Draft:Viscoelastic Fluids

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Scale Analysis of Linear Viscoelastic Fluids

Viscoelastic fluids are materials that exhibit both viscous and elastic characteristics when undergoing deformation. Unlike purely viscous fluids, such as water, or purely elastic materials, such as rubber, viscoelastic fluids demonstrate time-dependent stress-strain behavior. This dual nature makes them significant in various industrial and natural processes, including polymer processing, biological fluid dynamics, and the manufacturing of food and cosmetics.

Linear viscoelasticity is a specific regime where the material response remains proportional to the applied stress or strain, typically valid for small deformations or low strain rates. In this regime, the complex interactions between fluid motion and material elasticity can be modeled using linear constitutive equations. These equations simplify the description of the material's behavior, allowing for the use of mathematical tools such as scale analysis to further understand and predict fluid responses.

Scale analysis, in the context of linear viscoelastic fluids, involves the identification of key dimensionless parameters that govern the flow behavior and material response. By systematically comparing the relative magnitudes of different terms in the governing equations, scale analysis helps simplify complex fluid dynamics problems. This approach is especially useful for identifying dominant physical effects in specific applications, from small-scale biological flows to large-scale industrial processes.

Mathematical Modelling

Consider a hydrodynamically active viscoelastic fluid with constant properties overlying a passive gas in a destabilizing gravitational field. The fluid is assumed to be linearly viscoelastic for algebraic simplicity, while retaining the essential physics of the problem.

The displacement field in the viscoelastic fluid is described by the displacement vector $\mathbf {R}$, which represents the displacement of the position vector $\mathbf {x}$ in the current configuration from the position vector ${\boldsymbol {\zeta }}$ in the reference configuration . Mathematically, this is expressed as:

$\mathbf {x} ={\boldsymbol {\zeta }}+\mathbf {R} (\mathbf {x} )$.

For a linear viscoelastic fluid , the stress tensor $\mathbf {T}$ is given by:

$\mathbf {T} =-p\mathbf {I} +G\left(\nabla \mathbf {R} +\nabla \mathbf {R} ^{T}\right)+\mu _{g}\left(\nabla \mathbf {v} +\nabla \mathbf {v} ^{T}\right)$,

where $G$ is the shear modulus, $\mu _{g}$ is the viscosity of the fluid, and $\mathbf {v}$ is the velocity field. The velocity field is also expressed in terms of the displacement field, as given by:

$\mathbf {v} =\left(\mathbf {I} -\nabla \mathbf {R} ^{T}\right)^{-1}\cdot {\frac {\partial \mathbf {R} }{\partial t}}$.

The equations of motion for the viscoelastic medium are:

$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=\nabla \cdot \mathbf {T} +\rho \mathbf {g} _{z}$,

where $\mathbf {T}$ and $\mathbf {v}$ are as defined above, $\mathbf {g} _{z}$ is the acceleration due to gravity, and $\rho$ is the fluid density.

The viscoelastic fluid is taken to be incompressible, and the mass conservation law requires:

${\text{det}}(\mathbf {F} )=1$,

where $\mathbf {F}$ is the deformation tensor, given by:

$\mathbf {F} ={\frac {\partial {\boldsymbol {\zeta }}_{i}}{\partial x_{j}}}$,

and ${\boldsymbol {\zeta }}_{i}$ represent the components of the position vector in the reference configuration. By expanding this and using the displacement relation, we obtain:

${\frac {\partial X}{\partial x}}+{\frac {\partial Z}{\partial z}}-{\frac {\partial X}{\partial x}}{\frac {\partial Z}{\partial z}}+{\frac {\partial Z}{\partial x}}{\frac {\partial X}{\partial z}}=0$.

These governing equations are complemented by boundary conditions at both the rigid wall and the fluid interface.

At the rigid wall, the displacement fields are taken to be zero. At the interface, $z=h(x,t)$ , the normal and tangential components of the momentum balance hold:

$\mathbf {n} \cdot \mathbf {T} \cdot \mathbf {t} =0\quad {\text{and}}\quad \mathbf {n} \cdot \mathbf {T} \cdot \mathbf {n} =-\gamma \nabla \cdot \mathbf {n}$,

where the unit normal vector $\mathbf {n}$ and unit tangent vector $\mathbf {t}$ are given by:

$\mathbf {n} ={\frac {-{\frac {\partial h}{\partial x}}\mathbf {i} _{x}+\mathbf {i} _{z}}{\sqrt {1+\left({\frac {\partial h}{\partial x}}\right)^{2}}}}$ and $\mathbf {t} ={\frac {\mathbf {i} _{x}+{\frac {\partial h}{\partial x}}\mathbf {i} _{z}}{\sqrt {1+\left({\frac {\partial h}{\partial x}}\right)^{2}}}}$.

