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Transport Operations 2nd Edition Allen Stuart - Solutions

Parallel-disk viscometer (Fig. 3B.5), a fluid, whose viscosity is to be measured, is placed in the gap of thickness B between the two disks of radius R. One measures the torque Tz required to turn the upper disk at an angular velocity Ω. Develop the formula for deducing the viscosity from these
Circulating axial flow in an annulus (Fig. 3B.6), a rod of radius KR moves upward with a constant velocity v0 through a cylindrical container of inner radius R containing a Newtonian liquid, the liquid circulates in the cylinder, moving upward along the moving central rod and moving downward along
Momentum fluxes for creeping flow into a slot (Fig. 3.B-7). An incompressible Newtonian liquid is flowing very slowly into a thin slot of thickness 2B (in the y direction) and width W (in the z direction). The mass rate of flow in the slot is w. From the results of Problem 2B.3 it can be shown that
Velocity distribution for creeping flow toward a slot (Fig. 3B.7) it is desired to get the velocity distribution given for the upstream region in the previous problem. We postulate that vθ = 0, vz = 0, v, = vr (r = θ), and P = P(r, θ).(a) Show that the equation of continuity in cylindrical
Slow transverse flow around a cylinder (see Fig. 3.7-1). An incompressible Newtonian fluid approaches a stationary cylinder with a uniform, steady velocity v∞ in the positive x direction. When the equations of change are solved for creeping flow, the following expressions5 are found for the
Radial flow between parallel disks (Fig. 3B.10). A part of a lubrication system consists of two circular disks between which a lubricant flows radially. The flow takes place because of a modified pressure difference P1 – P2 between the inner and outer radii r1 and r2, respectively. (a) Write
Radial flow between two coaxial cylinders, consider an incompressible fluid, at constant temperature, flowing radially between two porous cylindrical shells with inner and outer radii kR and R. (a) Show that the equation of continuity leads to vr = C/r, where C is a constant. (b) Simplify
Pressure distribution in incompressible fluids, Penelope is staring at a beaker filled with a liquid, which for all practical purposes can be considered as incompressible; let its density be P0. She tells you she is trying to understand how the pressure in the liquid varies with depth. She has
Flow of a fluid through a sudden contraction(a) An incompressible liquid flows through a sudden contraction from a pipe of diameter D1 into a pipe of smaller diameter D2. What does the Bernoulli equation predict for P1 – P2, the difference between the modified pressures upstream and downstream of
Torricelli's equation for efflux from a tank (Fig. 3B.14) a large uncovered tank is filled with a liquid to a height h. Near the bottom of the tank, there is a hole that allows the fluid to exit to the atmosphere. Apply Bernoulli's equation to a streamline that extends from the surface of the
Shape of free surface in tangential annular flow (a) A liquid is in the annular space between two vertical cylinders of radii KR and R, and the liquid is open to the atmosphere at the top. Show that when the inner cylinder rotates with an angular velocity Ω, and the outer cylinder is fixed,
Flow in a slit with uniform cross flow (Fig. 3B.16). A fluid flows in the positive x-direction through a long flat duct of length L, width W, and thickness B, where L > > W >> B. The duct has porous walls at y = 0 and y = B, so that a constant cross flow can be maintained, with vy = v0, a constant,
Parallel-disk compression viscometer 6 (Fig. 3C-1) A fluid fills completely the region between two circular disks of radius R. The bottom disk is fixed, and the upper disk is made to approach the lower one very slowly with a constant speed v0, starting from a height H0 (and H0 < < R). The
Normal stresses at solid surfaces for compressible fluids. Extend example 3.1-1 to compressible fluids. Show that τzz|z = 0 = (4/3μ + k)(∂ in p/∂t)z=0 Discuss the physical significance of this result.
