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Homework answers / question archive / 1) For the linear generation, use the Robin condition at the surface rather than the Dirichlet condition used in the text

1) For the linear generation, use the Robin condition at the surface rather than the Dirichlet condition used in the text. Derive an expression for the temperature profile, the maximum temperature, and the stability condition.Verify the solutions in the text for the temperature distribution in a square slab with constant generation of heat. Generate illustrative contour plots for the temperature profiles.

2) Consider heat conduction with generation in a slab and a sphere geometries. Derive the solution similar to Eq. (8.18) for slab and sphere cases. For all three cases (slab, cylinder, and sphere) find the heat flow from the surface to the fluid and show that the results satisfy an overall heat balance.

3) Consider the case of linear heat generation with a linear variable-thermal conductivity. Express the governing equation in terms of dimensionless form. What are the number of dimensionless parameters needed to characterize the model? The numerical solution based on BVP4C introduced in Chapter 10 would be useful here.

4) Derive equations for the temperature in a slab if the thermal conductivity (a) is constant, (b) varies linearly as k(T) = k0 + a(T − T0), and (c) varies as a quadratic function k(T) = k0 + a(T − T0) + b(T − T0)2. State how the heat flow should be calculated for each of these cases. How should the “mean” temperature on which to base the mean conductivity be defined for each of these cases?

5) A turkey is being roasted in a microwave oven. How would you calculate the internal heat generation term, Q? V? Show that a variable transformation known as the Kirchhoff transformation,F(T) = 0∫T k(s)ds where s is a dummy variable, reduces the heat equation to ∇2F = 0 for the variable conductivity case.

6) Write in detail the expression for (v · ∇)T in spherical coordinates. Also write in detail the expression for the Laplacian, and thereby complete the temperature equation for spherical coordinates. What simplifications result for an axisymmetric case, i.e., when the temperature has no dependence on the φ direction?

7) For fully developed flow in a pipe the contributions to viscous dissipation are from τrz and dvz/dr. What is the form of the viscous generation term for a fully developed laminar flow of a Newtonian fluid in a pipe? How does it vary with radial position? Calculate this for (a) water and (b) crude oil flowing in a pipe of diameter 2 cm at a velocity of 5 cm/s.

8) Comment and elaborate on the following statement from the BSL book: for viscoelastic fluids the term τ˜ : ∇v does not have to be positive since some energy may be stored as elastic energy.How does the temperature equation simplify if there is no flow? Write this out in detail in all of the three coordinate systems.

9) Write out in detail all the terms for τ˜ : ∇v in rectangular Cartesian coordinates. Now assume a Newtonian fluid; use the generalized version of Newton’s law of viscosity for the stress tensor τ˜ and expand τ˜ : ∇v. Thus derive the expression for the viscous dissipation rate in Cartesian coordinates. How does it simplify for a simple-shear-driven flow between two rectangular channels? How does it simplify for a pressure-driven flow between two parallel channels?

10) Case-study problem: flow analysis of an Ellis fluid. The three-constant Ellis model (Eq. (5.43)) can describe a wide range of experimental data for many fluids. The paper by Matsuhisa and Bird (1965) provides a detailed analysis of this case. the various flow geometries were also analyzed in this paper, and hence this makes an interesting case study. Your goal is to reproduce the results and show some practical applications using rheological-property data for such a fluid.