Introduction: Food Rheology and Structure

Abstract

This chapter contains an introduction to the rheology of foods and the topics covered in the book. The equation of continuity (conservation of mass) and the equation of motion (conservation of momentum) are introduced with the primary objective to point out the origins of those useful relationships and the assumptions made in deriving them.

Appendix 1-A

Momentum and Heat Transport Equations for Incompressible Fluids

Transport Equations in Vector Notation

The equations of continuity, motion, and energy in vector notation are given here:

$$ \begin{aligned}& \nabla \cdot \text{v}=\text{0} \\& \rho \frac{\text{DV}}{\text{Dt}}=-\nabla p-[\nabla \cdot \tau ]+\rho \ g \\& k{{\nabla }^{2}}T=-[\nabla \cdot q]-[\tau :\nabla \text{V}] \\ \end{aligned} $$

The Equation of Continuity (Bird et al. 1960)

Cartesian coordinates (x, y, z):

$$ \frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x}(\rho {{\nu }_{x}})+\frac{\partial }{\partial y}(\rho {{\nu }_{y}})+\frac{\partial }{\partial z}(\rho {{\nu }_{z}})=0 $$

Source: Bird, R. B., Stewart, W.E., and Lightfort, E.N. 1960. Transport Phenomena, John Wiley and sons, New York.

Cylindrical coordinates (r, θ, z):

$$ \frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial }{\partial r}(\rho r{{v}_{r}})+\frac{1}{r}\frac{\partial }{\partial \theta }(\rho {{v}_{\theta }})+\frac{\partial }{\partial z}(\rho {{v}_{z}})=0 $$

Spherical coordinates (r, θ, ϕ):

$$\frac{\partial \rho }{\partial t}+\frac{1}{{{r}^{2}}}\frac{\partial }{\partial r}(\rho {{r}^{2}}{{v}_{r}})+\frac{1}{r\ \text{Sin}\theta }\frac{\partial }{\partial \theta }(\rho {{v}_{\theta }}\,\text{Sin}\theta )\text{+}\frac{\text{1}}{r\ \text{Sin}\theta }\frac{\partial }{\partial \phi }(\rho {{\text{v}}_{\phi }})\text{=0}$$

Equation of Motion in Rectangular Coordinates (x, y, z) in terms of σ (Bird et al. 1960)

x-component:

$$ \rho \left( \frac{\partial {{v}_{x}}}{\partial t}+{{v}_{x}}\frac{\partial {{v}_{x}}}{\partial x}+{{v}_{y}}\frac{\partial {{v}_{x}}}{\partial y}+{{v}_{z}}\frac{\partial {{v}_{x}}}{\partial z} \right)=-\frac{\partial p}{\partial x}+\left( \frac{\partial {{\sigma }_{xx}}}{\partial x}+\frac{\partial {{\sigma }_{yx}}}{\partial y}+\frac{\partial {{\sigma }_{zx}}}{\partial z} \right)+\rho {{g}_{x}} $$

(A)

y-component:

$$ \rho \left( \frac{\partial {{v}_{y}}}{\partial t}+{{v}_{x}}\frac{\partial {{v}_{y}}}{\partial x}+{{v}_{y}}\frac{\partial {{v}_{y}}}{\partial y}+{{v}_{z}}\frac{\partial {{v}_{y}}}{\partial z} \right)=-\frac{\partial p}{\partial y}+\left( \frac{\partial {{\sigma }_{xy}}}{\partial x}+\frac{\partial {{\sigma }_{yy}}}{\partial y}+\frac{\partial {{\sigma }_{zy}}}{\partial z} \right)+\rho {{g}_{y}} $$

(B)

z-component:

$$ \rho \left( \frac{\partial {{v}_{z}}}{\partial t}+{{v}_{x}}\frac{\partial {{v}_{z}}}{\partial x}+{{v}_{y}}\frac{\partial {{v}_{z}}}{\partial y}+{{v}_{z}}\frac{\partial {{v}_{z}}}{\partial z} \right)=-\frac{\partial p}{\partial z}+\left( \frac{\partial {{\sigma }_{xz}}}{\partial x}+\frac{\partial {{\sigma }_{yz}}}{\partial y}+\frac{\partial {{\sigma }_{zz}}}{\partial z} \right)+\rho {{g}_{y}} $$

(C)

The Equation of Motion in Spherical Coordinates (r, θ, ϕ) in terms of σ (Bird et al. 1960)

r-component:

$$\begin{array}{l}\rho \left( {\frac{{\partial {v_r}}}{{\partial t}} + {v_r}\frac{{\partial {v_r}}}{{\partial r}} + \frac{{{v_\theta }}}{r}\frac{{\partial {v_r}}}{{\partial \theta }} + \frac{{{v_\phi }}}{{r\sin \theta }}\frac{{\partial {v_r}}}{{\partial \phi }}-\frac{{v_\theta ^2 + v_\phi ^2}}{r}} \right)\\\quad = - \frac{{\partial p}}{{\partial r}} + \left( {\frac{1}{{{r^2}}}\;\frac{\partial }{{\partial r}}\left( {r{\sigma ^2}_{rr}} \right) + \frac{1}{{r\;\sin \;\theta }}\frac{\partial }{{\partial \theta }}\;({\sigma _{r\theta }}\;\sin \;\theta )} \right.\\\left. {\quad + \frac{1}{{r\sin \theta }}\frac{{\partial {\sigma_{r\phi }}}}{{\partial \phi }} - \frac{{{\sigma _{\theta \theta }}+ {\sigma _{\phi \phi }}}}{r} + \rho {g_r}} \right)\end{array}$$