Deformation of a fluid line (Fig. 3C.3). A fluid is contained in the annular space between two cylinders of radii KR and R. The inner cylinder is made to rotate with a constant angular velocity of ?r,. Consider a line of fluid particles in the plane z = 0 extending from the inner cylinder to the
Alternative methods of solving the Couette vis-cometer problem by use of angular momentum concepts (Fig. 3.6-1). (a) By making a shell angular-momentum balance on a thin shell of thickness ∆r, show that d/dr(r2τrθ) = 0 Next insert the appropriate expression for τrθ in the gradient
Two-phase interfacial boundary conditions, in S2.1, boundary conditions for solving viscous flow problems were given. At that point no mention was made of the role of interfacial tension. At the interface between two immiscible fluids, I and II, the following boundary condition should be
Derivation of the equations of change by integral theorems (Fig. 3D.1)(a) A fluid is flowing through some region of 3-dimensional space. Select an arbitrary "blob" of this fluid that is, a region that is bounded by some surface S(t) enclosing a volume V(t), whose elements move with the local fluid
The equation of change for vorticity (a) By taking the curl of the Navier-Stokes equation of motion (in either the D/Dt form or the ?/?t form), obtain an equation for the vorticity, w = [? x v] of the fluid; this equation may be written in two ways in which ? is a third-order tensor whose
Alternate form of the equation of motion show that, for an incompressible Newtonian fluid with constant viscosity, the equation of motion may be put into the form where
Time for attainment of steady flow in tube flow(a) A heavy oil with a kinematic viscosity of 3.45 x 10-4 m2/s, is at rest in a long vertical tube with a radius of 0.7 cm. The fluid is suddenly allowed to flow from the bottom of the tube by virtue of gravity. After what time will the velocity at the
Velocity near a moving sphere, a sphere of radius R is falling in creeping flow with a terminal velocity v∞ through a quiescent fluid of viscosity/μ. At what horizontal distance from the sphere does the velocity of the fluid fall to 1% of the terminal velocity of the sphere?
Construction of streamlines for the potential flow around a cylinder Plot the streamlines for the flow around a cylinder using the information in Example 4.3-1 by the following procedure: (a) Select a value of ψ = C (that is, select a streamline). (b) Plot Y = C + K (straight lines
Comparison of exact and approximate profiles for flow along a flat plate compares the values of y√v∞x: obtained from Eq. 4.4-18 with those from Fig. 4.4-3, at the following values of y√v∞/vx(a) 1.5,(b) 3.0,(c) 4.0. Express the results as the ratio of the approximate to the exact values.
Numerical demonstration of the von Karman momentum balance(a) Evaluate the integrals in Eq. 4.4-13 numerically for the Blasius velocity profile given in Fig. 4.4-3. (b) Use the results of (a) to determine the magnitude of the wall shear stress τyx|y = 0.(c) Calculate the total drag force, Fx,
Use of boundary-layer equations, air at 1 atm and 20oC flows tangentially on both sides of a thin, smooth flat plate of width W = 10 ft, and of length L = 3 ft in the direction of the flow. The velocity outside the boundary layer is constant at 20 ft/s. (a) Compute the local Reynolds number
Entrance flow in conduits (a) Estimate the entrance length for laminar flow in a circular tube. Assume that the boundary-layer thickness S is given adequately by Eq. 4.4-17, with v∞ of the flat-plate problem corresponding to vmax in the tube-flow problem. Assume further that the entrance
Flow of a fluid with a suddenly applied constant wall stress, in the system studied in Example 4.1-1, let the fluid be at rest before t = 0. At time t = 0 a constant force is applied to the fluid at the wall in the positive x direction, so that the shear stress τyx takes on a new constant value
Flow near a wall suddenly set in motion (approximate solution) (Fig. 4B.2). Applu procedure like that of Example 4.4-1 to get an approximate solution for Example 4.1.1(a) Integrate Eq. 4.4-1 over y to get make use of the boundary conditions and the Leibniz rule for differentiating an integral (Eq.
Creeping flow around a spherical bubble, when a liquid flows around a gas bubble, circulation takes place within the bubble. This circulation lowers the interfacial shear stress, and, to a first approximation, we may assume that it is entirely eliminated. Repeat the development of Ex. 4.2-1 for
Use of the vorticity equation(a) Work Problem 2B.3 using the y-component of the vorticity equation (Eq. 3D.2-1) and the following boundary conditions: at x = ±B, vz = 0 and at x = 0, vz = vz, max. Show that this leads to vz = vz, max [1 – (x)/B)2] then obtain the pressure distribution from the
Steady potential flow around a stationary sphere. 2 In Example 4.2-1 we worked through the creeping flow around a sphere. We now wish to consider the flow of an incompressible, inviscid fluid in irrotational flow around a sphere. For such a problem, we know that the velocity potential must satisfy
Potential flow near a stagnation point (Fig. 4B.6). (a) Show that the complex potential w = -v0z2 describes the flow near a plane stagnation point. (b) Find the velocity components vx(x, y) and vy(x, y). (c) Explain the physical significance of v0.
