified to include velocity variations across the inlet and outlet? Calculate the correction coefficient (α) for the case of fully developed laminar flow in a straight pipe (circular cross section). 1 Chapter 4 | Differential Analysis 4.1 A velocity field is described in a cartesian coordinate system xyz. Use the rule of chains to express the material (substantial) acceleration in local and convective terms. Give a physical description of each term. 4.2 Derive the differential form of the continuity equation using a cartesian coordinate system. After this, specialize to incompressible flow conditions. 4.3 Show that Ma2 ≪ 1 is a necessary condition for incompressible flow. Provide an engineering estimate of Ma for which the flow has to be treated as compressible. Hint: Local mass balance, steady flow ! @(ρu)=@x + @(ρv)=@y + @(ρw)=@z = 0. 4.4 (a) Define the stress tensor σij in a flowing fluid, expressed in terms of the viscous stress tensor τij and the pressure p. What symmetry condition is normally valid for σij? (b) Illustrate the components σ11, σ22, σ12 and σ21 in a figure (σ12 = σxy, . ). (c) Define τxy for a newtonian fluid using cartesian coordinates. 4.5 Using cartesian coordinates, write out the continuity equation as well as the x-component of the momentum equation for incompressible flow of a newtonian fluid with constant fluid properties. 4.6 Write out the divergence and rotation (curl), respectively, applied to the velocity field V = (u; v; w) in cartesian coordinates. What do they represent physically? 4.7 Consider, at zero time, a cubical fluid element in a flow field. Kinematically, the instantaneous effects on this element from the flow field can be sub-divided into four elementary motions. Define and describe these elementary motions. Illustrate with a figure. 4.8 (a) State the general definition of the vorticity vector at a point in a flow field. What does it describe physically? (b) For two-dimensional flow in a plane, show that the vorticity vector only has one component. (c) Show that the only component of the vorticity vector in plane two-dimensional flow can be interpreted as twice the instantaneous anti-clockwise rotational rate of the diagonal in a infinitesimal fluid element, initially being quadratic. (d) How is it possible for an initially irrotational flow to become locally rotational, i.e., to possess vorticity? State at least three mechanisms. 4.9 Consider two-dimensional incompressible flow in a plane, V = (u; v; 0), where u(x; y), v(x; y). (a) Define the stream function through implicit expressions using u(x; y) and v(x; y). Also show that satisfies the Laplace equation r2 = 0 when the flow is irrotational. (b) Show that = const: represents streamlines and that the difference in between two streamlines is equal to the volume rate per unit width. Chapter 5 | Dimensional Analysis and Similarity 5.1 (a) What is the principle of dimensional homogeneity, PDH? (b) What are the MLT Θ-dimensions for dynamic (absolute) viscosity µ, heat conductivity k and power P (work per unit time)? (c) State the Π-theorem, in words. 5.2 Define the Strouhal number (St) for periodic vortex shedding downstream of a circular cylinder of diameter d, exposed to a constant flow velocity V perpendicular to the cylinder axis. Assuming incompressible flow, state an approximate value of the Strouhal number over the interval Re = V d/ν = 300 − 105. 2 5.3 Explain briefly (a) homologous points and homologous times (b) geometric, kinematic and dynamic similarity 5.4 In practical cases concerning free-surface flows, most model tests have to renounce on the principle of dynamic similarity. Explain why. 5.5 For incompressible steady flow conditions, the drag D on a smooth spherical object depends only on the diameter d, fluid viscosity µ, fluid density ρ and flow velocity U. (a) Define the conventional drag coefficient CD, and show, using dimensional analysis, that CD only depends on the Reynolds number. (b) Suggest an alternative drag coefficient that does not include the flow velocity. Explain why this coefficient could be more appropriate to use, in some cases. 5.6 Show that the Reynolds, Froude and Weber number, respectively, each is a rough measure of a ratio between certain fluid forces in a flow field (mass times acceleration = inertia force; [Υ] = N=m). 5.7 Ekman (Ek) and Rossby (Ro) numbers are of importance in rotating flows.1 Ekman number is a rough measure of the ratio between frictional (viscous) and Coriolis (fictional) forces; Rossby number correspondingly between inertial and Coriolis forces. Define these two numbers if Ω is a characteristic angular rate of rotation (characteristic velocity U; characteristic length L). Coriolis acceleration: 2 Ω × V . Chapter 6 | Viscous Flows in Ducts 6.1 Make a sketch on the variation of pressure and the velocity profile development for laminar flow in a horizontal pipe connected to a large reservoir. The submerged inlet is rounded. State an approximate expression for the entrance length. 6.2 Define the hydraulic diameter Dh in duct flows. Find the hydraulic diameter for a circular and a square cross-section. In engineering calculations, state the commonly accepted Reynolds number above which the flow can be regarded as fully turbulent. 6.3 Define the Darcy-Weisbach friction factor f for pipe flow. How is the (major) pressure loss calculated? What can be stated generally for f in fully developed laminar duct flow, regardless of the cross section? 6.4 Consider steady, fully developed laminar flow in a horizontal and straight pipe of circular cross section. How does the pressure drop per unit length ∆p=L vary with the volume flow rate Q, the diameter D, and the fluid density ρ and viscosity µ, respectively? The answer can be given through exponents a − d in the expression ∆p=L / QaDbρcµd. How does the influence of the respective quantity change if the flow is instead turbulent at sufficiently high Reynolds number? Used relationships should be justified. 6.5 Make an account for the recommended scheme of obtaining the pressure loss over a certain length in a duct of constant but non-circular cross section, assuming fully developed turbulent flow. The expression for the corresponding pressure loss in fully developed laminar flow is known. 6.6 (a) Make a sketch of mean velocity profiles in fully developed laminar and turbulent flow in a pipe (circular cross section). For turbulent flow, what is the influence of Reynolds number? (b) State an approximate power-law expression for the mean velocity profile in fully developed turbulent pipe flow. Use this expression to determine the ratio between the average velocity and the mean velocity at the center of the pipe.
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