Finally, the kinematic relation at the impermeable interface is:

$\mathbf {v} \cdot \mathbf {n} ={\frac {\partial h}{\partial t}}{\sqrt {1+\left({\frac {\partial h}{\partial x}}\right)^{2}}}$.

Non-Dimensionalization of Governing Equations

The governing equations of motion for the viscoelastic fluid can be made dimensionless by introducing characteristic scales. The following characteristic scales are used:

• $x_{c}=H$: characteristic length scale in the horizontal direction

• $z_{c}=H$: characteristic length scale in the vertical direction

• $X_{c}=H$: characteristic horizontal displacement

• $Z_{c}=H$: characteristic vertical displacement

• $t_{c}={\frac {H}{U}}$: characteristic time scale

• $p_{c}={\frac {\mu U}{H}}$: characteristic pressure scale

Here, $U$ is a characteristic velocity scale. Using these characteristic scales, the governing equations are non-dimensionalized to yield the following form:

$Re\left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+{\frac {1}{Wi}}\nabla ^{2}\mathbf {R} +{\frac {1}{Wi}}\nabla (\nabla \cdot \mathbf {R} )+\nabla ^{2}\mathbf {v} +{\frac {Bo}{Ca}}\mathbf {i} _{z}$

Where the non-dimensional numbers are defined as follows:

• $Wi={\frac {\mu U}{GH}}$: the Weissenberg number, which measures the ratio of elastic to viscous forces.

• $Re={\frac {\rho HU}{\mu }}$: the Reynolds number, representing the ratio of inertial to viscous forces.

• $Bo={\frac {\rho gH^{2}}{\gamma }}$: the Bond number, characterizing the ratio of gravitational to surface tension forces.

• $Ca={\frac {\mu U}{\gamma }}$: the Capillary number, representing the ratio of viscous to surface tension forces.

These non-dimensional numbers capture the relative importance of different physical effects in the flow of a viscoelastic fluid. By analyzing the magnitudes of these numbers, one can determine the dominant forces under different flow regimes.

Effects of Dimensionless Numbers as They Approach Zero

When dimensionless numbers approach zero, they indicate dominance of certain physical forces and the suppression of others:

• Reynolds Number ($Re$ → 0): Inertia effects are negligible compared to viscous forces, resulting in slow, smooth, and laminar flow, where viscous forces dominate.
• Capillary Number ($Ca$ → 0): Surface tension dominates over viscous forces, resulting in stable, undeformed fluid interfaces.
• Weissenberg Number ($Wi$ → 0): Elastic effects vanish, and the fluid behaves like a Newtonian fluid, with viscous forces governing the flow.
• Bond Number ($Bo$ → 0): Surface tension overwhelms gravitational forces, leading to perfectly shaped droplets or bubbles unaffected by gravity.

Scale Analysis of the Navier-Stokes Equations for a Linear Viscoelastic Fluid

The governing equations for a linear viscoelastic fluid, expressed in non-dimensional form, are given by:

$Re\left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+{\frac {1}{Wi}}\nabla ^{2}\mathbf {R} +{\frac {1}{Wi}}\nabla (\nabla \cdot \mathbf {R} )+\nabla ^{2}\mathbf {v} +{\frac {Bo}{Ca}}\mathbf {i} _{z}$

To perform a scale analysis , we analyze each term in the equation using the following characteristic scales:

• $H$: characteristic length scale
• $U$: characteristic velocity scale
• $\mu$: dynamic viscosity
• $G$: shear modulus
• $\rho$: density
• $g$: gravitational acceleration
• $\gamma$: surface tension

1. Time Derivative Term

The first term on the left-hand side is the time derivative:

$Re{\frac {\partial \mathbf {v} }{\partial t}}$

Scaling:

• The velocity field $\mathbf {v}$ has a scale of $U$.
• The time $t$ has a scale of ${\frac {H}{U}}$.

