Kinematics and Equations of Balance

Abstract

In this chapter, we review the kinematics and the equations of balance (conservation equations), leaving the question of constitutive description to the next chapter.

Problems

Problem 3.1

Using $\mathbf {FF}^{-1}=\mathbf {I}$ for a deformation gradient $\mathbf {F},$ show that

$$ \frac{d}{{dt}}\mathbf {F}^{-1}=-\mathbf {F}^{-1}\mathbf {L},\;\;\;\mathbf {F} ^{-1}\left( 0\right) =\mathbf {I}. $$

Problem 3.2

For a simple shear flow, where the velocity field takes the form

$$\begin{aligned} u = \dot{\gamma }y,\;\;\;v = 0,\;\;\;w = 0, \end{aligned}$$

(3.72)

show that the velocity gradient and its exponent are given by

$$\begin{aligned} \left[ \mathbf {L} \right] = \left[ {\begin{array}{*{20}c} 0 &{} {\dot{\gamma }}&{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{array}} \right] ,\;\;\;\exp \left( { \mathbf {L}} \right) = \mathbf {I} + \mathbf {L}. \end{aligned}$$

(3.73)

Show that the path lines are given by

$$\begin{aligned} {\varvec{\xi }}\left( \tau \right) = \mathbf {x} + \left( {\tau - t} \right) \mathbf {Lx}. \end{aligned}$$

(3.74)

so that a fluid element dX can only be stretched linearly in time at most.

Problem 3.3

Repeat the same exercise for an elongational flow, where

$$\begin{aligned} u=ax,\;\;\;v=by,\;\;\;w=cz,\;\;\;a+b+c=0. \end{aligned}$$

(3.75)

In this case, show that

$$\begin{aligned} \left[ \mathbf {L}\right] =\left[ {\begin{array}{*{20}c}a &{} 0 &{} 0 \\ 0 &{} b &{} 0 \\ 0 &{} 0 &{} c \\ \end{array}}\right] ,\;\;\;\left[ {e^{\mathbf {L}}}\right] = \left[ {\begin{array}{*{20}c} {e^{a}}&{} 0 &{} 0 \\ 0 &{} {e^{b} } &{} 0 \\ 0 &{} 0 &{} {e^{c} } \\ \end{array}}\right] . \end{aligned}$$

(3.76)

Show that the path lines are given by

$$\begin{aligned} \left[ {{\varvec{\xi }}\left( \tau \right) }\right] =\left[ \begin{array}{l} \xi \\ \psi \\ \zeta \end{array} \right] =\left[ {\begin{array}{*{20}c} {e^{a(\tau - t)} }&{} 0 &{} 0 \\ 0 &{} {e^{b(\tau - t)} } &{} 0 \\ 0 &{} 0 &{} {e^{c(\tau - t)} }\\ \end{array}}\right] \left[ \begin{array}{l} x \\ y \\ z \end{array} \right] . \end{aligned}$$

(3.77)

Conclude that exponential flow can stretch the fluid element exponentially fast.

Problem 3.4

Consider a super-imposed oscillatory shear flow:

$$ u = \dot{\gamma }_{m} y,\;\;\;v = 0,\;\;\;w = \omega \gamma _{a} y\cos \omega t. $$

Show that the path lines are

$$\begin{aligned} \xi \left( \tau \right)&= x + \dot{\gamma }_{m} \left( {\tau - t} \right) y, \nonumber \\ \psi \left( \tau \right)&= y, \\ \zeta \left( \tau \right)&= z + \gamma _{0} y\left( {\sin \omega \tau - \sin \omega t} \right) . \nonumber \end{aligned}$$

(3.78)

Problem 3.5

Calculate the Rivlin–Ericksen tensors for the elongational flow (3.75).

Problem 3.6

Calculate the Rivlin–Ericksen tensors for the unsteady flow (3.78).

Problem 3.7

Write down, in component forms the conservation of mass and linear momentum equations, assuming the fluid is incompressible, in Cartesian, cylindrical and spherical coordinate systems.