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5.5 Show that the material derivative of the vorticity of the material contained in a volume V is given by di

5.5 Show that the material derivative of the vorticity of the material contained in a volume V is given by di

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5.5 Show that the material derivative of the vorticity of the material contained in a volume V is given by . mar- 5.6 Given the velocity field v, = ax, – bx, vz = bx, + ax,, vz = c/x} +x • 1999 by CRC Press LLC where a, b, and c are constants, determine (a) whether or not the continuity equation is satisfied (b) whether the motion is isochoric. Answers: (a) only when p= pe , (b) only if a = 0. 5.7 For a certain contiuum at rest, the stress is given by where p, is a constant. Use the continuity equation to show that for this case the stress power may be expressed as O,D = 5.8 Consider the motion x, = (1 + t/k)X, where k is a constant. From the conservation of mass and the initial condition p = P, at t = 0, determine p as a function of Por t, and k. Answer: p=- (k+)" 5.9 By combining Eqs 5.3-10b and 5.3-6, verify the result presented in Eq 4.11-6. 5.10 Using the identity Eu = 2(w, + w,vw -w, v) as well as the continuity equation, show that Ed+ 2w,v 2p dt
Transcribed Image Text:5.5 Show that the material derivative of the vorticity of the material contained in a volume V is given by . mar- 5.6 Given the velocity field v, = ax, – bx, vz = bx, + ax,, vz = c/x} +x • 1999 by CRC Press LLC where a, b, and c are constants, determine (a) whether or not the continuity equation is satisfied (b) whether the motion is isochoric. Answers: (a) only when p= pe , (b) only if a = 0. 5.7 For a certain contiuum at rest, the stress is given by where p, is a constant. Use the continuity equation to show that for this case the stress power may be expressed as O,D = 5.8 Consider the motion x, = (1 + t/k)X, where k is a constant. From the conservation of mass and the initial condition p = P, at t = 0, determine p as a function of Por t, and k. Answer: p=- (k+)" 5.9 By combining Eqs 5.3-10b and 5.3-6, verify the result presented in Eq 4.11-6. 5.10 Using the identity Eu = 2(w, + w,vw -w, v) as well as the continuity equation, show that Ed+ 2w,v 2p dt
Problems 5.1 Determine the material derivative of the flux of any vector property Qj through the spatial area S. Specifically, show that in agreement with Eq 5.2-5. 1999 by CRC Press LLC 5.2 Let the property P in Eq 5.2-1 be the scalar 1 so that the integral in that equation represents the instantaneous volume V. Show that in this case dV = 5.3 Verify the identity !! and, by using this identity as well as the result of Problem 5.1, prove that the material derivative of the vorticity flux equals one half the flux of the curl of the acceleration; that is, show that 5.4 Making use of the divergence theorem of Gauss together with the identity aw at show that w,v, - dt
Transcribed Image Text:Problems 5.1 Determine the material derivative of the flux of any vector property Qj through the spatial area S. Specifically, show that in agreement with Eq 5.2-5. 1999 by CRC Press LLC 5.2 Let the property P in Eq 5.2-1 be the scalar 1 so that the integral in that equation represents the instantaneous volume V. Show that in this case dV = 5.3 Verify the identity !! and, by using this identity as well as the result of Problem 5.1, prove that the material derivative of the vorticity flux equals one half the flux of the curl of the acceleration; that is, show that 5.4 Making use of the divergence theorem of Gauss together with the identity aw at show that w,v, - dt
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