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This paper investigates the impact of bulk viscosity within the framework of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></semantics></math></inline-formula> gravity. We consider a time-dependent viscosity model with a particular Hubble parameter expression. Here, we looked into the viability of well-motivated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></semantics></math></inline-formula> gravity model, which takes the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>=</mo><mi>α</mi><mspace width="3.33333pt"></mspace><mo form="prefix">log</mo><mo>(</mo><mi>B</mi><mo>)</mo><mo>+</mo><mi>β</mi><mspace width="3.33333pt"></mspace><mi>T</mi></mrow></semantics></math></inline-formula>, and has free parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. The 46 observational Hubble data (OHD) in the range <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>z</mi><mo>≤</mo><mn>2.36</mn></mrow></semantics></math></inline-formula> were used to constrain the model parameters to achieve the solution. We have used the Markov Chain Monte Carlo (MCMC) method to estimate model parameters and observe that the model appears to be in good agreement with the observations. In addition, we evaluate the effective viscous equation of state parameter for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></semantics></math></inline-formula> model. We have examined the characteristics of different energy conditions for the stability analysis. The model is valid based on the positive behavior of null energy conditions (NEC), weak energy conditions (WEC), and dominant energy conditions (DEC); however, strong energy conditions (SEC) are in violation, suggesting that the universe is expanding faster. Our model was found in the quintom region. We also discussed how the tachyon scalar field corresponds to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>T</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></semantics></math></inline-formula> gravity.