(A)

θ-component:

$$ \begin{array}{l}\rho \left( {\frac{{\partial v_\theta }}{{\partial t}} + {v_r}\frac{{\partial v_\theta }}{{\partial r}} + \frac{{{v_\theta }}}{r}\frac{{\partial {v_\theta }}}{{\partial \theta }} + \frac{{{v_\phi }}}{{r\sin \theta }}\frac{{\partial {v_\theta }}}{{\partial \phi }} + \frac{{{v_r}{v_\theta }}}{r} - \frac{{v_\phi ^2\cot \theta }}{r}} \right)\\ \quad= - \frac{1}{r}\frac{{\partial p}}{{\partial \theta }} + \left( {\frac{1}{{{r^2}}}\frac{\partial }{{\partial r}}\left( {{r^2}{\sigma _{r\theta }}} \right) + \frac{1}{{r\sin \theta }}\frac{\partial }{{\partial \theta }}} \right)({\sigma _{\theta \theta }}\sin \theta )\\\left. {\quad + \frac{1}{{r\sin \theta }}\frac{{\partial {\sigma_{\theta \phi }}}}{{\partial \phi }} + \frac{{{\sigma _{r\theta}}}}{r} - \frac{{\cot \theta }}{r}{\sigma _{\phi \phi }}} \right)+ \rho {g_\theta }\end{array} $$

(B)

ϕ-component:

$$\begin{array}{l}\rho \left( {\frac{{\partial {v_\phi }}}{{\partial t}} + {v_r}\frac{{\partial {v_\phi }}}{{\partial r}} + \frac{{{v_\theta }}}{r}\frac{{\partial {v_\phi }}}{{\partial \theta }} + \frac{{{v_\phi }}}{{r\sin \theta }}\frac{{\partial {v_\phi }}}{{\partial \phi }} + \frac{{{v_\phi }{v_r}}}{r} + \frac{{{v_\theta }{v_\phi }}}{r}\cot \theta } \right)\\ \quad= - \frac{1}{{r\sin \theta }}\frac{{\partial p}}{{\partial \phi }} + \left( {\frac{1}{{{r^{\rm{2}}}}}\frac{\partial }{{\partial r}}({r^{\rm{2}}}{\sigma _{r\phi }}) + \frac{1}{r}\;\frac{{\partial {\sigma _{\theta \phi }}}}{{\partial \theta }} + \frac{1}{{r\;\sin \theta }}\frac{{\partial {\sigma _{\phi \phi }}}}{{\partial \phi }}} \right.\\\quad\left. { + \frac{{{\sigma _r}_\phi }}{r} + \frac{{2\cot \theta}}{r}{\sigma _{\theta \phi }}} \right) + \rho {g_\phi }\end{array}$$

(C)

Equation of Motion in Cylindrical Coordinates (r, θ, z) in terms of σ (Bird et al. 1960)

r-component:

$$\begin{array}{l}\rho \left( {\frac{{\partial {v_r}}}{{\partial t}} + {v_r}\frac{{\partial {v_r}}}{{\partial r}} + \frac{{{v_\theta }}}{r}\frac{{\partial {v_r}}}{{\partial \theta }} - \frac{{v_\theta ^2}}{r} + {v_z}\frac{{\partial {v_r}}}{{\partial z}}} \right)\\ \quad= - \frac{{\partial p}}{{\partial r}} + \left( {\frac{1}{r}\frac{\partial }{{\partial r}}(r{\sigma _{rr}}) + \frac{1}{r}\frac{{\partial {\sigma _{r\theta }}}}{{\partial \theta }} - \frac{{{\sigma _{\theta \theta }}}}{{r\,}} + \frac{{\partial {\sigma _{rz}}}}{{\partial z}}} \right) + \rho {g_r}\end{array}$$

(A)

θ-component:

$$\begin{array}{l}\rho \left( {\frac{{\partial {v_\theta }}}{{\partial t}} + {v_r}\frac{{\partial {v_\theta }}}{{\partial r}} + \frac{{{v_\theta }}}{r}\frac{{\partial {v_\theta }}}{{\partial \theta }} + \frac{{{v_r}{v_\theta }}}{r} + {v_z}\frac{{\partial {v_\theta }}}{{\partial z}}} \right)\\\quad = - \frac{1}{r}\frac{{\partial p}}{{\partial \theta }} + \left( {\frac{1}{{{r^{\rm{2}}}}}\frac{\partial }{{\partial r}}({r^{\rm{2}}}{\sigma _{r\theta }}) + \frac{1}{r}\frac{{\partial {\sigma _{\theta \theta }}}}{{\partial \theta }} + \frac{{\partial {\sigma _{\theta z}}}}{{\partial z}}} \right) + \rho {g_\theta }\end{array}$$

(B)

z-component:

$$ \begin{array}{l}\rho \left( {\frac{{\partial {v_z}}}{{\partial t}} + {v_r}\frac{{\partial {v_z}}}{{\partial r}} + \frac{{{v_\theta }}}{r}\frac{{\partial {v_z}}}{{\partial \theta }} + {v_z}\frac{{\partial {v_z}}}{{\partial z}}} \right)\\ = - \frac{{\partial p}}{{\partial z}} + \left( {\frac{1}{r}\frac{\partial }{{\partial r}}(r{\sigma _{rz}}) + \frac{1}{r}\frac{{\partial {\sigma _{\theta z}}}}{{\partial \theta }} + \frac{{\partial {\sigma _{zz}}}}{{\partial z}}} \right) + \rho {g_z}\end{array} $$

(C)