Vortex flow(a) Show that the complex potential w = (iT/2π) in z describes the flow in a vortex. Verify that the tangential velocity is given by vθ = T/2ππ and that vr = 0. This type of flow is sometimes called a free vortex. (b) Compare the functional dependence of vθ on r in (a) with
The flow field about a line source considers the symmetric radial flow of an incompressible, inviscid fluid outward from an infinitely long uniform source, coincident with the z-axis of a cylindrical coordinate system. Fluid is being generated at a volumetric rate F per unit length of
Checking solutions to unsteady flow problems(a) Verify the solutions to the problems in Examples 4.1-1, 2, and 3 by showing that they satisfy the partial differential equations, initial conditions, and boundary conditions. To show that Eq. 4.1-15 satisfies the differential equation, one has to know
Radial flow through a porous medium (Fig. 4C.4), a fluid flows through a porous cylindrical shell with inner and outer radii R1 and R2, respectively. At these surfaces, the pressures are known to be P1 and P2, respectively. The length of the cylindrical shell is h. (a) Find the pressure
Flow near an oscillating wall? Show, by using Laplace tion to the problem stated in Eqs. 4.1-44 to 47 is transforms, that the complete solution to the problem stated in Eqs 4.1-44 to 47 is
Flows in the disk-and-tube system (Fig. 4D.3)9 (a) A fluid in a circular tube is caused to move tangentially by a tightly fitting rotating disk at the liquid surface at z = 0; the bottom of the tube is located at z = L. Find the steady-state velocity distribution vθ(r, z), when the angular
Unsteady annular flows? (a) Obtain a solution to the Navier-Stokes equation for the start-up of axial annular flow by a sudden impressed pressure gradient. Check your result against the published solution. (b) Solve the Navier-Stokes equation for the unsteady tangential flow in an
Stream functions for steady three-dimensional flow.(a) Show that the velocity functions pv = [∆ x A] and pv = [(∆ψ1) x (∆ψ2)] both satisfy the equation of continuity identically for steady compressible flow. The functions ψ1, ψ2, and A are arbitrary, except that their derivatives
Pressure drop needed for laminar-turbulent transition. A fluid with viscosity 18.3 cp and density 1.32 g/cm3 is flowing in a long horizontal tube of radius 1.05 in. (2.67 cm). For what pressure gradient will the flow become turbulent?
Velocity distribution in turbulent pipe flow water is flowing through a long, straight, level run of smooth 6.00 in. i.d. pipe, at a temperature of 68oF. The pressure gradient length of the pipe is 1.0 psi/mi. (a) Determine the wall shear stress τ0 in psi (lbf/in.2) and Pa. (b) Assume
Average flow velocity in turbulent tube flow. (a) For the turbulent flow in smooth circular tubes, the function 1 is sometimes useful for curve-fitting purposes: near Re = 4 x 103, n = 6; near Re = 1.1 x 105, n = 7; and near Re = 3.2 x 106, n = 10. Show that the ratio of average to maximum
Mass flow rate in a turbulent circular jet.(a) Verify that the velocity distributions in Eqs. 5.6-21 and 22 do indeed satisfy the differential equations and boundary conditions.(b) Verify that Eq. 5.6-25 follows from Eq. 5.6-21.
The eddy viscosity expression in the viscous sublayer, verify that Eq. 5.4-2 for the eddy viscosity comes directly from the Taylor series expression in Eq. 5.3-13.
