Continuum Mechanics by D. S. Chandrasekharaiah, Lokenath Debnath | 9780121678807 | 0121678806 | Holooly

Continuum Mechanics

Continuum Mechanics

246 SOLVED PROBLEMS

Question: 2.3.3

Let a and b be vectors with components ai and bi· and A be a tensor with components aij. Show that aibi and aii are scalar invariants. ...

Verified Answer:

If

a^{,}_{i}

and

b_{i}[/late...

Question: 10.14.9

For an incompressible viscous fluid moving under a conservative body force, show that the circulation Ic round a circuit c moving with the fluid is not constant in general. Deduce that Ic is constant if and only if curl w = ▽£, for some ξ. ...

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For the given flow, the Navier-Stokes equation is ...

Question: 10.14.8

Show that the rate of decrease in kinetic energy due to viscosity in a finite volume V of an incompressible fluid is given by W=μ ∫V w² dV -μ ∫S n⋅(v × w) dS (10.14.58) where 5 is the boundary of V. If S is a rigid solid surface at rest, deduce that W=μ ∫V w² dV = ∫V Φ dV (10.14.59) ...

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From (8.6.1), we recall that the kinetic energy of...

Question: 10.14.7

For a flow of an incompressible viscous fluid under conservative body force, show that the vorticity equation is given by ...

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For the given flow, the Navier-Stokes equation is ...

Question: 10.14.6

For an incompressible viscous fluid moving under a conservative body force, prove the following: (i) ∇² (p/ρ + χ +1/2 v²)=div (v × w) (10.14.41) (ii) ∇² (p/ρ + χ )=1/2 w² -D⋅D (10.14.42) Further, if the motion is irrotational, deduce that ∇²v² = 2D⋅D ≥ 0 (10.14.43) ...

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(i) For an incompressible viscous fluid moving und...

Question: 10.14.5

For a nonsteady flow of an incompressible viscous fluid under conservative body force with curl w = ∇ξ for some scalar function ξ show that the Navier-Stokes equation becomes ∂v/∂t + w × v = −∇H* (10.14.39) where H*=p/ρ +1/2 v² + χ + vξ (10.14.40) Deduce the following. (i) If the flow is of ...

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For an incompressible viscous fluid moving under a...

Question: 10.14.4

Consider a steady motion of an incompressible viscous fluid under a conservative body force. If H0 = 1/2 v² +p/ρ + χ (10.14.33) prove the following. (i) H0 is constant along the field lines of the vector f= (v × w) × curl w (10.14.34) (ii) v⋅∇H0 =V (∇²H0 -w² ) (10.14.35) ...

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For the motion considered, the relation (10.14.17)...

Question: 10.14.3

A fluid motion for which the Reynolds number is small (so that nonlinear terms in velocity are negligible) is known as a creeping flow or Stokes’s flow. For a steady creeping flow of an incompressible viscous fluid under zero body force, show that p is a harmonic function. Deduce that ψ defined by ...

Verified Answer:

For creeping flow,

\frac{Dv}{Dt}=\frac{\par...

Question: 10.14.2

For steady flow of an incompressible viscous fluid under a conservative body force, prove the following: ...

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(i) For an incompressible viscous fluid moving und...

Question: 10.14.1

Find the pressure distribution such that the velocity field given by v1 = k (x1²-x2²), v2 = 2 k x1 x2,v3 = 0, (k = constant) (10.14.10) satisfies the Navier-Stokes equation for an incompressible fluid in the absence of body force. ...

Verified Answer:

When written in the component form, the Navier-Sto...

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