Module 2 – Solutions of the Newtonian viscous-flow equations
Analyse the laminar Poiseuille flow with slip, see Problem 3-52 of the book. Comment on the effect of the slip on the volume flow rate: does it increase or decrease because of the slip (for given pressure gradient and geometry)? Do you find this outcome logical?
Sheet 2a-9: Verify the computation of the integral properties of the suction boundary layer velocity profile. Note that as the velocity reaches the “edge value” asymptotically, the upper integration boundary must be set at infinity!
Sheet 2a-9: show that the result for the shear stress can also be derived with an integral momentum balance (control volume approach).
Sheet 2a-11: Compute (numerically) the integral boundary layer properties of the impulsively started flow.
Verify the expression for the pressure coefficient in the stagnation point on Sheet 2b-11. Make a numerical assessment of the viscous effect for a cylinder of 1 mm radius in an airflow of 1 m/s.
Axisymmetric stagnation flow (sheet 2b-16). Consider the axisymmetric potential flow around a sphere. In spherical coordinates the velocity potential and velocity components are given by:
Verify that the local solution near the stagnation point can be written as (see Eq.3-163):
;2uBxvBy==− with: 32UBR∞=⋅
NB: For the definition of the local coordinate system with xRθ= and yrR=−, compare the figure on sheet 2b-5.
Make the exercises on Sheet 2b-24, related to the Jeffery-Hamel flow, which is characterised by: max(,)()()rururfθη=⋅, with max()constant/urr= and /ηθα=.
Sheet 2a-12: Determine the shear stress on an infinite plate which is started from rest, in case the plate velocity Uincreases quadratically in time: 2(*)(*)Utat=⋅, where a is a constant.
Problem 3-15 in the book: A film of constant thickness h is flowing steadily due to gravity down a plane inclined at an angle θ. The atmosphere exerts constant pressure and negligible shear on the free surface. Determine the velocity distribution ()uy across the film.
VISCOUS FLOWS – AE 4120 – Homework Suggestions
The flow over an infinite plate which is linearly accelerated from rest, ()Utat=⋅, see sheet 2a-12, also allows a similarity solution to be formulated. Using the same transformation as for the impulsively started plate, i.e.: (,)()()uytUtfη=⋅ with /2ytην=, determine the resulting ordinary differential equation for ()fη. Show (without actually solving the differential equation!) that this confirms that the wall shear stress varies in proportion to t.
Sheet 2b-16 (book section 3-8.1.3): Verify the derivation of the governing differential equation for the axisymmetric stagnation flow, eq. (3-165). Determine also the expression for the pressure field (see sheets 2b-6 and 11 for the planar stagnation flow). For these derivations you need the x- and y-momentum equations for an axisymmetric flow:
Consider a unidirectional flow, with (,)uuxy= and 0v=, that is governed by the following momentum equation:
The external flow is given by 0()euxU= where 0U is a constant. To verify if the problem allows a self-similar solution, assume:
()euufη=⋅ with ()yLxη=
Evaluate the transformation and determine how ()Lx needs to be defined, and what the resulting ordinary differential equation for ()fη is. (Hint: consult the explanation on sheet 2b-3)
Repeat Homework 2-12, but now with the external flow given by ()euxKx=⋅, where K is a constant.
Repeat Homework 2-12, but now with the external flow given by ()meuxKx=⋅, where K and m are constants. Verify that Homework 2-12 and 2-13 are special cases of this.
Determine for a small spherical gas bubble of 0.1 mm diameter that is rising in water the velocity at which the buoyancy force is exactly in equilibrium with the viscous drag. Assume a Stokes flow and check the value of the Reynolds number to verify if this is justified