Derivation of the equation of change for the Reynolds stresses. At the end of S5.2 it was pointed out that there is an equation of change for the Reynolds stresses. This can be derived by (a) Multiplying the ith component of the vector form of Eq. 5.2-5 by vj and time smoothing, (b)
Kinetic energy of turbulence, by taking the trace of Eq. 5D.1-1 obtains the following: Interpret the equation
Pressure drop required for a pipe with fittings. What pressure drop is needed for pumping water at 20oC through a pipe of 25 cm diameter and 1234 m length at a rate of 1.97 m/s? The pipe is at the same elevation through out and contains four standard radius 90o elbows and two 45o elbows. The
Pressure difference required for flow in pipe with elevation change (Fig. 6A.2). Water at 68?F is to be pumped through 95 ft of standard 3-in. pipe (internal diameter 3.068 in.) into an overhead reservoir. (a) What pressure is required at the outlet of the pump supply water to the overhead
Flow rate for a given pressure drop. How many gal/hr of water at 68o F can be delivered through a 1320-ft length of smooth 6.00-in. i. d. pipe under a pressure difference of 0.25 psi? Assume that the pipe is "hydraulically smooth."(a) Solve by Method A of Example 6.2-2.(b) Solve by Method B of
Motion of a sphere in a liquid, a hollow sphere, 5.00 mm in diameter, with a mass of 0.0500 g, is released in a column of liquid and attains a terminal velocity of 0.500 cm/s. The liquid density is 0.900 g/cm3. The local gravitational acceleration is 980.7 cm/sec2. The sphere is far enough from the
Sphere diameter for a given terminal velocity. (a) Explain how to find the sphere diameter D corresponding to given values of v∞, p, ps, µ, and g by making a direct construction on Fig. 6.3-1. (b) Rework Problem 2A.4 by using Fig. 6.3-1. (c) Rework (b) when the gas velocity is 10
Estimation of void fraction of a packed column, a tube of 146 sq. in. cross section and 73 in. height is packed with spherical particles of diameter 2 mm. When a pressure difference of 158 psi is maintained across the column, a 60% aqueous sucrose solution at 20oC flows through the bed at a rate of
Force on a water tower in a gale, a water tower has a spherical storage tank 40 ft in diameter. In a 100-mph gale what is the force of the wind on the spherical tank at 0°C? Take the density of air to be 1.29 g/liter or 0.08 lb/ft3 and the viscosity to be 0.017 cp.
Flow of gas through a packed column, a horizontal tube with diameter 4 in. and length 5.5 ft is packed with glass spheres of diameter 1/16 in., and the void fraction is 0.41. Carbon dioxide is to be pumped through the tube at 300K, at which temperature its viscosity is known to be 1.495 x 10-4 g/cm
Effect of error in friction factor calculations, in a calculation using the Blasius formula for turbulent flow in pipes, the Reynolds number used was too low by 4%. Calculate the resulting error in the friction factor.
Friction factor for flow along a flat plate2 (a) An expression for the drag force on a flat plate, wetted on both sides, is given in Eq. 4.4-30. This equation was derived by using laminar boundary, layer theory and is known to be in good agreement with experimental data. Define a friction
Friction factor for laminar flow in a slit, use the results of Problem 2B.3 to show that for the laminar flow in a thin slit of thickness 2B the friction factor is f = 12/Re, if the Reynolds number is defined as Re = 2B(vz)p/µ. Compare this result for f with what one would get from the mean
Friction factor for a rotating disk, a thin circular disk of radius R is immersed in a large body of fluid with density p and viscosity µ. If a torque Tz is required to make the disk rotate at an angular velocity 12, then a friction factor f may be defined analogously to Eq. 6.1-1 as follows, Tz/R
Turbulent flow in horizontal pipes, a fluid is flowing with a mass flow rate w in a smooth horizontal pipe of length L and diameter D as the result of a pressure difference P0 - PL. The flow is known to be turbulent. The pipe is to be replaced by one of diameter D/2 but with the same length. The
Inadequacy of mean hydraulic radius for laminar flow(a) For laminar flow in an annulus with radii kR and Rr use Eqs. 6.2-17 and 18 to get an expression for the average velocity in terms of the pressure difference analogous to the exact expression given in Eq. 2.4-16.(b) What is the percentage of
Falling sphere in Newton's drag-law region, a sphere initially at rest at z = 0 falls under the influence of gravity. Conditions are such that, after a negligible interval, the sphere falls with a resisting force proportional to the square of the velocity.(a) Find the distance z that the sphere
Design of an experiment to verify the f vs. Re chart for spheres, it is desired to design an experiment to test the friction factor chart in Fig. 6.3-1 for flow around a sphere. Specifically, we want to test the plotted value f = 1 at Re = 100. This is to be done by dropping bronze spheres (Psph =