Thus, the scale for the time derivative is:

${\frac {\partial \mathbf {v} }{\partial t}}\sim {\frac {U^{2}}{H}}$

Therefore, the scaled term becomes:

$Re{\frac {\partial \mathbf {v} }{\partial t}}\sim Re\cdot {\frac {U^{2}}{H}}$

2. Convective Term

The second term on the left-hand side is the convective term:

$Re\left(\mathbf {v} \cdot \nabla \mathbf {v} \right)$

Scaling:

• The velocity $\mathbf {v} \sim U$
• The gradient $\nabla \mathbf {v}$ scales as ${\frac {U}{H}}$

Thus, the scale for the convective term is:

$\mathbf {v} \cdot \nabla \mathbf {v} \sim {\frac {U^{2}}{H}}$

Therefore, the scaled term becomes:

$Re\left(\mathbf {v} \cdot \nabla \mathbf {v} \right)\sim Re\cdot {\frac {U^{2}}{H}}$

3. Pressure Gradient Term

The pressure gradient term is given by:

$-\nabla p$

Scaling:

• The pressure $p$ has a scale of $\mu {\frac {U}{H}}$.
• The gradient $\nabla p$ scales as ${\frac {p}{H}}$.

Thus, the scaled term becomes:

$-\nabla p\sim {\frac {\mu U}{H^{2}}}$

4. Viscoelastic Terms

The first viscoelastic term is:

${\frac {1}{Wi}}\nabla ^{2}\mathbf {R}$

Scaling:

• The displacement $\mathbf {R} \sim H$.
• The Laplacian $\nabla ^{2}\mathbf {R} \sim {\frac {1}{H}}$.

Thus, we have:

$\nabla ^{2}\mathbf {R} \sim {\frac {1}{H}}$

Therefore, the scaled term becomes:

${\frac {1}{Wi}}\nabla ^{2}\mathbf {R} \sim {\frac {1}{Wi}}\cdot {\frac {1}{H}}$

The second viscoelastic term is:

${\frac {1}{Wi}}\nabla (\nabla \cdot \mathbf {R} )$

Scaling:

• The divergence $\nabla \cdot \mathbf {R} \sim {\frac {1}{H}}$.
• Therefore, $\nabla (\nabla \cdot \mathbf {R} )\sim {\frac {1}{H^{2}}}$.

Thus, we have:

${\frac {1}{Wi}}\nabla (\nabla \cdot \mathbf {R} )\sim {\frac {1}{Wi}}\cdot {\frac {1}{H^{2}}}$

5. Viscous Term

The viscous term is given by:

$\nabla ^{2}\mathbf {v}$

Scaling:

• The Laplacian $\nabla ^{2}\mathbf {v} \sim {\frac {U}{H^{2}}}$.

Thus, we have:

$\nabla ^{2}\mathbf {v} \sim {\frac {U}{H^{2}}}$

6. Gravitational Term

The gravitational term is expressed as:

${\frac {Bo}{Ca}}\mathbf {i} _{z}$

Scaling:

• The Bond number $Bo\sim {\frac {\rho gH^{2}}{\gamma }}$ and the Capillary number $Ca\sim {\frac {\mu U}{\gamma }}$.

Thus, we have:

${\frac {Bo}{Ca}}\sim {\frac {\frac {\rho gH^{2}}{\gamma }}{\frac {\mu U}{\gamma }}}={\frac {\rho gH^{2}}{\mu U}}$

Summary of Scaled Terms

In summary, the scale analysis reveals the following characteristic orders of magnitude for each term:

• Time Derivative Term: $Re\cdot {\frac {U^{2}}{H}}$

• Convective Term: $Re\cdot {\frac {U^{2}}{H}}$

• Pressure Gradient Term: $-{\frac {\mu U}{H^{2}}}$

• Viscoelastic Term 1: ${\frac {1}{Wi}}\cdot {\frac {1}{H}}$

• Viscoelastic Term 2: ${\frac {1}{Wi}}\cdot {\frac {1}{H^{2}}}$

• Viscous Term: ${\frac {U}{H^{2}}}$

• Gravitational Term: ${\frac {\rho gH^{2}}{\mu U}}$

This scale analysis helps to identify which terms are dominant under different flow regimes by comparing the magnitudes of these scaled terms. The dominant terms depend on the physical situation being modeled, which simplifies the governing equations and enhances our understanding of the fluid’s dynamics.

Article Prepared by

• Aryan Kumar (Roll no. 21135030), IIT (BHU), Varanasi
• Pratik Mishra (Roll no. 21135100), IIT (BHU), Varanasi
• Shreyam Gupta (Roll no. 21135126), IIT (BHU), Varanasi
• Sourav (Roll no. 21135131), IIT (BHU), Varanasi
• Ashwani Benjwal (Roll no. 21135152), IIT (BHU), Varanasi