Shear Rate Calculator

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Shear Rate Calculator

Result:

Shear Rate: — s-1

function calculateShearRate() { var velocity = parseFloat(document.getElementById("velocity").value); var distance = parseFloat(document.getElementById("distance").value); var shearRateResultElement = document.getElementById("shearRateResult"); if (isNaN(velocity) || isNaN(distance) || distance === 0) { shearRateResultElement.textContent = "Invalid input"; return; } // Shear Rate (γ̇) = Velocity Gradient (dv/dy) // In this simplified calculator, we assume velocity gradient is directly input as velocity / distance. // If you have a velocity gradient directly, you would only need that input. // For typical flow profiles where dv/dy is approximated, the inputs here represent that approximation. // Example: For a fluid between two parallel plates moving at a relative velocity, // where one plate is stationary and the other moves at velocity 'v' over a distance 'h', // the velocity gradient is approximately v/h. var shearRate = velocity / distance; shearRateResultElement.textContent = shearRate.toFixed(2); } .calculator-container { font-family: sans-serif; border: 1px solid #ccc; padding: 20px; border-radius: 8px; max-width: 500px; margin: 20px auto; background-color: #f9f9f9; } .calculator-form h2 { text-align: center; margin-bottom: 20px; color: #333; } .form-group { margin-bottom: 15px; } .form-group label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .form-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-form button { display: block; width: 100%; padding: 10px 15px; background-color: #4CAF50; color: white; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; transition: background-color 0.3s ease; } .calculator-form button:hover { background-color: #45a049; } .calculator-result { margin-top: 25px; padding: 15px; background-color: #e9e9e9; border-radius: 4px; text-align: center; } .calculator-result h3 { margin-top: 0; color: #333; } #result p { font-size: 1.1em; color: #666; } #shearRateResult { font-weight: bold; color: #007bff; }

Understanding Shear Rate in Fluid Dynamics

Shear rate, often denoted by the symbol $\dot{\gamma}$ (gamma dot), is a fundamental concept in fluid dynamics, particularly crucial when dealing with non-Newtonian fluids. It quantifies how quickly the fluid layers deform or slide past each other. Mathematically, shear rate is defined as the rate of change of velocity with respect to distance, perpendicular to the flow direction. In simpler terms, it measures the velocity difference between adjacent fluid layers divided by the distance separating them.

The Mathematical Definition

For a one-dimensional flow, the shear rate is expressed as:

$\dot{\gamma} = \frac{dv}{dy}$

Where:

  • $\dot{\gamma}$ is the shear rate, typically measured in units of inverse seconds (s-1).
  • $v$ is the velocity of the fluid.
  • $y$ is the distance perpendicular to the direction of flow.

In many practical scenarios, especially with simple flow geometries like flow between parallel plates or in pipes, the velocity gradient $\frac{dv}{dy}$ can be approximated by $\frac{\Delta v}{\Delta y}$, where $\Delta v$ is the difference in velocity between two points and $\Delta y$ is the distance between those points. Our calculator uses this approximation, where 'Velocity Gradient' can be thought of as the velocity difference across a specific distance.

Why is Shear Rate Important?

The significance of shear rate lies in its direct relationship with shear stress ($\tau$), which is the force per unit area acting parallel to the fluid's surface. The relationship between shear stress and shear rate defines the rheological behavior of a fluid:

  • Newtonian Fluids: For these fluids, such as water or air under normal conditions, the shear stress is directly proportional to the shear rate. The constant of proportionality is the fluid's dynamic viscosity ($\mu$). So, $\tau = \mu \dot{\gamma}$. This means viscosity is constant regardless of the shear rate.
  • Non-Newtonian Fluids: Many common substances, like ketchup, paint, blood, and polymer solutions, are non-Newtonian. Their viscosity changes with the applied shear rate.
    • Shear-thinning (Pseudoplastic): Viscosity decreases as shear rate increases (e.g., paint, ketchup).
    • Shear-thickening (Dilatant): Viscosity increases as shear rate increases (e.g., cornstarch and water mixture).
    • Bingham Plastics: Require a minimum shear stress to flow, after which they behave like a Newtonian fluid.

Understanding and controlling shear rate is vital in various industrial processes, including mixing, pumping, coating, extrusion, and even in biological systems where blood flow and cellular deformation are involved.

How to Use the Calculator

This calculator helps you determine the shear rate when you know the velocity difference across a certain distance within a fluid flow. This is a simplified model, often used when the velocity gradient is linear or can be approximated as such.

  • Velocity Gradient (m/s per m): Enter the difference in velocity between two points in the fluid, divided by the distance separating those points. If you are calculating the velocity gradient between two parallel plates, one stationary and one moving at a velocity $V$ relative to the stationary plate, and the distance between them is $H$, you would input $V$ for the velocity and $H$ for the distance. The calculator will compute $V/H$.
  • Distance (m): Enter the perpendicular distance over which the velocity difference occurs.

Clicking "Calculate Shear Rate" will provide you with the shear rate in s-1.

Example Calculation:

Consider a fluid flowing between two parallel plates. The top plate is moving at a velocity of 0.2 m/s, and the bottom plate is stationary. The distance between the plates is 0.05 meters. Assuming a linear velocity profile, the velocity gradient is approximately the velocity of the top plate divided by the distance between the plates.

  • Velocity Gradient (input as Velocity Difference / Distance): 0.2 m/s / 0.05 m = 4 m/s per m
  • Distance: 0.05 m

If you input "0.2" for "Velocity Gradient (m/s per m)" and "0.05" for "Distance (m)" into our calculator (interpreting the first input as the velocity difference and the second as the distance over which that difference occurs), the result will be:

Shear Rate = 0.2 m/s / 0.05 m = 4.00 s-1

This value of shear rate is important for determining the shear stress and understanding the fluid's behavior, especially if it's a non-Newtonian fluid.

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