Friction factor for flow past an infinite cylinder, the flow past a long cylinder is very different from the flow past a sphere, and the method introduced in S4.2 can-not be used to describe this system. It is found that, when the fluid approaches with a velocity v∞, the kinetic force acting on a
Two-dimensional particle trajectories, a sphere of radius R is fired horizontally (in the x direction) at high velocity in still air above level ground. As it leaves the propelling device, an identical sphere is dropped from the same height above the ground (in the y direction). (a) Develop
Friction factor for a bubble in a clean liquid, when a gas bubble moves through a liquid, the bulk of the liquid behaves as if it were in potential flow; that is, the flow field in the liquid phase is very nearly given by Eqs.4B.5-2 and 3. The drag force is closely related to the energy dissipation
Pressure rise in a sudden enlargement (Fig. 7.6-1). An aqueous salt solution is flowing through a sudden enlargement at a rate of 450 U. So gal/min = 0.0384 m3/s. The inside diameter of the smaller pipe is 5 in. and that of the large pipe is 9 in. What is the pressure rise in pounds per square inch
Pumping a hydrochloric acid solution (Fig. 7A.2), a dilute HC1 solution of constant density and viscosity (p = 62.4lbm/ft3, µ = 1 cp) is to be pumped from tank I to tank 2 with no overall change in elevation. The pressures in the gas spaces of the two tanks are p1 = 1 arm and P2 = 4 arm. The pipe
Compressible gas flow in a cylindrical pipe, gaseous nitrogen is in isothermal turbulent flow at 25°C through a straight length of horizontal pipe with 3-in. inside diameter at a rate of 0.28lbm/s. The absolute pressures at the inlet and outlet are 2atm and 1atm, respectively. Evaluate the viscous
Incompressible flow in an annulus, Water at 60°F is being delivered from a pump through a coaxial annular conduit 20.3 ft long at a rate of 241 U.S. gal/min, the inner and outer radii of the annular space are 3 in. and 7 in. The inlet is 5 ft lower than the outlet. Determine the power output
Force on a U-bend (Fig. 7A.5). Water at 68°F (p = 62.4lbm/ft3, µ = 1 cp) is flowing in turbulent flow in a U-shaped pipe bend at 3 ft3/s. What is the horizontal force exerted by the water on the U-bend?
Flowerate calculation (Fig. 7A.6) for the system shown in the figure, calculate the volume flow rate of water at 68°F.
Evaluation of various velocity averages from Pitot tube data. Following are some experimental data1 for a Pitot tube traverse for the flow of water in a pipe of internal radius 3.06 in.: Plot these data and find out whether the flow is laminar or turbulent. Then use Simpson's rule for numerical
Velocity averages from the 1/7 power law. Evaluate the velocity ratios in Problem cording to the velocity distribution in Eq. 5.1-4.
Relation between force and viscous loss for flow in conduits of variable cross section equation 7.5-6 gives the relation Ff→s = pSEv between the drag force and viscous loss for straight conduits of arbitrary, but constant, cross section. Here we consider a straight channel whose cross section
Flow through a sudden enlargement (Fig. 7.6-1). A fluid is flowing through a sudden enlargement, in which the initial and final diameters are D1 and D2 respectively. At what ratio D2/D1 will the pressure rise P2 – P1 be a maximum for a given value of v1?
Flow between two tanks (Fig. 7B.4). Case I: A fluid flows between two tanks A and B because PA > PB. The tanks are at the same elevation and there is no pump in the line. The connecting line has a cross-sectional area Si and the mass rate of flow is w for a pressure drop of (pA – pB)t. Case II:
Revised design of an air duct (Fig. 7B.5), a straight, horizontal air duct was to be installed in a factory. The duct was supposed to be 4 ft x 4 ft in cross section. Because of an obstruction, the duct may be only 2 ft high, but it may have any width. How wide should the duct be to have the same
Multiple discharges into a common conduit (Fig. 7B.6) extend Example 7.6-1 to an incompressible fluid discharging from several tubes into a larger tube with a net increase in cross section. Such systems are important in heat exchangers of certain types, for which the expansion and contraction
Inventory variations in a gas reservoir, a natural gas reservoir is to be supplied from a pipeline at a steady rate of go w1 lbm/hr. During a 24-hour period, the fuel demand from the reservoir, w2, varies approximately as follows, w2 = A + B cos wt where wt is a dimensionless time measured from the
Change in liquid height with time (Fig. 7.1-1). (a) Derive Eq. 7.1-4 by using integral calculus. (b) In Example 7.1-1, obtain the expression for the liquid height h as a function of time t. (c) Make a graph of Eqo 7.1-8 using dimensionless quantities. Is this useful?
