Page 1 Of 3 ME 240A Midterm 2 Fall 2016 Instructor Marko Pri

Page 1 Of 3me 240a Midterm 2 Fall 2016 Instructor Marko Princevac

Calculate the flow field, flow rate, and shear stresses on the upper plate and the slope in a laminar, incompressible flow scenario with a moving plate and inclined slope. Also, sketch the velocity field and specify fluid viscosity, density, plate velocity, and slope angle.

Analyze a liquid film draining down a vertical rod, deriving the velocity profile vₙ(r) and the relation between the flow rate Q and the film radius b, considering fully developed flow where shear balances gravity. Calculate the entrance length lₑ proportional to the Reynolds number based on the channel width and fluid properties. Determine the distance positions where the velocity boundary layer and thermal boundary layer reach a size proportional to L, and find their ratio.

Assess the drag force on a sphere, deriving the terminal velocity U_T given the drag force expression. For an initially stationary sphere, determine the time and distance to reach terminal velocity and comment on the results.

Paper For Above instruction

The comprehensive analysis of laminar flows and related phenomena in fluid mechanics involves understanding various flow configurations and deriving essential parameters such as velocity profiles, flow rates, boundary layer thicknesses, and terminal velocities. This essay discusses these topics in detail, integrating theoretical foundations with calculations supported by fundamental principles.

Flow in a System with Moving Plate and Inclined Slope

Consider a flow confined between a moving upper plate and an inclined slope, forming a laminar, incompressible flow. The flow field can be described by solving the Navier-Stokes equations under the appropriate boundary conditions. Assuming steady, laminar flow and neglecting inertial terms due to low Reynolds number, the velocity profile u(y) can be obtained by integrating the simplified momentum equations. The boundary conditions specify no-slip at the plate and the slope surface, with the upper plate moving at velocity U₀. The velocity distribution u(y) is linear if the slope's influence is incorporated as a boundary condition or as a body force component, leading to an expression like u(y) = (U₀/h) y, where h is the characteristic height of the flow region. The shear stress exerted on the upper plate is τ = μ du/dy |_{y=h} = μ U₀/h, and on the slope, it depends on the local velocity gradient at the inclined surface, requiring geometric considerations involving the slope angle a and its effect on the velocity distribution through coordinate transformation.

The flow rate Q can be computed by integrating the velocity profile across the cross-sectional area, resulting in Q = ∫A u(y) dA, which simplifies in a rectangular domain to Q = U₀ * cross-sectional area, provided the velocity profile is uniform or known.

Flow of Liquid Film over a Vertical Rod

When a liquid film drains down a vertical rod, the velocity profile v_z(r) is shaped by the balance between gravity and viscous forces. Starting from the Navier-Stokes equations in cylindrical coordinates, and assuming steady, fully developed flow with no radial motion, the momentum balance simplifies to the viscous term balancing gravity. The differential equation becomes μ (1/r) d/dr (r dv_z/dr) = ρ g, whose solution considering boundary conditions at the film surface (no shear stress at the free surface) and the solid surface (no-slip at r=a) yields:

v_z(r) = (ρ g / 4 μ) (b² - r²), where b is the radius of the film at the location of interest. The total flow rate Q is obtained by integrating v_z(r) over the cross-sectional area, leading to Q = ∫_{a}^{b} 2π r v_z(r) dr, which results in an expression relating Q to b and a: Q = (π ρ g / 8 μ) (b⁴ - a⁴).

Furthermore, the entrance length lₑ, i.e., the axial distance required for flow to become fully developed, is proportional to the Reynolds number Re = ρ U b / μ, highlighting the transition's dependence on the flow's inertial and viscous forces. This relationship is expressed as lₑ/d ≈ 0.05 Re for a pipe or channel with diameter d, emphasizing that higher Re implies longer entrance lengths.

Thermal and Velocity Boundary Layer Development

In a free stream approaching a flat plate, the velocity boundary layer δ_v and thermal boundary layer δ_T evolve with downstream distance x. Under laminar conditions and high Prandtl number (Pr=10), the thermal boundary layer is thinner than the velocity boundary layer. Using boundary layer theory, the approximate location where δ_v ≈ L and δ_T ≈ L are given by x_v ≈ (ν x) / U and x_T ≈ (α x) / U, respectively. The ratio x_T / x_v = Pr^{1/3} ≈ 2.15, indicating thermal boundary layer thickness grows faster in thermal diffusion relative to velocity diffusion under these conditions.

Drag and Terminal Velocity of a Sphere

The drag force on a sphere moving at velocity U in a viscous fluid is given by a Stokesian approximation: F_D = 3π μ D U, where D is the sphere diameter. Setting the drag force equal to the gravitational force minus buoyancy, i.e., F_D = (ρ_s - ρ_f) g (π D³/6), we rearrange to solve for the terminal velocity U_T:

U_T = [(ρ_s - ρ_f) g D²] / (18 μ). For a sphere initially at rest, the time to reach terminal velocity can be derived from the equation of motion considering acceleration due to the net force minus drag. The characteristic time t ≈ (ρ_s D²) / (18 μ) × ln(U_T / U_i), assuming initial velocity U_i=0, and the distance traveled before reaching U_T is approximately s ≈ U_T t / 2, indicating that the sphere quickly attains terminal velocity in high-viscosity fluids.

This comprehensive analysis illustrates the pivotal concepts of laminar flow regimes, boundary layer development, and particle sedimentation velocities, integrating fundamental principles with calculations for practical applications in fluid mechanics.

References

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