Draining of a cylindrical tank with exit pipe (Fig. 7B.9)(a) Rework Example 7.1-1, but with a cylindrical tank instead of a spherical tank. Use the quasi-steady-state approach; that is, use the unsteady-state mass balance along with the Hagen-Poiseuille equation for the laminar flow in the
Efflux time for draining a conical tank (Fig. 7B.10), a conical tank, with dimensions given in the figure, is initially filled with a liquid. The liquid is allowed to drain out by gravity. Determine the efflux time. In parts (a)-(c) take the liquid in the cone to be the "system." (a) First use
Disintegration of wood chips (Figure 7B.11). In the manufacture of paper pulp the cellulose fibers of wood chips are freed from the lignin binder by heating in alkaline solutions under pressure in large cylindrical tanks called digesters. At the end of the "cooking" period, a small port in one end
Criterion for Vapor-Free Flow in a Pipeline, to ensure that a pipeline is completely liquid-filled, it is necessary that p > Pvap at every point. Apply this criterion to the system in Fig. 7.5-1, by using mechanical energy balances over appropriate portions of the system.
End corrections in tube viscometers (Fig. 7C.1)? In analyzing tube-flow viscometric data to determine viscosity, one compares pressure drop versus flow rate data with the theoretical expression (the Hagen–Poiseuille equation of Eq. 2.3-21). The latter assumes that the flow is fully developed in
Derivation of the macroscopic balances from the equations of change, derive the macroscopic mass and momentum balances by integrating the equations of continuity and motion over the flow system of Fig. 7.0-1. Follow the procedure given in S7.8 for the macroscopic mechanical energy balance, using
Flow of a polyisoprene solution in a pipe, a 13.5% (by weight) solution of polyisoprene in isopentane has the following power law parameters at 323 K: n = 0.2 and m = 5 x 103 Pa ∙ sn. It is being pumped (in laminar flow) through a horizontal pipe that has a length of 10.2 m and an internal
Pumping of a polyethylene oxide solution a 1% aqueous solution of polyethylene oxide at 333 K has power law parameters n = 0.6 and m = 0.50 Pa ∙ sn. The solution is being pumped between two tanks, with the first tank at pressure Pl and the second at pressure P2. The pipe carrying the solution has
Flow of a polymeric film work the problem in S2.2 for the power law fluid. Show that the result simplifies properly to the Newtonian result.
Power law flow in a narrow slit, in Example 8.3-2 show how to derive the velocity distribution for the region – < x < 0, is it possible to combine this result with that in Eq. 8.3-13 into one equation?
Non-Newtonian flow in an annulus, rework Problem 2B.7 for the annular flow of a power law fluid with the flow being driven by the axial motion of the inner cylinder. (a) Show that the velocity distribution for the fluid is ? (b) Verify that the result in (a) simplifies to the Newtonian result when
Flow of a polymeric liquid in a tapered tribe work Probtem 2B.10 for a power law fluid, using the lubrication approximation.
Derivation of the Buckingham-Reiner equation2?rework Example 8.3-1 for the Bingham model, first find the velocity distribution. Then show that the mass rate of flow is given by in which ?R = (P0 ? PL)2L is the shear stress at the tube wall. This expression is valid only ?R???R.?
The complex-viscosity components for the Jeffreys fitrid,(a) Work Example 8.4-1 for the Jeffreys model of Eq. 8.4-4, and show that the results are Eqs. 8.5-12 and 13. How are these results related to Eqs. (G) and (H) of Table 8.5-17(b) Obtain the complex-viscosity components for the Jeffreys model
Stress relaxation after cessation of shear flow. A viscoelastic fluid is in steady-state flow between a pair of parallel plates, with vx = ?y If the flow is suddenly stopped (i.e., ?/becomes zero), the stresses do not go to zero as would be the case for a NewtonJan fluid. Explore this stress
Draining of a tank with an exit pipe (Fig. 7B.9) rework Problem 7B.9 (a) for the power law fluid
The Giesekus model(a) Use the results in Table 8.5-1 to get the limiting values for the non-Newtonian viscosity and the normal stress differences as the shear rate goes to zero.(b) Find the limiting expressions for the non-Newtonian viscosity and the two normal-stress coefficients in the limit as
Squeezing flow between parallel disks (Fig. 3C.1).4 Rework Problem 3C.l (g) for the power law fluid. This device can be useful for determining the power law parameters for materials that are highly viscous. Show that the power law analog of Eq. 3C.1-16is.

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